3.2.64 \(\int \frac {x^{5/2} (A+B x^3)}{(a+b x^3)^2} \, dx\) [164]

Optimal. Leaf size=312 \[ -\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}-\frac {(A b-7 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}} \]

[Out]

1/3*(A*b-B*a)*x^(7/2)/a/b/(b*x^3+a)+1/9*(A*b-7*B*a)*arctan(b^(1/6)*x^(1/2)/a^(1/6))/a^(5/6)/b^(13/6)+1/18*(A*b
-7*B*a)*arctan(-3^(1/2)+2*b^(1/6)*x^(1/2)/a^(1/6))/a^(5/6)/b^(13/6)+1/18*(A*b-7*B*a)*arctan(3^(1/2)+2*b^(1/6)*
x^(1/2)/a^(1/6))/a^(5/6)/b^(13/6)-1/36*(A*b-7*B*a)*ln(a^(1/3)+b^(1/3)*x-a^(1/6)*b^(1/6)*3^(1/2)*x^(1/2))/a^(5/
6)/b^(13/6)*3^(1/2)+1/36*(A*b-7*B*a)*ln(a^(1/3)+b^(1/3)*x+a^(1/6)*b^(1/6)*3^(1/2)*x^(1/2))/a^(5/6)/b^(13/6)*3^
(1/2)-1/3*(A*b-7*B*a)*x^(1/2)/a/b^2

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Rubi [A]
time = 0.35, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {468, 327, 335, 215, 648, 632, 210, 642, 211} \begin {gather*} -\frac {(A b-7 a B) \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \text {ArcTan}\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}+\sqrt {3}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \text {ArcTan}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}-\frac {\sqrt {x} (A b-7 a B)}{3 a b^2}+\frac {x^{7/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(5/2)*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

-1/3*((A*b - 7*a*B)*Sqrt[x])/(a*b^2) + ((A*b - a*B)*x^(7/2))/(3*a*b*(a + b*x^3)) - ((A*b - 7*a*B)*ArcTan[Sqrt[
3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(5/6)*b^(13/6)) + ((A*b - 7*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])
/a^(1/6)])/(18*a^(5/6)*b^(13/6)) + ((A*b - 7*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(5/6)*b^(13/6)) - ((
A*b - 7*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(5/6)*b^(13/6)) + ((A*b
- 7*a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(5/6)*b^(13/6))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 215

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] +
Int[(r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; 2*(r^2/(a*n))*Int[1/
(r^2 + s^2*x^2), x] + Dist[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {\left (-\frac {A b}{2}+\frac {7 a B}{2}\right ) \int \frac {x^{5/2}}{a+b x^3} \, dx}{3 a b}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-7 a B) \int \frac {1}{\sqrt {x} \left (a+b x^3\right )} \, dx}{6 b^2}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{a+b x^6} \, dx,x,\sqrt {x}\right )}{3 b^2}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {\sqrt [6]{a}-\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{9 a^{5/6} b^2}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {\sqrt [6]{a}+\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{9 a^{5/6} b^2}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{9 a^{2/3} b^2}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{36 a^{2/3} b^2}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{36 a^{2/3} b^2}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{18 \sqrt {3} a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{18 \sqrt {3} a^{5/6} b^{13/6}}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}-\frac {(A b-7 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}\\ \end {align*}

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Mathematica [A]
time = 0.47, size = 181, normalized size = 0.58 \begin {gather*} \frac {\frac {6 \sqrt [6]{b} \sqrt {x} \left (-A b+7 a B+6 b B x^3\right )}{a+b x^3}+\frac {2 (A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{a^{5/6}}+\frac {(-A b+7 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )}{a^{5/6}}+\frac {\sqrt {3} (A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{a^{5/6}}}{18 b^{13/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(5/2)*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

((6*b^(1/6)*Sqrt[x]*(-(A*b) + 7*a*B + 6*b*B*x^3))/(a + b*x^3) + (2*(A*b - 7*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1
/6)])/a^(5/6) + ((-(A*b) + 7*a*B)*ArcTan[(a^(1/3) - b^(1/3)*x)/(a^(1/6)*b^(1/6)*Sqrt[x])])/a^(5/6) + (Sqrt[3]*
(A*b - 7*a*B)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x])/(a^(1/3) + b^(1/3)*x)])/a^(5/6))/(18*b^(13/6))

