3.494 \(\int \frac{1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=676 \[ \frac{b^{11/4} (7 b c-15 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} (b c-a d)^3}-\frac{b^{11/4} (7 b c-15 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} (b c-a d)^3}+\frac{b^{11/4} (7 b c-15 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} (b c-a d)^3}-\frac{b^{11/4} (7 b c-15 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4} (b c-a d)^3}-\frac{7 a^2 d^2-8 a b c d+7 b^2 c^2}{6 a^2 c^2 x^{3/2} (b c-a d)^2}+\frac{d^{11/4} (15 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{11/4} (15 b c-7 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} (b c-a d)^3}+\frac{d^{11/4} (15 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{11/4} (15 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{11/4} (b c-a d)^3}+\frac{b}{2 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d (a d+b c)}{2 a c x^{3/2} \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

-(7*b^2*c^2 - 8*a*b*c*d + 7*a^2*d^2)/(6*a^2*c^2*(b*c - a*d)^2*x^(3/2)) + (d*(b*c
+ a*d))/(2*a*c*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)) + b/(2*a*(b*c - a*d)*x^(3/2)*
(a + b*x^2)*(c + d*x^2)) + (b^(11/4)*(7*b*c - 15*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) - (b^(11/4)*(7*b*c - 15*
a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*(b*c - a
*d)^3) + (d^(11/4)*(15*b*c - 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)
])/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) - (d^(11/4)*(15*b*c - 7*a*d)*ArcTan[1 + (S
qrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) + (b^(11/4)
*(7*b*c - 15*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8
*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) - (b^(11/4)*(7*b*c - 15*a*d)*Log[Sqrt[a] + Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) + (
d^(11/4)*(15*b*c - 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d
]*x])/(8*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) - (d^(11/4)*(15*b*c - 7*a*d)*Log[Sqrt[c
] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(b*c - a*d
)^3)

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Rubi [A]  time = 2.177, antiderivative size = 676, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ \frac{b^{11/4} (7 b c-15 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} (b c-a d)^3}-\frac{b^{11/4} (7 b c-15 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} (b c-a d)^3}+\frac{b^{11/4} (7 b c-15 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} (b c-a d)^3}-\frac{b^{11/4} (7 b c-15 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4} (b c-a d)^3}-\frac{7 a^2 d^2-8 a b c d+7 b^2 c^2}{6 a^2 c^2 x^{3/2} (b c-a d)^2}+\frac{d^{11/4} (15 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{11/4} (15 b c-7 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} (b c-a d)^3}+\frac{d^{11/4} (15 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{11/4} (15 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{11/4} (b c-a d)^3}+\frac{b}{2 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d (a d+b c)}{2 a c x^{3/2} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

-(7*b^2*c^2 - 8*a*b*c*d + 7*a^2*d^2)/(6*a^2*c^2*(b*c - a*d)^2*x^(3/2)) + (d*(b*c
+ a*d))/(2*a*c*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)) + b/(2*a*(b*c - a*d)*x^(3/2)*
(a + b*x^2)*(c + d*x^2)) + (b^(11/4)*(7*b*c - 15*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) - (b^(11/4)*(7*b*c - 15*
a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*(b*c - a
*d)^3) + (d^(11/4)*(15*b*c - 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)
])/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) - (d^(11/4)*(15*b*c - 7*a*d)*ArcTan[1 + (S
qrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) + (b^(11/4)
*(7*b*c - 15*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8
*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) - (b^(11/4)*(7*b*c - 15*a*d)*Log[Sqrt[a] + Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c - a*d)^3) + (
d^(11/4)*(15*b*c - 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d
]*x])/(8*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) - (d^(11/4)*(15*b*c - 7*a*d)*Log[Sqrt[c
] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(b*c - a*d
)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 3.75799, size = 610, normalized size = 0.9 \[ \frac{1}{48} \left (\frac{3 \sqrt{2} b^{11/4} (15 a d-7 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} (a d-b c)^3}+\frac{3 \sqrt{2} b^{11/4} (15 a d-7 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} (b c-a d)^3}+\frac{6 \sqrt{2} b^{11/4} (15 a d-7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{11/4} (a d-b c)^3}+\frac{6 \sqrt{2} b^{11/4} (15 a d-7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{11/4} (b c-a d)^3}-\frac{24 b^3 \sqrt{x}}{a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{32}{a^2 c^2 x^{3/2}}+\frac{3 \sqrt{2} d^{11/4} (15 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{11/4} (b c-a d)^3}+\frac{3 \sqrt{2} d^{11/4} (15 b c-7 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{11/4} (a d-b c)^3}+\frac{6 \sqrt{2} d^{11/4} (15 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{11/4} (b c-a d)^3}+\frac{6 \sqrt{2} d^{11/4} (7 a d-15 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{11/4} (b c-a d)^3}-\frac{24 d^3 \sqrt{x}}{c^2 \left (c+d x^2\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

