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

Optimal. Leaf size=557 \[ \frac{9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^{5/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d (d x)^{3/2}}{1024 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(45*d*(d*x)^(3/2))/(1024*a^3*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(3/2)
)/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (d*(d*x)^(3/2))/(32*a*b*
(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (9*d*(d*x)^(3/2))/(256*a^2*b*(a
+ b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*d^(5/2)*(a + b*x^2)*ArcTan[1 -
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(13/4)*b^(7/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*d^(5/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^
(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(13/4)*b^(7/4)*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4]) + (45*d^(5/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(13/4)*b^(7/4)*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*d^(5/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqr
t[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(13/4)*b^(7
/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.973764, antiderivative size = 557, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^{5/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{13/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d (d x)^{3/2}}{1024 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(45*d*(d*x)^(3/2))/(1024*a^3*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(3/2)
)/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (d*(d*x)^(3/2))/(32*a*b*
(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (9*d*(d*x)^(3/2))/(256*a^2*b*(a
+ b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*d^(5/2)*(a + b*x^2)*ArcTan[1 -
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(13/4)*b^(7/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*d^(5/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^
(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(13/4)*b^(7/4)*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4]) + (45*d^(5/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(13/4)*b^(7/4)*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*d^(5/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqr
t[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(13/4)*b^(7
/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)**(5/2)/(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 = 0.36038, size = 324, normalized size = 0.58 \[ \frac{(d x)^{5/2} \left (a+b x^2\right ) \left (-1024 a^{13/4} b^{3/4} x^{3/2}+288 a^{5/4} b^{3/4} x^{3/2} \left (a+b x^2\right )^2+256 a^{9/4} b^{3/4} x^{3/2} \left (a+b x^2\right )+360 \sqrt [4]{a} b^{3/4} x^{3/2} \left (a+b x^2\right )^3+45 \sqrt{2} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-45 \sqrt{2} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-90 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+90 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{8192 a^{13/4} b^{7/4} x^{5/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((d*x)^(5/2)*(a + b*x^2)*(-1024*a^(13/4)*b^(3/4)*x^(3/2) + 256*a^(9/4)*b^(3/4)*x
^(3/2)*(a + b*x^2) + 288*a^(5/4)*b^(3/4)*x^(3/2)*(a + b*x^2)^2 + 360*a^(1/4)*b^(
3/4)*x^(3/2)*(a + b*x^2)^3 - 90*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)] + 90*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[
x])/a^(1/4)] + 45*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sq
rt[x] + Sqrt[b]*x] - 45*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1
/4)*Sqrt[x] + Sqrt[b]*x]))/(8192*a^(13/4)*b^(7/4)*x^(5/2)*((a + b*x^2)^2)^(5/2))

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Maple [B]  time = 0.029, size = 1046, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8192*(45*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2))
/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*x^8*b^4*d^8+90*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^8-90*
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
^8+360*(a*d^2/b)^(1/4)*(d*x)^(15/2)*b^4+180*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(
1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^
2/b)^(1/2)))*x^6*a*b^3*d^8+360*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^8-360*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^8+1368*(a*d^2/b)^(1/4)*(d*x)^(11/2)*a*