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Maple [A]
time = 0.39, size = 216, normalized size = 0.69

method	result	size
derivativedivides	\(\frac {2 B \sqrt {x}}{b^{2}}+\frac {\frac {2 \left (-\frac {A b}{6}+\frac {B a}{6}\right ) \sqrt {x}}{b \,x^{3}+a}+\frac {\left (A b -7 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}\right )}{3}}{b^{2}}\)	\(216\)
default	\(\frac {2 B \sqrt {x}}{b^{2}}+\frac {\frac {2 \left (-\frac {A b}{6}+\frac {B a}{6}\right ) \sqrt {x}}{b \,x^{3}+a}+\frac {\left (A b -7 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}\right )}{3}}{b^{2}}\)	\(216\)
risch	\(\frac {2 B \sqrt {x}}{b^{2}}-\frac {\sqrt {x}\, A}{3 b \left (b \,x^{3}+a \right )}+\frac {\sqrt {x}\, B a}{3 b^{2} \left (b \,x^{3}+a \right )}+\frac {A \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 b a}+\frac {A \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{18 b a}-\frac {A \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 b a}+\frac {A \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 b a}+\frac {A \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 b a}-\frac {7 B \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 b^{2}}-\frac {7 B \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{18 b^{2}}+\frac {7 B \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 b^{2}}-\frac {7 B \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 b^{2}}-\frac {7 B \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 b^{2}}\)	\(405\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x^3+A)/(b*x^3+a)^2,x,method=_RETURNVERBOSE)

[Out]

2*B/b^2*x^(1/2)+2/b^2*((-1/6*A*b+1/6*B*a)*x^(1/2)/(b*x^3+a)+1/6*(A*b-7*B*a)*(1/3/a*(a/b)^(1/6)*arctan(x^(1/2)/
(a/b)^(1/6))-1/12/a*3^(1/2)*(a/b)^(1/6)*ln(3^(1/2)*(a/b)^(1/6)*x^(1/2)-x-(a/b)^(1/3))+1/6/a*(a/b)^(1/6)*arctan
(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))+1/12/a*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+1/6/
a*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))))

