3.2.78 \(\int \frac {A+B x^3}{x^{7/2} (a+b x^3)^3} \, dx\) [178]

Optimal. Leaf size=351 \[ -\frac {11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac {17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}+\frac {11 (17 A b-5 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}+\frac {11 (17 A b-5 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{23/6} \sqrt [6]{b}} \]

[Out]

-11/180*(17*A*b-5*B*a)/a^3/b/x^(5/2)+1/6*(A*b-B*a)/a/b/x^(5/2)/(b*x^3+a)^2+1/36*(17*A*b-5*B*a)/a^2/b/x^(5/2)/(
b*x^3+a)-11/108*(17*A*b-5*B*a)*arctan(b^(1/6)*x^(1/2)/a^(1/6))/a^(23/6)/b^(1/6)-11/216*(17*A*b-5*B*a)*arctan(-
3^(1/2)+2*b^(1/6)*x^(1/2)/a^(1/6))/a^(23/6)/b^(1/6)-11/216*(17*A*b-5*B*a)*arctan(3^(1/2)+2*b^(1/6)*x^(1/2)/a^(
1/6))/a^(23/6)/b^(1/6)+11/432*(17*A*b-5*B*a)*ln(a^(1/3)+b^(1/3)*x-a^(1/6)*b^(1/6)*3^(1/2)*x^(1/2))/a^(23/6)/b^
(1/6)*3^(1/2)-11/432*(17*A*b-5*B*a)*ln(a^(1/3)+b^(1/3)*x+a^(1/6)*b^(1/6)*3^(1/2)*x^(1/2))/a^(23/6)/b^(1/6)*3^(
1/2)

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Rubi [A]
time = 0.38, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {468, 296, 331, 335, 215, 648, 632, 210, 642, 211} \begin {gather*} \frac {11 (17 A b-5 a B) \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \text {ArcTan}\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}+\sqrt {3}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \text {ArcTan}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}+\frac {11 (17 A b-5 a B) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac {17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^3),x]

[Out]

(-11*(17*A*b - 5*a*B))/(180*a^3*b*x^(5/2)) + (A*b - a*B)/(6*a*b*x^(5/2)*(a + b*x^3)^2) + (17*A*b - 5*a*B)/(36*
a^2*b*x^(5/2)*(a + b*x^3)) + (11*(17*A*b - 5*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(23/6)
*b^(1/6)) - (11*(17*A*b - 5*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(23/6)*b^(1/6)) - (11*(
17*A*b - 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(108*a^(23/6)*b^(1/6)) + (11*(17*A*b - 5*a*B)*Log[a^(1/3) -
Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqrt[3]*a^(23/6)*b^(1/6)) - (11*(17*A*b - 5*a*B)*Log[a^(1/
3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqrt[3]*a^(23/6)*b^(1/6))

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 296

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

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 {A+B x^3}{x^{7/2} \left (a+b x^3\right )^3} \, dx &=\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac {\left (\frac {17 A b}{2}-\frac {5 a B}{2}\right ) \int \frac {1}{x^{7/2} \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac {17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}+\frac {(11 (17 A b-5 a B)) \int \frac {1}{x^{7/2} \left (a+b x^3\right )} \, dx}{72 a^2 b}\\ &=-\frac {11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac {17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac {(11 (17 A b-5 a B)) \int \frac {1}{\sqrt {x} \left (a+b x^3\right )} \, dx}{72 a^3}\\ &=-\frac {11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac {17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac {(11 (17 A b-5 a B)) \text {Subst}\left (\int \frac {1}{a+b x^6} \, dx,x,\sqrt {x}\right )}{36 a^3}\\ &=-\frac {11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac {17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac {(11 (17 A b-5 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 )}{108 a^{23/6}}-\frac {(11 (17 A b-5 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 )}{108 a^{23/6}}-\frac {(11 (17 A b-5 a B)) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{108 a^{11/3}}\\ &=-\frac {11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac {17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac {11 (17 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}-\frac {(11 (17 A b-5 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 )}{432 a^{11/3}}-\frac {(11 (17 A b-5 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 )}{432 a^{11/3}}+\frac {(11 (17 A b-5 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 )}{144 \sqrt {3} a^{23/6} \sqrt [6]{b}}-\frac {(11 (17 A b-5 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 )}{144 \sqrt {3} a^{23/6} \sqrt [6]{b}}\\ &=-\frac {11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac {17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}-\frac {11 (17 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}+\frac {11 (17 A b-5 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{23/6} \sqrt [6]{b}}-\frac {(11 (17 A b-5 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 )}{216 \sqrt {3} a^{23/6} \sqrt [6]{b}}+\frac {(11 (17 A b-5 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 )}{216 \sqrt {3} a^{23/6} \sqrt [6]{b}}\\ &=-\frac {11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac {A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2}+\frac {17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}+\frac {11 (17 A b-5 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}+\frac {11 (17 A b-5 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{23/6} \sqrt [6]{b}}-\frac {11 (17 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{23/6} \sqrt [6]{b}}\\ \end {align*}