b^3*d^2+270*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)
)/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*x^4*a^2*b^2*d^8+540
*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^8-540*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^8+1912*(a*d^2/b)^(1/4)*(d*x)^(7/2)*a^2*b^2*d^4+180*2^(1/2)*ln(-(
(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d
*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*x^2*a^3*b*d^8+360*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^8-360*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^8-120*(a*d^2/b)
^(1/4)*(d*x)^(3/2)*a^3*b*d^6+45*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)
-d*x-(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))
*a^4*d^8+90*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^8-90*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^8)/d^5*(b*x^2+a)/(a*d^2/b)^(1/4)/b^2/a^3/((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)^(5/2)/(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 = 0.301588, size = 583, normalized size = 1.05 \[ \frac{180 \,{\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac{d^{10}}{a^{13} b^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{91125 \, a^{10} b^{5} \left (-\frac{d^{10}}{a^{13} b^{7}}\right )^{\frac{3}{4}}}{91125 \, \sqrt{d x} d^{7} + \sqrt{-8303765625 \, a^{7} b^{3} d^{10} \sqrt{-\frac{d^{10}}{a^{13} b^{7}}} + 8303765625 \, d^{15} x}}\right ) + 45 \,{\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac{d^{10}}{a^{13} b^{7}}\right )^{\frac{1}{4}} \log \left (91125 \, a^{10} b^{5} \left (-\frac{d^{10}}{a^{13} b^{7}}\right )^{\frac{3}{4}} + 91125 \, \sqrt{d x} d^{7}\right ) - 45 \,{\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac{d^{10}}{a^{13} b^{7}}\right )^{\frac{1}{4}} \log \left (-91125 \, a^{10} b^{5} \left (-\frac{d^{10}}{a^{13} b^{7}}\right )^{\frac{3}{4}} + 91125 \, \sqrt{d x} d^{7}\right ) + 4 \,{\left (45 \, b^{3} d^{2} x^{7} + 171 \, a b^{2} d^{2} x^{5} + 239 \, a^{2} b d^{2} x^{3} - 15 \, a^{3} d^{2} x\right )} \sqrt{d x}}{4096 \,{\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4096*(180*(a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4 + 4*a^6*b^2*x^2 + a^7*b
)*(-d^10/(a^13*b^7))^(1/4)*arctan(91125*a^10*b^5*(-d^10/(a^13*b^7))^(3/4)/(91125
*sqrt(d*x)*d^7 + sqrt(-8303765625*a^7*b^3*d^10*sqrt(-d^10/(a^13*b^7)) + 83037656
25*d^15*x))) + 45*(a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4 + 4*a^6*b^2*x^2 +
a^7*b)*(-d^10/(a^13*b^7))^(1/4)*log(91125*a^10*b^5*(-d^10/(a^13*b^7))^(3/4) + 9
1125*sqrt(d*x)*d^7) - 45*(a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4 + 4*a^6*b^
2*x^2 + a^7*b)*(-d^10/(a^13*b^7))^(1/4)*log(-91125*a^10*b^5*(-d^10/(a^13*b^7))^(
3/4) + 91125*sqrt(d*x)*d^7) + 4*(45*b^3*d^2*x^7 + 171*a*b^2*d^2*x^5 + 239*a^2*b*
d^2*x^3 - 15*a^3*d^2*x)*sqrt(d*x))/(a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4
+ 4*a^6*b^2*x^2 + a^7*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{\frac{5}{2}}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**(5/2)/((a + b*x**2)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.296103, size = 549, normalized size = 0.99 \[ \frac{1}{8192} \, d{\left (\frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{4} b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{4} b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{4} b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{4} b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{8 \,{\left (45 \, \sqrt{d x} b^{3} d^{9} x^{7} + 171 \, \sqrt{d x} a b^{2} d^{9} x^{5} + 239 \, \sqrt{d x} a^{2} b d^{9} x^{3} - 15 \, \sqrt{d x} a^{3} d^{9} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{3} b{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8192*d*(90*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/
4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^4*b^4*sign(b*d^4*x^2 + a*d^4)) + 90*sqrt(2
)*(a*b^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/
(a*d^2/b)^(1/4))/(a^4*b^4*sign(b*d^4*x^2 + a*d^4)) - 45*sqrt(2)*(a*b^3*d^2)^(3/4
)*ln(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^4*b^4*sign(b*d^
4*x^2 + a*d^4)) + 45*sqrt(2)*(a*b^3*d^2)^(3/4)*ln(d*x - sqrt(2)*(a*d^2/b)^(1/4)*
sqrt(d*x) + sqrt(a*d^2/b))/(a^4*b^4*sign(b*d^4*x^2 + a*d^4)) + 8*(45*sqrt(d*x)*b
^3*d^9*x^7 + 171*sqrt(d*x)*a*b^2*d^9*x^5 + 239*sqrt(d*x)*a^2*b*d^9*x^3 - 15*sqrt
(d*x)*a^3*d^9*x)/((b*d^2*x^2 + a*d^2)^4*a^3*b*sign(b*d^4*x^2 + a*d^4)))