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Maxima [A]
time = 0.52, size = 311, normalized size = 1.00 \begin {gather*} \frac {{\left (B a - A b\right )} \sqrt {x}}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} + \frac {2 \, B \sqrt {x}}{b^{2}} - \frac {\frac {\sqrt {3} {\left (7 \, B a - A b\right )} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} {\left (7 \, B a - A b\right )} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, {\left (7 \, B a b^{\frac {1}{3}} - A b^{\frac {4}{3}}\right )} \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (7 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} - A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (7 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} - A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{36 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/3*(B*a - A*b)*sqrt(x)/(b^3*x^3 + a*b^2) + 2*B*sqrt(x)/b^2 - 1/36*(sqrt(3)*(7*B*a - A*b)*log(sqrt(3)*a^(1/6)*
b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1/3))/(a^(5/6)*b^(1/6)) - sqrt(3)*(7*B*a - A*b)*log(-sqrt(3)*a^(1/6)*b^(1/6)*
sqrt(x) + b^(1/3)*x + a^(1/3))/(a^(5/6)*b^(1/6)) + 4*(7*B*a*b^(1/3) - A*b^(4/3))*arctan(b^(1/3)*sqrt(x)/sqrt(a
^(1/3)*b^(1/3)))/(a^(2/3)*b^(1/3)*sqrt(a^(1/3)*b^(1/3))) + 2*(7*B*a^(4/3)*b^(1/3) - A*a^(1/3)*b^(4/3))*arctan(
(sqrt(3)*a^(1/6)*b^(1/6) + 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(1/3)))/(a*b^(1/3)*sqrt(a^(1/3)*b^(1/3))) + 2*(7*
B*a^(4/3)*b^(1/3) - A*a^(1/3)*b^(4/3))*arctan(-(sqrt(3)*a^(1/6)*b^(1/6) - 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(1
/3)))/(a*b^(1/3)*sqrt(a^(1/3)*b^(1/3))))/b^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2566 vs. \(2 (232) = 464\).
time = 1.69, size = 2566, normalized size = 8.22 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/36*(4*sqrt(3)*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^
3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^2*b
^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4
- 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/3) + (49*B^2*a^2 - 14*A*B*a*b + A^2*b^2)*x + (7*B*a^2*b^2 - A*a*b^
3)*sqrt(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2
*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6))*a^4*b^11*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 3601
5*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(5/6) +
2*sqrt(3)*(7*B*a^5*b^11 - A*a^4*b^12)*sqrt(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2
- 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(5/6) - sqrt(3)*(117649*B
^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*
b^5 + A^6*b^6))/(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*
B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)) + 4*sqrt(3)*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b
+ 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(
1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^2*b^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*
A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/3) + (49*B^2*a^2 - 14*A*B*a*b
+ A^2*b^2)*x - (7*B*a^2*b^2 - A*a*b^3)*sqrt(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2
- 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6))*a^4*b^11*(-(11764
9*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B
*a*b^5 + A^6*b^6)/(a^5*b^13))^(5/6) + 2*sqrt(3)*(7*B*a^5*b^11 - A*a^4*b^12)*sqrt(x)*(-(117649*B^6*a^6 - 100842
*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/
(a^5*b^13))^(5/6) + sqrt(3)*(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^
3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6))/(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*
b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)) - (b^3*x^3 + a*b^2)*(-(117649*B^
6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b
^5 + A^6*b^6)/(a^5*b^13))^(1/6)*log(a^2*b^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6
860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/3) + (49*B^2*a^2 - 14*A*B
*a*b + A^2*b^2)*x + (7*B*a^2*b^2 - A*a*b^3)*sqrt(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4
*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)) + (b^3*x^3 +
a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2
*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)*log(a^2*b^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*
A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/3) + (
49*B^2*a^2 - 14*A*B*a*b + A^2*b^2)*x - (7*B*a^2*b^2 - A*a*b^3)*sqrt(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b
+ 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(
1/6)) + 2*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*
b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)*log(a*b^2*(-(117649*B^6*a^6 - 100842*A
*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a
^5*b^13))^(1/6) - (7*B*a - A*b)*sqrt(x)) - 2*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*
A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)*log
(-a*b^2*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^
2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6) - (7*B*a - A*b)*sqrt(x)) + 12*(6*B*b*x^3 + 7*B*a - A*b)*sq
rt(x))/(b^3*x^3 + a*b^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1658 vs. \(2 (299) = 598\).
time = 185.02, size = 1658, normalized size = 5.31 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} + 2 B \sqrt {x}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {13}{2}}}{13}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} + 2 B \sqrt {x}}{b^{2}} & \text {for}\: a = 0 \\- \frac {12 A a b \sqrt {x}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {2 A a b \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 A a b \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {A a b \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {A a b \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 \sqrt {3} A a b \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 \sqrt {3} A a b \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {2 A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 \sqrt {3} A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 \sqrt {3} A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {84 B a^{2} \sqrt {x}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {14 B a^{2} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 B a^{2} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {7 B a^{2} \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {7 B a^{2} \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 \sqrt {3} B a^{2} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 \sqrt {3} B a^{2} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {72 B a b x^{\frac {7}{2}}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {14 B a b x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 B a b x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {7 B a b x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {7 B a b x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 \sqrt {3} B a b x^{3} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 \sqrt {3} B a b x^{3} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