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Mathematica [A]
time = 0.56, size = 209, normalized size = 0.60 \begin {gather*} \frac {-\frac {6 a^{5/6} \left (187 A b^2 x^6+a^2 \left (72 A-85 B x^3\right )+a b x^3 \left (289 A-55 B x^3\right )\right )}{x^{5/2} \left (a+b x^3\right )^2}+\frac {110 (-17 A b+5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}+\frac {55 (17 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )}{\sqrt [6]{b}}+\frac {55 \sqrt {3} (-17 A b+5 a B) \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{\sqrt [6]{b}}}{1080 a^{23/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^3),x]

[Out]

((-6*a^(5/6)*(187*A*b^2*x^6 + a^2*(72*A - 85*B*x^3) + a*b*x^3*(289*A - 55*B*x^3)))/(x^(5/2)*(a + b*x^3)^2) + (
110*(-17*A*b + 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/b^(1/6) + (55*(17*A*b - 5*a*B)*ArcTan[(a^(1/3) - b^(1
/3)*x)/(a^(1/6)*b^(1/6)*Sqrt[x])])/b^(1/6) + (55*Sqrt[3]*(-17*A*b + 5*a*B)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*Sq
rt[x])/(a^(1/3) + b^(1/3)*x)])/b^(1/6))/(1080*a^(23/6))

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Maple [A]
time = 0.37, size = 237, normalized size = 0.68

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {\left (\frac {23}{72} b^{2} A -\frac {11}{72} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (29 A b -17 B a \right ) \sqrt {x}}{72}}{\left (b \,x^{3}+a \right )^{2}}+\frac {11 \left (17 A b -5 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}\right )}{72}\right )}{a^{3}}-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}\)	\(237\)
default	\(-\frac {2 \left (\frac {\left (\frac {23}{72} b^{2} A -\frac {11}{72} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (29 A b -17 B a \right ) \sqrt {x}}{72}}{\left (b \,x^{3}+a \right )^{2}}+\frac {11 \left (17 A b -5 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}\right )}{72}\right )}{a^{3}}-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}\)	\(237\)
risch	\(-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {23 x^{\frac {7}{2}} b^{2} A}{36 a^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {11 x^{\frac {7}{2}} b B}{36 a^{2} \left (b \,x^{3}+a \right )^{2}}-\frac {29 A \sqrt {x}\, b}{36 a^{2} \left (b \,x^{3}+a \right )^{2}}+\frac {17 B \sqrt {x}}{36 a \left (b \,x^{3}+a \right )^{2}}-\frac {187 A 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 )}{432 a^{4}}-\frac {187 A 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 )}{216 a^{4}}+\frac {187 A 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 )}{432 a^{4}}-\frac {187 A 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 )}{216 a^{4}}-\frac {187 A b \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{108 a^{4}}+\frac {55 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 )}{432 a^{3}}+\frac {55 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 )}{216 a^{3}}-\frac {55 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 )}{432 a^{3}}+\frac {55 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 )}{216 a^{3}}+\frac {55 B \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{108 a^{3}}\)	\(435\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

-2/a^3*(((23/72*b^2*A-11/72*a*b*B)*x^(7/2)+1/72*a*(29*A*b-17*B*a)*x^(1/2))/(b*x^3+a)^2+11/72*(17*A*b-5*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))))-2/5*A/a^3/x^(5/2)

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Maxima [A]
time = 0.51, size = 346, normalized size = 0.99 \begin {gather*} \frac {11 \, {\left (5 \, B a b - 17 \, A b^{2}\right )} x^{6} + 17 \, {\left (5 \, B a^{2} - 17 \, A a b\right )} x^{3} - 72 \, A a^{2}}{180 \, {\left (a^{3} b^{2} x^{\frac {17}{2}} + 2 \, a^{4} b x^{\frac {11}{2}} + a^{5} x^{\frac {5}{2}}\right )}} + \frac {11 \, {\left (\frac {\sqrt {3} {\left (5 \, B a - 17 \, 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 (5 \, B a - 17 \, 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 (5 \, B a b^{\frac {1}{3}} - 17 \, 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 (5 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} - 17 \, 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 (5 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} - 17 \, 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}}}}\right )}}{432 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/180*(11*(5*B*a*b - 17*A*b^2)*x^6 + 17*(5*B*a^2 - 17*A*a*b)*x^3 - 72*A*a^2)/(a^3*b^2*x^(17/2) + 2*a^4*b*x^(11
/2) + a^5*x^(5/2)) + 11/432*(sqrt(3)*(5*B*a - 17*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)*(5*B*a - 17*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*(5*B*a*b^(1/3) - 17*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*(5*B*a^(4/3)*b^(1/3) - 17*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*(5*B*a^(4/3)*b^(1/3) - 17*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))))/a^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2690 vs. \(2 (261) = 522\).
time = 3.32, size = 2690, normalized size = 7.66 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

