\(\int \frac {x^{7/2} (A+B x^2)}{(a+b x^2)^3} \, dx\) [383]

Optimal result

Integrand size = 22, antiderivative size = 316 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {5 (A b-9 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}-\frac {5 (A b-9 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}} \]

[Out]

1/4*(A*b-B*a)*x^(9/2)/a/b/(b*x^2+a)^2+1/16*(A*b-9*B*a)*x^(5/2)/a/b^2/(b*x^2+a)-5/64*(A*b-9*B*a)*arctan(1-b^(1/
4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/b^(13/4)*2^(1/2)+5/64*(A*b-9*B*a)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4)
)/a^(3/4)/b^(13/4)*2^(1/2)-5/128*(A*b-9*B*a)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/b^(
13/4)*2^(1/2)+5/128*(A*b-9*B*a)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/b^(13/4)*2^(1/2)
-5/16*(A*b-9*B*a)*x^(1/2)/a/b^3

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {468, 294, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {5 (A b-9 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}-\frac {5 (A b-9 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}-\frac {5 \sqrt {x} (A b-9 a B)}{16 a b^3}+\frac {x^{5/2} (A b-9 a B)}{16 a b^2 \left (a+b x^2\right )}+\frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

[In]

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

[Out]

(-5*(A*b - 9*a*B)*Sqrt[x])/(16*a*b^3) + ((A*b - a*B)*x^(9/2))/(4*a*b*(a + b*x^2)^2) + ((A*b - 9*a*B)*x^(5/2))/
(16*a*b^2*(a + b*x^2)) - (5*(A*b - 9*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*b
^(13/4)) + (5*(A*b - 9*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*b^(13/4)) - (5*
(A*b - 9*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(3/4)*b^(13/4)) + (5*(
A*b - 9*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(3/4)*b^(13/4))

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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {\left (-\frac {A b}{2}+\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx}{4 a b} \\ & = \frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {(5 (A b-9 a B)) \int \frac {x^{3/2}}{a+b x^2} \, dx}{32 a b^2} \\ & = -\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(5 (A b-9 a B)) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{32 b^3} \\ & = -\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 b^3} \\ & = -\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {a} b^3}+\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {a} b^3} \\ & = -\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {a} b^{7/2}}+\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {a} b^{7/2}}-\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}-\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} a^{3/4} b^{13/4}} \\ & = -\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {5 (A b-9 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}-\frac {(5 (A b-9 a B)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}} \\ & = -\frac {5 (A b-9 a B) \sqrt {x}}{16 a b^3}+\frac {(A b-a B) x^{9/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b-9 a B) x^{5/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {5 (A b-9 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}-\frac {5 (A b-9 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{13/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.58 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {4 \sqrt [4]{b} \sqrt {x} \left (-5 a A b+45 a^2 B-9 A b^2 x^2+81 a b B x^2+32 b^2 B x^4\right )}{\left (a+b x^2\right )^2}+\frac {5 \sqrt {2} (-A b+9 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {5 \sqrt {2} (A b-9 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}}{64 b^{13/4}} \]

[In]

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

[Out]

((4*b^(1/4)*Sqrt[x]*(-5*a*A*b + 45*a^2*B - 9*A*b^2*x^2 + 81*a*b*B*x^2 + 32*b^2*B*x^4))/(a + b*x^2)^2 + (5*Sqrt
[2]*(-(A*b) + 9*a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(3/4) + (5*Sqrt[2]*(A*
b - 9*a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(3/4))/(64*b^(13/4))

Maple [A] (verified)

Time = 2.90 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.54

method	result	size
derivativedivides	\(\frac {2 B \sqrt {x}}{b^{3}}+\frac {\frac {2 \left (\left (-\frac {9}{32} b^{2} A +\frac {17}{32} a b B \right ) x^{\frac {5}{2}}-\frac {a \left (5 A b -13 B a \right ) \sqrt {x}}{32}\right )}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (A b -9 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}}{b^{3}}\)	\(172\)
default	\(\frac {2 B \sqrt {x}}{b^{3}}+\frac {\frac {2 \left (\left (-\frac {9}{32} b^{2} A +\frac {17}{32} a b B \right ) x^{\frac {5}{2}}-\frac {a \left (5 A b -13 B a \right ) \sqrt {x}}{32}\right )}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (A b -9 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}}{b^{3}}\)	\(172\)
risch	\(\frac {2 B \sqrt {x}}{b^{3}}+\frac {\frac {2 \left (-\frac {9}{32} b^{2} A +\frac {17}{32} a b B \right ) x^{\frac {5}{2}}-\frac {a \left (5 A b -13 B a \right ) \sqrt {x}}{16}}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (A b -9 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}}{b^{3}}\)	\(172\)