Piecewise((zoo*(-2*A/(5*x**(5/2)) + 2*B*sqrt(x)), Eq(a, 0) & Eq(b, 0)), ((2*A*x**(7/2)/7 + 2*B*x**(13/2)/13)/a
**2, Eq(b, 0)), ((-2*A/(5*x**(5/2)) + 2*B*sqrt(x))/b**2, Eq(a, 0)), (-12*A*a*b*sqrt(x)/(36*a**2*b**2 + 36*a*b*
*3*x**3) - 2*A*a*b*(-a/b)**(1/6)*log(sqrt(x) - (-a/b)**(1/6))/(36*a**2*b**2 + 36*a*b**3*x**3) + 2*A*a*b*(-a/b)
**(1/6)*log(sqrt(x) + (-a/b)**(1/6))/(36*a**2*b**2 + 36*a*b**3*x**3) - A*a*b*(-a/b)**(1/6)*log(-4*sqrt(x)*(-a/
b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**2*b**2 + 36*a*b**3*x**3) + A*a*b*(-a/b)**(1/6)*log(4*sqrt(x)*(-a/b)*
*(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**2*b**2 + 36*a*b**3*x**3) + 2*sqrt(3)*A*a*b*(-a/b)**(1/6)*atan(2*sqrt(3)
*sqrt(x)/(3*(-a/b)**(1/6)) - sqrt(3)/3)/(36*a**2*b**2 + 36*a*b**3*x**3) + 2*sqrt(3)*A*a*b*(-a/b)**(1/6)*atan(2
*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) + sqrt(3)/3)/(36*a**2*b**2 + 36*a*b**3*x**3) - 2*A*b**2*x**3*(-a/b)**(1/6)*
log(sqrt(x) - (-a/b)**(1/6))/(36*a**2*b**2 + 36*a*b**3*x**3) + 2*A*b**2*x**3*(-a/b)**(1/6)*log(sqrt(x) + (-a/b
)**(1/6))/(36*a**2*b**2 + 36*a*b**3*x**3) - A*b**2*x**3*(-a/b)**(1/6)*log(-4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(
-a/b)**(1/3))/(36*a**2*b**2 + 36*a*b**3*x**3) + A*b**2*x**3*(-a/b)**(1/6)*log(4*sqrt(x)*(-a/b)**(1/6) + 4*x +
4*(-a/b)**(1/3))/(36*a**2*b**2 + 36*a*b**3*x**3) + 2*sqrt(3)*A*b**2*x**3*(-a/b)**(1/6)*atan(2*sqrt(3)*sqrt(x)/
(3*(-a/b)**(1/6)) - sqrt(3)/3)/(36*a**2*b**2 + 36*a*b**3*x**3) + 2*sqrt(3)*A*b**2*x**3*(-a/b)**(1/6)*atan(2*sq
rt(3)*sqrt(x)/(3*(-a/b)**(1/6)) + sqrt(3)/3)/(36*a**2*b**2 + 36*a*b**3*x**3) + 84*B*a**2*sqrt(x)/(36*a**2*b**2
+ 36*a*b**3*x**3) + 14*B*a**2*(-a/b)**(1/6)*log(sqrt(x) - (-a/b)**(1/6))/(36*a**2*b**2 + 36*a*b**3*x**3) - 14
*B*a**2*(-a/b)**(1/6)*log(sqrt(x) + (-a/b)**(1/6))/(36*a**2*b**2 + 36*a*b**3*x**3) + 7*B*a**2*(-a/b)**(1/6)*lo
g(-4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**2*b**2 + 36*a*b**3*x**3) - 7*B*a**2*(-a/b)**(1/6)*l
og(4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**2*b**2 + 36*a*b**3*x**3) - 14*sqrt(3)*B*a**2*(-a/b)
**(1/6)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) - sqrt(3)/3)/(36*a**2*b**2 + 36*a*b**3*x**3) - 14*sqrt(3)*B*a
**2*(-a/b)**(1/6)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) + sqrt(3)/3)/(36*a**2*b**2 + 36*a*b**3*x**3) + 72*B
*a*b*x**(7/2)/(36*a**2*b**2 + 36*a*b**3*x**3) + 14*B*a*b*x**3*(-a/b)**(1/6)*log(sqrt(x) - (-a/b)**(1/6))/(36*a
**2*b**2 + 36*a*b**3*x**3) - 14*B*a*b*x**3*(-a/b)**(1/6)*log(sqrt(x) + (-a/b)**(1/6))/(36*a**2*b**2 + 36*a*b**
3*x**3) + 7*B*a*b*x**3*(-a/b)**(1/6)*log(-4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**2*b**2 + 36*
a*b**3*x**3) - 7*B*a*b*x**3*(-a/b)**(1/6)*log(4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**2*b**2 +
36*a*b**3*x**3) - 14*sqrt(3)*B*a*b*x**3*(-a/b)**(1/6)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) - sqrt(3)/3)/(
36*a**2*b**2 + 36*a*b**3*x**3) - 14*sqrt(3)*B*a*b*x**3*(-a/b)**(1/6)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6))
+ sqrt(3)/3)/(36*a**2*b**2 + 36*a*b**3*x**3), True))

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Giac [A]
time = 0.59, size = 313, normalized size = 1.00 \begin {gather*} \frac {2 \, B \sqrt {x}}{b^{2}} - \frac {\sqrt {3} {\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 \, a b^{3}} + \frac {\sqrt {3} {\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 \, a b^{3}} + \frac {B a \sqrt {x} - A b \sqrt {x}}{3 \, {\left (b x^{3} + a\right )} b^{2}} - \frac {{\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a b^{3}} - \frac {{\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a b^{3}} - \frac {{\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 \, a b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

2*B*sqrt(x)/b^2 - 1/36*sqrt(3)*(7*(a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*log(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x +
(a/b)^(1/3))/(a*b^3) + 1/36*sqrt(3)*(7*(a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b)^(1/6
) + x + (a/b)^(1/3))/(a*b^3) + 1/3*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^3 + a)*b^2) - 1/18*(7*(a*b^5)^(1/6)*B*a -
(a*b^5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6))/(a*b^3) - 1/18*(7*(a*b^5)^(1/6)*B*a
- (a*b^5)^(1/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2*sqrt(x))/(a/b)^(1/6))/(a*b^3) - 1/9*(7*(a*b^5)^(1/6)*B*a
- (a*b^5)^(1/6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a*b^3)