-1/2160*(220*sqrt(3)*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2
*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/
(a^23*b))^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^8*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2
- 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1
/3) + (25*B^2*a^2 - 170*A*B*a*b + 289*A^2*b^2)*x + (5*B*a^5 - 17*A*a^4*b)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*
B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b
^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6))*a^19*b*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2
- 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(5
/6) + 2*sqrt(3)*(5*B*a^20*b - 17*A*a^19*b^2)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a
^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*
b))^(5/6) - sqrt(3)*(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 +
31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6))/(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 27
09375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*
A^6*b^6)) + 220*sqrt(3)*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*
A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^
6)/(a^23*b))^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^8*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*
b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))
^(1/3) + (25*B^2*a^2 - 170*A*B*a*b + 289*A^2*b^2)*x - (5*B*a^5 - 17*A*a^4*b)*sqrt(x)*(-(15625*B^6*a^6 - 318750
*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*
a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6))*a^19*b*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*
b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))
^(5/6) + 2*sqrt(3)*(5*B*a^20*b - 17*A*a^19*b^2)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^
4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^
23*b))^(5/6) + sqrt(3)*(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^
3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6))/(15625*B^6*a^6 - 318750*A*B^5*a^5*b +
2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 241375
69*A^6*b^6)) - 55*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^
4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^
23*b))^(1/6)*log(121*a^8*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^
3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/3) + 121*(25*B^2*a^2
- 170*A*B*a*b + 289*A^2*b^2)*x + 121*(5*B*a^5 - 17*A*a^4*b)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 27
09375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*
A^6*b^6)/(a^23*b))^(1/6)) + 55*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2
709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569
*A^6*b^6)/(a^23*b))^(1/6)*log(121*a^8*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 122825
00*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/3) + 121
*(25*B^2*a^2 - 170*A*B*a*b + 289*A^2*b^2)*x - 121*(5*B*a^5 - 17*A*a^4*b)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B
^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^
5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)) + 110*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A
*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*
b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)*log(11*a^4*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4
*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b)
)^(1/6) - 11*(5*B*a - 17*A*b)*sqrt(x)) - 110*(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3)*(-(15625*B^6*a^6 - 318750*A
*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*
b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)*log(-11*a^4*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^
4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**3+A)/x**(7/2)/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]
time = 1.21, size = 334, normalized size = 0.95 \begin {gather*} \frac {11 \, \sqrt {3} {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 17 \, \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 )}{432 \, a^{4} b} - \frac {11 \, \sqrt {3} {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 17 \, \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 )}{432 \, a^{4} b} + \frac {11 \, {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 17 \, \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 )}{216 \, a^{4} b} + \frac {11 \, {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 17 \, \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 )}{216 \, a^{4} b} + \frac {11 \, {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 17 \, \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 )}{108 \, a^{4} b} + \frac {11 \, B a b x^{\frac {7}{2}} - 23 \, A b^{2} x^{\frac {7}{2}} + 17 \, B a^{2} \sqrt {x} - 29 \, A a b \sqrt {x}}{36 \, {\left (b x^{3} + a\right )}^{2} a^{3}} - \frac {2 \, A}{5 \, a^{3} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^(7/2)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

11/432*sqrt(3)*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*log(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))
/(a^4*b) - 11/432*sqrt(3)*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x +
(a/b)^(1/3))/(a^4*b) + 11/216*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqr
t(x))/(a/b)^(1/6))/(a^4*b) + 11/216*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6)
- 2*sqrt(x))/(a/b)^(1/6))/(a^4*b) + 11/108*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*arctan(sqrt(x)/(a/b)^(
1/6))/(a^4*b) + 1/36*(11*B*a*b*x^(7/2) - 23*A*b^2*x^(7/2) + 17*B*a^2*sqrt(x) - 29*A*a*b*sqrt(x))/((b*x^3 + a)^
2*a^3) - 2/5*A/(a^3*x^(5/2))