(-32/(a^2*c^2*x^(3/2)) - (24*b^3*Sqrt[x])/(a^2*(b*c - a*d)^2*(a + b*x^2)) - (24*
d^3*Sqrt[x])/(c^2*(b*c - a*d)^2*(c + d*x^2)) + (6*Sqrt[2]*b^(11/4)*(-7*b*c + 15*
a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*(-(b*c) + a*d)^3)
+ (6*Sqrt[2]*b^(11/4)*(-7*b*c + 15*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(
1/4)])/(a^(11/4)*(b*c - a*d)^3) + (6*Sqrt[2]*d^(11/4)*(15*b*c - 7*a*d)*ArcTan[1
- (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(11/4)*(b*c - a*d)^3) + (6*Sqrt[2]*d^(1
1/4)*(-15*b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(11/4)*
(b*c - a*d)^3) + (3*Sqrt[2]*b^(11/4)*(-7*b*c + 15*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(
1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*(-(b*c) + a*d)^3) + (3*Sqrt[2]*b^(1
1/4)*(-7*b*c + 15*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/(a^(11/4)*(b*c - a*d)^3) + (3*Sqrt[2]*d^(11/4)*(15*b*c - 7*a*d)*Log[Sqrt[c] -
Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4)*(b*c - a*d)^3) + (3*Sqr
t[2]*d^(11/4)*(15*b*c - 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + S
qrt[d]*x])/(c^(11/4)*(-(b*c) + a*d)^3))/48

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Maple [A]  time = 0.034, size = 825, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

-1/2*d^4/c^2/(a*d-b*c)^3*x^(1/2)/(d*x^2+c)*a+1/2*d^3/c/(a*d-b*c)^3*x^(1/2)/(d*x^
2+c)*b-7/8*d^4/c^3/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^
(1/2)+1)*a+15/8*d^3/c^2/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/
4)*x^(1/2)+1)*b-7/8*d^4/c^3/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)
^(1/4)*x^(1/2)-1)*a+15/8*d^3/c^2/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(c/d)^(1/4)*x^(1/2)-1)*b-7/16*d^4/c^3/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d
)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))
)*a+15/16*d^3/c^2/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d)^(1/4)*x^(1/2)*2^(1
/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))*b-2/3/a^2/c^2/x^(3
/2)-1/2*b^3/a/(a*d-b*c)^3*x^(1/2)/(b*x^2+a)*d+1/2*b^4/a^2/(a*d-b*c)^3*x^(1/2)/(b
*x^2+a)*c-15/8*b^3/a^2/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4
)*x^(1/2)+1)*d+7/8*b^4/a^3/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^
(1/4)*x^(1/2)+1)*c-15/8*b^3/a^2/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
a/b)^(1/4)*x^(1/2)-1)*d+7/8*b^4/a^3/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/
2)/(a/b)^(1/4)*x^(1/2)-1)*c-15/16*b^3/a^2/(a*d-b*c)^3*(a/b)^(1/4)*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)))*d+7/16*b^4/a^3/(a*d-b*c)^3*(a/b)^(1/4)*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)))*c

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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(1/((b*x^2 + a)^2*(d*x^2 + c)^2*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2*x^(5/2)),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{2} x^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2*x^(5/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2*x^(5/2)), x)