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Mupad [B]
time = 2.89, size = 1884, normalized size = 6.04 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(5/2)*(A + B*x^3))/(a + b*x^3)^2,x)

[Out]

(2*B*x^(1/2))/b^2 - (x^(1/2)*((A*b)/3 - (B*a)/3))/(a*b^2 + b^3*x^3) - (atan(((((2*x^(1/2)*(A^4*b^4 + 2401*B^4*
a^4 + 294*A^2*B^2*a^2*b^2 - 1372*A*B^3*a^3*b - 28*A^3*B*a*b^3))/(27*b^3) - (2*(A*b - 7*B*a)*(343*B^3*a^4 - A^3
*a*b^3 - 147*A*B^2*a^3*b + 21*A^2*B*a^2*b^2))/(27*(-a)^(5/6)*b^(19/6)))*(A*b - 7*B*a)*1i)/(18*(-a)^(5/6)*b^(13
/6)) + (((2*x^(1/2)*(A^4*b^4 + 2401*B^4*a^4 + 294*A^2*B^2*a^2*b^2 - 1372*A*B^3*a^3*b - 28*A^3*B*a*b^3))/(27*b^
3) + (2*(A*b - 7*B*a)*(343*B^3*a^4 - A^3*a*b^3 - 147*A*B^2*a^3*b + 21*A^2*B*a^2*b^2))/(27*(-a)^(5/6)*b^(19/6))
)*(A*b - 7*B*a)*1i)/(18*(-a)^(5/6)*b^(13/6)))/((((2*x^(1/2)*(A^4*b^4 + 2401*B^4*a^4 + 294*A^2*B^2*a^2*b^2 - 13
72*A*B^3*a^3*b - 28*A^3*B*a*b^3))/(27*b^3) - (2*(A*b - 7*B*a)*(343*B^3*a^4 - A^3*a*b^3 - 147*A*B^2*a^3*b + 21*
A^2*B*a^2*b^2))/(27*(-a)^(5/6)*b^(19/6)))*(A*b - 7*B*a))/(18*(-a)^(5/6)*b^(13/6)) - (((2*x^(1/2)*(A^4*b^4 + 24
01*B^4*a^4 + 294*A^2*B^2*a^2*b^2 - 1372*A*B^3*a^3*b - 28*A^3*B*a*b^3))/(27*b^3) + (2*(A*b - 7*B*a)*(343*B^3*a^
4 - A^3*a*b^3 - 147*A*B^2*a^3*b + 21*A^2*B*a^2*b^2))/(27*(-a)^(5/6)*b^(19/6)))*(A*b - 7*B*a))/(18*(-a)^(5/6)*b
^(13/6))))*(A*b - 7*B*a)*1i)/(9*(-a)^(5/6)*b^(13/6)) - (atan(((((3^(1/2)*1i)/2 - 1/2)*((2*x^(1/2)*(A^4*b^4 + 2
401*B^4*a^4 + 294*A^2*B^2*a^2*b^2 - 1372*A*B^3*a^3*b - 28*A^3*B*a*b^3))/(27*b^3) - (2*((3^(1/2)*1i)/2 - 1/2)*(
A*b - 7*B*a)*(343*B^3*a^4 - A^3*a*b^3 - 147*A*B^2*a^3*b + 21*A^2*B*a^2*b^2))/(27*(-a)^(5/6)*b^(19/6)))*(A*b -
7*B*a)*1i)/(18*(-a)^(5/6)*b^(13/6)) + (((3^(1/2)*1i)/2 - 1/2)*((2*x^(1/2)*(A^4*b^4 + 2401*B^4*a^4 + 294*A^2*B^
2*a^2*b^2 - 1372*A*B^3*a^3*b - 28*A^3*B*a*b^3))/(27*b^3) + (2*((3^(1/2)*1i)/2 - 1/2)*(A*b - 7*B*a)*(343*B^3*a^
4 - A^3*a*b^3 - 147*A*B^2*a^3*b + 21*A^2*B*a^2*b^2))/(27*(-a)^(5/6)*b^(19/6)))*(A*b - 7*B*a)*1i)/(18*(-a)^(5/6