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Mupad [B]
time = 2.96, size = 2500, normalized size = 7.12 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^3)/(x^(7/2)*(a + b*x^3)^3),x)

[Out]

- ((2*A)/(5*a) + (17*x^3*(17*A*b - 5*B*a))/(180*a^2) + (11*b*x^6*(17*A*b - 5*B*a))/(180*a^3))/(a^2*x^(5/2) + b
^2*x^(17/2) + 2*a*b*x^(11/2)) - (atan((((x^(1/2)*(443639472636450816*A^4*a^15*b^9 + 3319819810560000*B^4*a^19*
b^5 + 230262702060441600*A^2*B^2*a^17*b^7 - 45149549423616000*A*B^3*a^18*b^6 - 521928791337000960*A^3*B*a^16*b
^8) - (11*(17*A*b - 5*B*a)*(512439176949055488*A^3*a^19*b^8 - 13037837801472000*B^3*a^22*b^5 + 132985945575014
400*A*B^2*a^21*b^6 - 452152214955048960*A^2*B*a^20*b^7))/(216*(-a)^(23/6)*b^(1/6)))*(17*A*b - 5*B*a)*11i)/(216
*(-a)^(23/6)*b^(1/6)) + ((x^(1/2)*(443639472636450816*A^4*a^15*b^9 + 3319819810560000*B^4*a^19*b^5 + 230262702
060441600*A^2*B^2*a^17*b^7 - 45149549423616000*A*B^3*a^18*b^6 - 521928791337000960*A^3*B*a^16*b^8) + (11*(17*A
*b - 5*B*a)*(512439176949055488*A^3*a^19*b^8 - 13037837801472000*B^3*a^22*b^5 + 132985945575014400*A*B^2*a^21*
b^6 - 452152214955048960*A^2*B*a^20*b^7))/(216*(-a)^(23/6)*b^(1/6)))*(17*A*b - 5*B*a)*11i)/(216*(-a)^(23/6)*b^
(1/6)))/((11*(x^(1/2)*(443639472636450816*A^4*a^15*b^9 + 3319819810560000*B^4*a^19*b^5 + 230262702060441600*A^
2*B^2*a^17*b^7 - 45149549423616000*A*B^3*a^18*b^6 - 521928791337000960*A^3*B*a^16*b^8) - (11*(17*A*b - 5*B*a)*
(512439176949055488*A^3*a^19*b^8 - 13037837801472000*B^3*a^22*b^5 + 132985945575014400*A*B^2*a^21*b^6 - 452152
214955048960*A^2*B*a^20*b^7))/(216*(-a)^(23/6)*b^(1/6)))*(17*A*b - 5*B*a))/(216*(-a)^(23/6)*b^(1/6)) - (11*(x^
(1/2)*(443639472636450816*A^4*a^15*b^9 + 3319819810560000*B^4*a^19*b^5 + 230262702060441600*A^2*B^2*a^17*b^7 -
45149549423616000*A*B^3*a^18*b^6 - 521928791337000960*A^3*B*a^16*b^8) + (11*(17*A*b - 5*B*a)*(512439176949055
488*A^3*a^19*b^8 - 13037837801472000*B^3*a^22*b^5 + 132985945575014400*A*B^2*a^21*b^6 - 452152214955048960*A^2
*B*a^20*b^7))/(216*(-a)^(23/6)*b^(1/6)))*(17*A*b - 5*B*a))/(216*(-a)^(23/6)*b^(1/6))))*(17*A*b - 5*B*a)*11i)/(
108*(-a)^(23/6)*b^(1/6)) - (atan(((((3^(1/2)*1i)/2 - 1/2)*(17*A*b - 5*B*a)*(x^(1/2)*(443639472636450816*A^4*a^
15*b^9 + 3319819810560000*B^4*a^19*b^5 + 230262702060441600*A^2*B^2*a^17*b^7 - 45149549423616000*A*B^3*a^18*b^
6 - 521928791337000960*A^3*B*a^16*b^8) - (11*((3^(1/2)*1i)/2 - 1/2)*(17*A*b - 5*B*a)*(512439176949055488*A^3*a
^19*b^8 - 13037837801472000*B^3*a^22*b^5 + 132985945575014400*A*B^2*a^21*b^6 - 452152214955048960*A^2*B*a^20*b
^7))/(216*(-a)^(23/6)*b^(1/6)))*11i)/(216*(-a)^(23/6)*b^(1/6)) + (((3^(1/2)*1i)/2 - 1/2)*(17*A*b - 5*B*a)*(x^(
