3.777 \(\int \frac{(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=560 \[ \frac{35 d^3 \sqrt{d x}}{3072 a^2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^3 \sqrt{d x}}{768 a b^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 \sqrt{d x}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{5/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 d^{7/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(35*d^3*Sqrt[d*x])/(3072*a^2*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(5/
2))/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (5*d^3*Sqrt[d*x])/(96*
b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (5*d^3*Sqrt[d*x])/(768*a*b^
2*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (35*d^(7/2)*(a + b*x^2)*ArcTan[
1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(11/4)*b^(9/
4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (35*d^(7/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2
]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(11/4)*b^(9/4)*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) - (35*d^(7/2)*(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^(11/4)*b^(9/4)*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (35*d^(7/2)*(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^(11/4)*
b^(9/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.98409, antiderivative size = 560, 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 \[ \frac{35 d^3 \sqrt{d x}}{3072 a^2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^3 \sqrt{d x}}{768 a b^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 \sqrt{d x}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{5/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 d^{7/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{11/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(35*d^3*Sqrt[d*x])/(3072*a^2*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(5/
2))/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (5*d^3*Sqrt[d*x])/(96*
b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (5*d^3*Sqrt[d*x])/(768*a*b^
2*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (35*d^(7/2)*(a + b*x^2)*ArcTan[
1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(11/4)*b^(9/
4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (35*d^(7/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2
]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(11/4)*b^(9/4)*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) - (35*d^(7/2)*(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^(11/4)*b^(9/4)*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (35*d^(7/2)*(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^(11/4)*
b^(9/4)*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)**(7/2)/(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 = 0.323296, size = 324, normalized size = 0.58 \[ \frac{(d x)^{7/2} \left (a+b x^2\right ) \left (280 a^{3/4} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^3+160 a^{7/4} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2-4352 a^{11/4} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )+3072 a^{15/4} \sqrt [4]{b} \sqrt{x}-105 \sqrt{2} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+105 \sqrt{2} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-210 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+210 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{24576 a^{11/4} b^{9/4} x^{7/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

((d*x)^(7/2)*(a + b*x^2)*(3072*a^(15/4)*b^(1/4)*Sqrt[x] - 4352*a^(11/4)*b^(1/4)*
Sqrt[x]*(a + b*x^2) + 160*a^(7/4)*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 280*a^(3/4)*b^
(1/4)*Sqrt[x]*(a + b*x^2)^3 - 210*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1
/4)*Sqrt[x])/a^(1/4)] + 210*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sq
rt[x])/a^(1/4)] - 105*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x] + 105*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^(11/4)*b^(9/4)*x^(7/2)*((a + b*x^2)^2)^
(5/2))