[In]

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

[Out]

2*B/b^3*x^(1/2)+2/b^3*(((-9/32*b^2*A+17/32*a*b*B)*x^(5/2)-1/32*a*(5*A*b-13*B*a)*x^(1/2))/(b*x^2+a)^2+5/256*(A*
b-9*B*a)*(a/b)^(1/4)/a*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+
(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 748, normalized size of antiderivative = 2.37 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {5 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} \log \left (5 \, a b^{3} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} - 5 \, {\left (9 \, B a - A b\right )} \sqrt {x}\right ) - 5 \, {\left (-i \, b^{5} x^{4} - 2 i \, a b^{4} x^{2} - i \, a^{2} b^{3}\right )} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} \log \left (5 i \, a b^{3} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} - 5 \, {\left (9 \, B a - A b\right )} \sqrt {x}\right ) - 5 \, {\left (i \, b^{5} x^{4} + 2 i \, a b^{4} x^{2} + i \, a^{2} b^{3}\right )} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} \log \left (-5 i \, a b^{3} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} - 5 \, {\left (9 \, B a - A b\right )} \sqrt {x}\right ) - 5 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} \log \left (-5 \, a b^{3} \left (-\frac {6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac {1}{4}} - 5 \, {\left (9 \, B a - A b\right )} \sqrt {x}\right ) + 4 \, {\left (32 \, B b^{2} x^{4} + 45 \, B a^{2} - 5 \, A a b + 9 \, {\left (9 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]

[In]

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

[Out]

1/64*(5*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*
a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4)*log(5*a*b^3*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A
^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4) - 5*(9*B*a - A*b)*sqrt(x)) - 5*(-I*b^5*x^4 - 2*I*a*b^4*x^2 - I*a^2*b^3
)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4)*log(5
*I*a*b^3*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4
) - 5*(9*B*a - A*b)*sqrt(x)) - 5*(I*b^5*x^4 + 2*I*a*b^4*x^2 + I*a^2*b^3)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b +
486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4)*log(-5*I*a*b^3*(-(6561*B^4*a^4 - 2916*A*B^3*
a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4) - 5*(9*B*a - A*b)*sqrt(x)) - 5*(b^5*
x^4 + 2*a*b^4*x^2 + a^2*b^3)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b
^4)/(a^3*b^13))^(1/4)*log(-5*a*b^3*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 +
A^4*b^4)/(a^3*b^13))^(1/4) - 5*(9*B*a - A*b)*sqrt(x)) + 4*(32*B*b^2*x^4 + 45*B*a^2 - 5*A*a*b + 9*(9*B*a*b - A
*b^2)*x^2)*sqrt(x))/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.90 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (17 \, B a b - 9 \, A b^{2}\right )} x^{\frac {5}{2}} + {\left (13 \, B a^{2} - 5 \, A a b\right )} \sqrt {x}}{16 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {2 \, B \sqrt {x}}{b^{3}} - \frac {5 \, {\left (\frac {2 \, \sqrt {2} {\left (9 \, B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (9 \, B a - A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (9 \, B a - A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (9 \, B a - A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{128 \, b^{3}} \]

[In]

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

[Out]

1/16*((17*B*a*b - 9*A*b^2)*x^(5/2) + (13*B*a^2 - 5*A*a*b)*sqrt(x))/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3) + 2*B*sqr
t(x)/b^3 - 5/128*(2*sqrt(2)*(9*B*a - A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqr
t(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(9*B*a - A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(
1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(9*B*a - A*
b)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(9*B*a - A*b)*log(-s
qrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/b^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.96 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {2 \, B \sqrt {x}}{b^{3}} - \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a b^{4}} - \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a b^{4}} - \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a b^{4}} + \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a b^{4}} + \frac {17 \, B a b x^{\frac {5}{2}} - 9 \, A b^{2} x^{\frac {5}{2}} + 13 \, B a^{2} \sqrt {x} - 5 \, A a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} b^{3}} \]

[In]

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

[Out]

2*B*sqrt(x)/b^3 - 5/64*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/
4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^4) - 5/64*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt
(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^4) - 5/128*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4
)*A*b)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^4) + 5/128*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - (a*b^3)
^(1/4)*A*b)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^4) + 1/16*(17*B*a*b*x^(5/2) - 9*A*b^2*x^(5/
2) + 13*B*a^2*sqrt(x) - 5*A*a*b*sqrt(x))/((b*x^2 + a)^2*b^3)