1/2)*(443639472636450816*A^4*a^15*b^9 + 3319819810560000*B^4*a^19*b^5 + 230262702060441600*A^2*B^2*a^17*b^7 -
45149549423616000*A*B^3*a^18*b^6 - 521928791337000960*A^3*B*a^16*b^8) + (11*((3^(1/2)*1i)/2 - 1/2)*(17*A*b - 5
*B*a)*(512439176949055488*A^3*a^19*b^8 - 13037837801472000*B^3*a^22*b^5 + 132985945575014400*A*B^2*a^21*b^6 -
452152214955048960*A^2*B*a^20*b^7))/(216*(-a)^(23/6)*b^(1/6)))*11i)/(216*(-a)^(23/6)*b^(1/6)))/((11*((3^(1/2)*
1i)/2 - 1/2)*(17*A*b - 5*B*a)*(x^(1/2)*(443639472636450816*A^4*a^15*b^9 + 3319819810560000*B^4*a^19*b^5 + 2302
62702060441600*A^2*B^2*a^17*b^7 - 45149549423616000*A*B^3*a^18*b^6 - 521928791337000960*A^3*B*a^16*b^8) - (11*
((3^(1/2)*1i)/2 - 1/2)*(17*A*b - 5*B*a)*(512439176949055488*A^3*a^19*b^8 - 13037837801472000*B^3*a^22*b^5 + 13
2985945575014400*A*B^2*a^21*b^6 - 452152214955048960*A^2*B*a^20*b^7))/(216*(-a)^(23/6)*b^(1/6))))/(216*(-a)^(2
3/6)*b^(1/6)) - (11*((3^(1/2)*1i)/2 - 1/2)*(17*A*b - 5*B*a)*(x^(1/2)*(443639472636450816*A^4*a^15*b^9 + 331981
9810560000*B^4*a^19*b^5 + 230262702060441600*A^2*B^2*a^17*b^7 - 45149549423616000*A*B^3*a^18*b^6 - 52192879133
7000960*A^3*B*a^16*b^8) + (11*((3^(1/2)*1i)/2 - 1/2)*(17*A*b - 5*B*a)*(512439176949055488*A^3*a^19*b^8 - 13037
837801472000*B^3*a^22*b^5 + 132985945575014400*A*B^2*a^21*b^6 - 452152214955048960*A^2*B*a^20*b^7))/(216*(-a)^
(23/6)*b^(1/6))))/(216*(-a)^(23/6)*b^(1/6))))*((3^(1/2)*1i)/2 - 1/2)*(17*A*b - 5*B*a)*11i)/(108*(-a)^(23/6)*b^
(1/6)) - (atan(((((3^(1/2)*1i)/2 + 1/2)*(17*A*b - 5*B*a)*(x^(1/2)*(443639472636450816*A^4*a^15*b^9 + 331981981
0560000*B^4*a^19*b^5 + 230262702060441600*A^2*B^2*a^17*b^7 - 45149549423616000*A*B^3*a^18*b^6 - 52192879133700
0960*A^3*B*a^16*b^8) - (11*((3^(1/2)*1i)/2 + 1/2)*(17*A*b - 5*B*a)*(512439176949055488*A^3*a^19*b^8 - 13037837
801472000*B^3*a^22*b^5 + 132985945575014400*A*B^2*a^21*b^6 - 452152214955048960*A^2*B*a^20*b^7))/(216*(-a)^(23
/6)*b^(1/6)))*11i)/(216*(-a)^(23/6)*b^(1/6)) + (((3^(1/2)*1i)/2 + 1/2)*(17*A*b - 5*B*a)*(x^(1/2)*(443639472636
450816*A^4*a^15*b^9 + 3319819810560000*B^4*a^19*b^5 + 230262702060441600*A^2*B^2*a^17*b^7 - 45149549423616000*
A*B^3*a^18*b^6 - 521928791337000960*A^3*B*a^16*b^8) + (11*((3^(1/2)*1i)/2 + 1/2)*(17*A*b - 5*B*a)*(51243917694
9055488*A^3*a^19*b^8 - 13037837801472000*B^3*a^22*b^5 + 132985945575014400*A*B^2*a^21*b^6 - 452152214955048960
*A^2*B*a^20*b^7))/(216*(-a)^(23/6)*b^(1/6)))*11i)/(216*(-a)^(23/6)*b^(1/6)))/((11*((3^(1/2)*1i)/2 + 1/2)*(17*A
*b - 5*B*a)*(x^(1/2)*(443639472636450816*A^4*a^15*b^9 + 3319819810560000*B^4*a^19*b^5 + 230262702060441600*A^2
*B^2*a^17*b^7 - 45149549423616000*A*B^3*a^18*b^...

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