)*b^(13/6)))/((((3^(1/2)*1i)/2 - 1/2)*((2*x^(1/2)*(A^4*b^4 + 2401*B^4*a^4 + 294*A^2*B^2*a^2*b^2 - 1372*A*B^3*a
^3*b - 28*A^3*B*a*b^3))/(27*b^3) - (2*((3^(1/2)*1i)/2 - 1/2)*(A*b - 7*B*a)*(343*B^3*a^4 - A^3*a*b^3 - 147*A*B^
2*a^3*b + 21*A^2*B*a^2*b^2))/(27*(-a)^(5/6)*b^(19/6)))*(A*b - 7*B*a))/(18*(-a)^(5/6)*b^(13/6)) - (((3^(1/2)*1i
)/2 - 1/2)*((2*x^(1/2)*(A^4*b^4 + 2401*B^4*a^4 + 294*A^2*B^2*a^2*b^2 - 1372*A*B^3*a^3*b - 28*A^3*B*a*b^3))/(27
*b^3) + (2*((3^(1/2)*1i)/2 - 1/2)*(A*b - 7*B*a)*(343*B^3*a^4 - A^3*a*b^3 - 147*A*B^2*a^3*b + 21*A^2*B*a^2*b^2)
)/(27*(-a)^(5/6)*b^(19/6)))*(A*b - 7*B*a))/(18*(-a)^(5/6)*b^(13/6))))*((3^(1/2)*1i)/2 - 1/2)*(A*b - 7*B*a)*1i)
/(9*(-a)^(5/6)*b^(13/6)) - (atan(((((3^(1/2)*1i)/2 + 1/2)*((2*x^(1/2)*(A^4*b^4 + 2401*B^4*a^4 + 294*A^2*B^2*a^
2*b^2 - 1372*A*B^3*a^3*b - 28*A^3*B*a*b^3))/(27*b^3) - (2*((3^(1/2)*1i)/2 + 1/2)*(A*b - 7*B*a)*(343*B^3*a^4 -
A^3*a*b^3 - 147*A*B^2*a^3*b + 21*A^2*B*a^2*b^2))/(27*(-a)^(5/6)*b^(19/6)))*(A*b - 7*B*a)*1i)/(18*(-a)^(5/6)*b^
(13/6)) + (((3^(1/2)*1i)/2 + 1/2)*((2*x^(1/2)*(A^4*b^4 + 2401*B^4*a^4 + 294*A^2*B^2*a^2*b^2 - 1372*A*B^3*a^3*b
- 28*A^3*B*a*b^3))/(27*b^3) + (2*((3^(1/2)*1i)/2 + 1/2)*(A*b - 7*B*a)*(343*B^3*a^4 - A^3*a*b^3 - 147*A*B^2*a^
3*b + 21*A^2*B*a^2*b^2))/(27*(-a)^(5/6)*b^(19/6)))*(A*b - 7*B*a)*1i)/(18*(-a)^(5/6)*b^(13/6)))/((((3^(1/2)*1i)
/2 + 1/2)*((2*x^(1/2)*(A^4*b^4 + 2401*B^4*a^4 + 294*A^2*B^2*a^2*b^2 - 1372*A*B^3*a^3*b - 28*A^3*B*a*b^3))/(27*
b^3) - (2*((3^(1/2)*1i)/2 + 1/2)*(A*b - 7*B*a)*(343*B^3*a^4 - A^3*a*b^3 - 147*A*B^2*a^3*b + 21*A^2*B*a^2*b^2))
/(27*(-a)^(5/6)*b^(19/6)))*(A*b - 7*B*a))/(18*(-a)^(5/6)*b^(13/6)) - (((3^(1/2)*1i)/2 + 1/2)*((2*x^(1/2)*(A^4*
b^4 + 2401*B^4*a^4 + 294*A^2*B^2*a^2*b^2 - 1372*A*B^3*a^3*b - 28*A^3*B*a*b^3))/(27*b^3) + (2*((3^(1/2)*1i)/2 +
1/2)*(A*b - 7*B*a)*(343*B^3*a^4 - A^3*a*b^3 - 147*A*B^2*a^3*b + 21*A^2*B*a^2*b^2))/(27*(-a)^(5/6)*b^(19/6)))*
(A*b - 7*B*a))/(18*(-a)^(5/6)*b^(13/6))))*((3^(1/2)*1i)/2 + 1/2)*(A*b - 7*B*a)*1i)/(9*(-a)^(5/6)*b^(13/6))

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