3.773 \(\int \frac{(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=554 \[ -\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(-195*d^7*Sqrt[d*x])/(1024*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(13/2
))/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (13*d^3*(d*x)^(9/2))/(9
6*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (39*d^5*(d*x)^(5/2))/(256
*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (195*d^(15/2)*(a + b*x^2)*Ar
cTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(3/4)*b
^(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (195*d^(15/2)*(a + b*x^2)*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(3/4)*b^(17/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (195*d^(15/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d]
+ Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(3/4)*
b^(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (195*d^(15/2)*(a + b*x^2)*Log[Sqrt[a
]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2
]*a^(3/4)*b^(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.951387, antiderivative size = 554, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(-195*d^7*Sqrt[d*x])/(1024*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(13/2
))/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (13*d^3*(d*x)^(9/2))/(9
6*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (39*d^5*(d*x)^(5/2))/(256
*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (195*d^(15/2)*(a + b*x^2)*Ar
cTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(3/4)*b
^(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (195*d^(15/2)*(a + b*x^2)*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(3/4)*b^(17/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (195*d^(15/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d]
+ Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(3/4)*
b^(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (195*d^(15/2)*(a + b*x^2)*Log[Sqrt[a
]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2
]*a^(3/4)*b^(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**(15/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.343842, size = 324, normalized size = 0.58 \[ \frac{(d x)^{15/2} \left (a+b x^2\right ) \left (-14824 a^{3/4} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^3+19616 a^{7/4} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2-12544 a^{11/4} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )+3072 a^{15/4} \sqrt [4]{b} \sqrt{x}-585 \sqrt{2} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+585 \sqrt{2} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-1170 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+1170 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{24576 a^{3/4} b^{17/4} x^{15/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

((d*x)^(15/2)*(a + b*x^2)*(3072*a^(15/4)*b^(1/4)*Sqrt[x] - 12544*a^(11/4)*b^(1/4
)*Sqrt[x]*(a + b*x^2) + 19616*a^(7/4)*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 - 14824*a^(3
/4)*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 - 1170*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 - (Sqrt[
2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 1170*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^
(1/4)*Sqrt[x])/a^(1/4)] - 585*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 585*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(24576*a^(3/4)*b^(17/4)*x^(15/2)*((a + b
*x^2)^2)^(5/2))