Mupad [B] (verification not implemented)

Time = 5.24 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.41 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\sqrt {x}\,\left (\frac {13\,B\,a^2}{16}-\frac {5\,A\,a\,b}{16}\right )-x^{5/2}\,\left (\frac {9\,A\,b^2}{16}-\frac {17\,B\,a\,b}{16}\right )}{a^2\,b^3+2\,a\,b^4\,x^2+b^5\,x^4}+\frac {2\,B\,\sqrt {x}}{b^3}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}-\frac {5\,\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}+\frac {\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}+\frac {5\,\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}}{\frac {5\,\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}-\frac {5\,\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}-\frac {5\,\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}+\frac {5\,\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}}\right )\,\left (A\,b-9\,B\,a\right )\,5{}\mathrm {i}}{32\,{\left (-a\right )}^{3/4}\,b^{13/4}}-\frac {5\,\mathrm {atan}\left (\frac {\frac {5\,\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}-\frac {\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}+\frac {5\,\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}+\frac {\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}}{\frac {\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}-\frac {\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}-\frac {\left (A\,b-9\,B\,a\right )\,\left (\frac {25\,\sqrt {x}\,\left (A^2\,b^2-18\,A\,B\,a\,b+81\,B^2\,a^2\right )}{64\,b^3}+\frac {\left (45\,B\,a^2-5\,A\,a\,b\right )\,\left (A\,b-9\,B\,a\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,5{}\mathrm {i}}{64\,{\left (-a\right )}^{3/4}\,b^{13/4}}}\right )\,\left (A\,b-9\,B\,a\right )}{32\,{\left (-a\right )}^{3/4}\,b^{13/4}} \]

[In]

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

[Out]

(x^(1/2)*((13*B*a^2)/16 - (5*A*a*b)/16) - x^(5/2)*((9*A*b^2)/16 - (17*B*a*b)/16))/(a^2*b^3 + b^5*x^4 + 2*a*b^4
*x^2) + (2*B*x^(1/2))/b^3 - (atan((((A*b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) -
(5*(45*B*a^2 - 5*A*a*b)*(A*b - 9*B*a))/(64*(-a)^(3/4)*b^(13/4)))*5i)/(64*(-a)^(3/4)*b^(13/4)) + ((A*b - 9*B*a
)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) + (5*(45*B*a^2 - 5*A*a*b)*(A*b - 9*B*a))/(64*(-a)
^(3/4)*b^(13/4)))*5i)/(64*(-a)^(3/4)*b^(13/4)))/((5*(A*b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*
a*b))/(64*b^3) - (5*(45*B*a^2 - 5*A*a*b)*(A*b - 9*B*a))/(64*(-a)^(3/4)*b^(13/4))))/(64*(-a)^(3/4)*b^(13/4)) -
(5*(A*b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) + (5*(45*B*a^2 - 5*A*a*b)*(A*b - 9
*B*a))/(64*(-a)^(3/4)*b^(13/4))))/(64*(-a)^(3/4)*b^(13/4))))*(A*b - 9*B*a)*5i)/(32*(-a)^(3/4)*b^(13/4)) - (5*a
tan(((5*(A*b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) - ((45*B*a^2 - 5*A*a*b)*(A*b
- 9*B*a)*5i)/(64*(-a)^(3/4)*b^(13/4))))/(64*(-a)^(3/4)*b^(13/4)) + (5*(A*b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81
*B^2*a^2 - 18*A*B*a*b))/(64*b^3) + ((45*B*a^2 - 5*A*a*b)*(A*b - 9*B*a)*5i)/(64*(-a)^(3/4)*b^(13/4))))/(64*(-a)
^(3/4)*b^(13/4)))/(((A*b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) - ((45*B*a^2 - 5*
A*a*b)*(A*b - 9*B*a)*5i)/(64*(-a)^(3/4)*b^(13/4)))*5i)/(64*(-a)^(3/4)*b^(13/4)) - ((A*b - 9*B*a)*((25*x^(1/2)*
(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) + ((45*B*a^2 - 5*A*a*b)*(A*b - 9*B*a)*5i)/(64*(-a)^(3/4)*b^(13/4
)))*5i)/(64*(-a)^(3/4)*b^(13/4))))*(A*b - 9*B*a))/(32*(-a)^(3/4)*b^(13/4))