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Maple [B]  time = 0.03, size = 1146, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/24576*(105*(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+210*(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^6-210*(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+420*(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+840*(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^
6-840*(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+630*(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+1260*(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^6-1260*(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+280*(d*x)^(13/2)*a*b^3+420*(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+840*(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^6-840*(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+1000*(d*x)^(9/2)*a^2*b^2*d^2+105*(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+210*(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^6-210*(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^6-3192*(d*x)^(5/2
)*a^3*b*d^4-840*(d*x)^(1/2)*a^4*d^6)/d^3*(b*x^2+a)/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)^(7/2)/(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 = 0.301979, size = 583, normalized size = 1.04 \[ -\frac{420 \,{\left (a^{2} b^{6} x^{8} + 4 \, a^{3} b^{5} x^{6} + 6 \, a^{4} b^{4} x^{4} + 4 \, a^{5} b^{3} x^{2} + a^{6} b^{2}\right )} \left (-\frac{d^{14}}{a^{11} b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{3} b^{2} \left (-\frac{d^{14}}{a^{11} b^{9}}\right )^{\frac{1}{4}}}{\sqrt{d x} d^{3} + \sqrt{a^{6} b^{4} \sqrt{-\frac{d^{14}}{a^{11} b^{9}}} + d^{7} x}}\right ) - 105 \,{\left (a^{2} b^{6} x^{8} + 4 \, a^{3} b^{5} x^{6} + 6 \, a^{4} b^{4} x^{4} + 4 \, a^{5} b^{3} x^{2} + a^{6} b^{2}\right )} \left (-\frac{d^{14}}{a^{11} b^{9}}\right )^{\frac{1}{4}} \log \left (35 \, a^{3} b^{2} \left (-\frac{d^{14}}{a^{11} b^{9}}\right )^{\frac{1}{4}} + 35 \, \sqrt{d x} d^{3}\right ) + 105 \,{\left (a^{2} b^{6} x^{8} + 4 \, a^{3} b^{5} x^{6} + 6 \, a^{4} b^{4} x^{4} + 4 \, a^{5} b^{3} x^{2} + a^{6} b^{2}\right )} \left (-\frac{d^{14}}{a^{11} b^{9}}\right )^{\frac{1}{4}} \log \left (-35 \, a^{3} b^{2} \left (-\frac{d^{14}}{a^{11} b^{9}}\right )^{\frac{1}{4}} + 35 \, \sqrt{d x} d^{3}\right ) - 4 \,{\left (35 \, b^{3} d^{3} x^{6} + 125 \, a b^{2} d^{3} x^{4} - 399 \, a^{2} b d^{3} x^{2} - 105 \, a^{3} d^{3}\right )} \sqrt{d x}}{12288 \,{\left (a^{2} b^{6} x^{8} + 4 \, a^{3} b^{5} x^{6} + 6 \, a^{4} b^{4} x^{4} + 4 \, a^{5} b^{3} x^{2} + a^{6} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(7/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/12288*(420*(a^2*b^6*x^8 + 4*a^3*b^5*x^6 + 6*a^4*b^4*x^4 + 4*a^5*b^3*x^2 + a^6
*b^2)*(-d^14/(a^11*b^9))^(1/4)*arctan(a^3*b^2*(-d^14/(a^11*b^9))^(1/4)/(sqrt(d*x
)*d^3 + sqrt(a^6*b^4*sqrt(-d^14/(a^11*b^9)) + d^7*x))) - 105*(a^2*b^6*x^8 + 4*a^
3*b^5*x^6 + 6*a^4*b^4*x^4 + 4*a^5*b^3*x^2 + a^6*b^2)*(-d^14/(a^11*b^9))^(1/4)*lo
g(35*a^3*b^2*(-d^14/(a^11*b^9))^(1/4) + 35*sqrt(d*x)*d^3) + 105*(a^2*b^6*x^8 + 4
*a^3*b^5*x^6 + 6*a^4*b^4*x^4 + 4*a^5*b^3*x^2 + a^6*b^2)*(-d^14/(a^11*b^9))^(1/4)
*log(-35*a^3*b^2*(-d^14/(a^11*b^9))^(1/4) + 35*sqrt(d*x)*d^3) - 4*(35*b^3*d^3*x^
6 + 125*a*b^2*d^3*x^4 - 399*a^2*b*d^3*x^2 - 105*a^3*d^3)*sqrt(d*x))/(a^2*b^6*x^8
+ 4*a^3*b^5*x^6 + 6*a^4*b^4*x^4 + 4*a^5*b^3*x^2 + a^6*b^2)

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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)**(7/2)/(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 = 0.292966, size = 556, normalized size = 0.99 \[ \frac{1}{24576} \, d^{2}{\left (\frac{210 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{3} b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{210 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{3} b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{105 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{3} b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{105 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{3} b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{8 \,{\left (35 \, \sqrt{d x} b^{3} d^{9} x^{6} + 125 \, \sqrt{d x} a b^{2} d^{9} x^{4} - 399 \, \sqrt{d x} a^{2} b d^{9} x^{2} - 105 \, \sqrt{d x} a^{3} d^{9}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{2} b^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(7/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="giac")

[Out]

1/24576*d^2*(210*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^3*b^3*sign(b*d^4*x^2 + a*d^4)) + 210
*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^3*b^3*sign(b*d^4*x^2 + a*d^4)) + 105*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^3*b
^3*sign(b*d^4*x^2 + a*d^4)) - 105*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^3*b^3*sign(b*d^4*x^2 + a*d^4)) + 8*
(35*sqrt(d*x)*b^3*d^9*x^6 + 125*sqrt(d*x)*a*b^2*d^9*x^4 - 399*sqrt(d*x)*a^2*b*d^
9*x^2 - 105*sqrt(d*x)*a^3*d^9)/((b*d^2*x^2 + a*d^2)^4*a^2*b^2*sign(b*d^4*x^2 + a
*d^4)))