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Maple [B]  time = 0.032, size = 1144, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/24576*(585*(a*d^2/b)^(1/4)*2^(1/2)*ln(-(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2
)+(a*d^2/b)^(1/2))/((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*x^
8*b^4*d^6+1170*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/
4))/(a*d^2/b)^(1/4))*x^8*b^4*d^6-1170*(a*d^2/b)^(1/4)*2^(1/2)*arctan((-2^(1/2)*(
d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^8*b^4*d^6+2340*(a*d^2/b)^(1/4)*2^
(1/2)*ln(-(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/((a*d^2/b)^(
1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*x^6*a*b^3*d^6+4680*(a*d^2/b)^(1/4
)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^6*a*b^
3*d^6-4680*(a*d^2/b)^(1/4)*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))
/(a*d^2/b)^(1/4))*x^6*a*b^3*d^6+3510*(a*d^2/b)^(1/4)*2^(1/2)*ln(-(d*x+(a*d^2/b)^
(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-
d*x-(a*d^2/b)^(1/2)))*x^4*a^2*b^2*d^6+7020*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/
2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^4*a^2*b^2*d^6-7020*(a*d^2/b)^
(1/4)*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^4
*a^2*b^2*d^6-14824*(d*x)^(13/2)*a*b^3+2340*(a*d^2/b)^(1/4)*2^(1/2)*ln(-(d*x+(a*d
^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^
(1/2)-d*x-(a*d^2/b)^(1/2)))*x^2*a^3*b*d^6+4680*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2
^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^2*a^3*b*d^6-4680*(a*d^2/b
)^(1/4)*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x
^2*a^3*b*d^6-24856*(d*x)^(9/2)*a^2*b^2*d^2+585*(a*d^2/b)^(1/4)*2^(1/2)*ln(-(d*x+
(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/((a*d^2/b)^(1/4)*(d*x)^(1/2
)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*a^4*d^6+1170*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(
1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^4*d^6-1170*(a*d^2/b)^(1/4)*
2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^4*d^6-1
7784*(d*x)^(5/2)*a^3*b*d^4-4680*(d*x)^(1/2)*a^4*d^6)*d*(b*x^2+a)/a/b^4/((b*x^2+a
)^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(15/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30072, size = 548, normalized size = 0.99 \[ -\frac{2340 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} a b^{4}}{\sqrt{d x} d^{7} + \sqrt{d^{15} x + \sqrt{-\frac{d^{30}}{a^{3} b^{17}}} a^{2} b^{8}}}\right ) - 585 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} \log \left (195 \, \sqrt{d x} d^{7} + 195 \, \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} a b^{4}\right ) + 585 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} \log \left (195 \, \sqrt{d x} d^{7} - 195 \, \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} a b^{4}\right ) + 4 \,{\left (1853 \, b^{3} d^{7} x^{6} + 3107 \, a b^{2} d^{7} x^{4} + 2223 \, a^{2} b d^{7} x^{2} + 585 \, a^{3} d^{7}\right )} \sqrt{d x}}{12288 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(15/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/12288*(2340*(b^8*x^8 + 4*a*b^7*x^6 + 6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)
*(-d^30/(a^3*b^17))^(1/4)*arctan((-d^30/(a^3*b^17))^(1/4)*a*b^4/(sqrt(d*x)*d^7 +
sqrt(d^15*x + sqrt(-d^30/(a^3*b^17))*a^2*b^8))) - 585*(b^8*x^8 + 4*a*b^7*x^6 +
6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)*(-d^30/(a^3*b^17))^(1/4)*log(195*sqrt(d
*x)*d^7 + 195*(-d^30/(a^3*b^17))^(1/4)*a*b^4) + 585*(b^8*x^8 + 4*a*b^7*x^6 + 6*a
^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)*(-d^30/(a^3*b^17))^(1/4)*log(195*sqrt(d*x)
*d^7 - 195*(-d^30/(a^3*b^17))^(1/4)*a*b^4) + 4*(1853*b^3*d^7*x^6 + 3107*a*b^2*d^
7*x^4 + 2223*a^2*b*d^7*x^2 + 585*a^3*d^7)*sqrt(d*x))/(b^8*x^8 + 4*a*b^7*x^6 + 6*
a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**(15/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293368, size = 552, normalized size = 1. \[ \frac{1}{24576} \, d^{6}{\left (\frac{1170 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{1170 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{585 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{585 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{8 \,{\left (1853 \, \sqrt{d x} b^{3} d^{9} x^{6} + 3107 \, \sqrt{d x} a b^{2} d^{9} x^{4} + 2223 \, \sqrt{d x} a^{2} b d^{9} x^{2} + 585 \, \sqrt{d x} a^{3} d^{9}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(15/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="giac")

[Out]

1/24576*d^6*(1170*sqrt(2)*(a*b^3*d^2)^(1/4)*d*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2
/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a*b^5*sign(b*d^4*x^2 + a*d^4)) + 1170
*sqrt(2)*(a*b^3*d^2)^(1/4)*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sq
rt(d*x))/(a*d^2/b)^(1/4))/(a*b^5*sign(b*d^4*x^2 + a*d^4)) + 585*sqrt(2)*(a*b^3*d
^2)^(1/4)*d*ln(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a*b^5*s
ign(b*d^4*x^2 + a*d^4)) - 585*sqrt(2)*(a*b^3*d^2)^(1/4)*d*ln(d*x - sqrt(2)*(a*d^
2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a*b^5*sign(b*d^4*x^2 + a*d^4)) - 8*(1853*
sqrt(d*x)*b^3*d^9*x^6 + 3107*sqrt(d*x)*a*b^2*d^9*x^4 + 2223*sqrt(d*x)*a^2*b*d^9*
x^2 + 585*sqrt(d*x)*a^3*d^9)/((b*d^2*x^2 + a*d^2)^4*b^4*sign(b*d^4*x^2 + a*d^4))
)