3.495 \(\int \frac{1}{x^{7/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=731 \[ \frac{b^{13/4} (9 b c-17 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^3}-\frac{b^{13/4} (9 b c-17 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^3}-\frac{b^{13/4} (9 b c-17 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} (b c-a d)^3}+\frac{b^{13/4} (9 b c-17 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{13/4} (b c-a d)^3}-\frac{9 a^2 d^2-8 a b c d+9 b^2 c^2}{10 a^2 c^2 x^{5/2} (b c-a d)^2}+\frac{(a d+b c) \left (9 a^2 d^2-17 a b c d+9 b^2 c^2\right )}{2 a^3 c^3 \sqrt{x} (b c-a d)^2}+\frac{d^{13/4} (17 b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{13/4} (17 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{13/4} (17 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^3}+\frac{d^{13/4} (17 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^3}+\frac{b}{2 a x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d (a d+b c)}{2 a c x^{5/2} \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

-(9*b^2*c^2 - 8*a*b*c*d + 9*a^2*d^2)/(10*a^2*c^2*(b*c - a*d)^2*x^(5/2)) + ((b*c
+ a*d)*(9*b^2*c^2 - 17*a*b*c*d + 9*a^2*d^2))/(2*a^3*c^3*(b*c - a*d)^2*Sqrt[x]) +
(d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)) + b/(2*a*(b*c - a*d)*
x^(5/2)*(a + b*x^2)*(c + d*x^2)) - (b^(13/4)*(9*b*c - 17*a*d)*ArcTan[1 - (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) + (b^(13/4)*(9*b
*c - 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*
(b*c - a*d)^3) - (d^(13/4)*(17*b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])
/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) + (d^(13/4)*(17*b*c - 9*a*d)*ArcTa
n[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) + (
b^(13/4)*(9*b*c - 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b
]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) - (b^(13/4)*(9*b*c - 17*a*d)*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d
)^3) + (d^(13/4)*(17*b*c - 9*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
+ Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) - (d^(13/4)*(17*b*c - 9*a*d)*Lo
g[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(b
*c - a*d)^3)

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Rubi [A]  time = 2.84229, antiderivative size = 731, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ \frac{b^{13/4} (9 b c-17 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^3}-\frac{b^{13/4} (9 b c-17 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^3}-\frac{b^{13/4} (9 b c-17 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} (b c-a d)^3}+\frac{b^{13/4} (9 b c-17 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{13/4} (b c-a d)^3}-\frac{9 a^2 d^2-8 a b c d+9 b^2 c^2}{10 a^2 c^2 x^{5/2} (b c-a d)^2}+\frac{(a d+b c) \left (9 a^2 d^2-17 a b c d+9 b^2 c^2\right )}{2 a^3 c^3 \sqrt{x} (b c-a d)^2}+\frac{d^{13/4} (17 b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{13/4} (17 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{13/4} (17 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^3}+\frac{d^{13/4} (17 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^3}+\frac{b}{2 a x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d (a d+b c)}{2 a c x^{5/2} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

-(9*b^2*c^2 - 8*a*b*c*d + 9*a^2*d^2)/(10*a^2*c^2*(b*c - a*d)^2*x^(5/2)) + ((b*c
+ a*d)*(9*b^2*c^2 - 17*a*b*c*d + 9*a^2*d^2))/(2*a^3*c^3*(b*c - a*d)^2*Sqrt[x]) +
(d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)) + b/(2*a*(b*c - a*d)*
x^(5/2)*(a + b*x^2)*(c + d*x^2)) - (b^(13/4)*(9*b*c - 17*a*d)*ArcTan[1 - (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) + (b^(13/4)*(9*b
*c - 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*
(b*c - a*d)^3) - (d^(13/4)*(17*b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])
/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) + (d^(13/4)*(17*b*c - 9*a*d)*ArcTa
n[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) + (
b^(13/4)*(9*b*c - 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b
]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) - (b^(13/4)*(9*b*c - 17*a*d)*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d
)^3) + (d^(13/4)*(17*b*c - 9*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
+ Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) - (d^(13/4)*(17*b*c - 9*a*d)*Lo
g[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(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**(7/2)/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 2.7427, size = 630, normalized size = 0.86 \[ \frac{1}{80} \left (\frac{5 \sqrt{2} b^{13/4} (17 a d-9 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4} (a d-b c)^3}+\frac{5 \sqrt{2} b^{13/4} (17 a d-9 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4} (b c-a d)^3}+\frac{10 \sqrt{2} b^{13/4} (17 a d-9 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{13/4} (b c-a d)^3}+\frac{10 \sqrt{2} b^{13/4} (17 a d-9 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{13/4} (a d-b c)^3}+\frac{40 b^4 x^{3/2}}{a^3 \left (a+b x^2\right ) (b c-a d)^2}+\frac{320 (a d+b c)}{a^3 c^3 \sqrt{x}}-\frac{32}{a^2 c^2 x^{5/2}}+\frac{5 \sqrt{2} d^{13/4} (17 b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (b c-a d)^3}+\frac{5 \sqrt{2} d^{13/4} (17 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (a d-b c)^3}+\frac{10 \sqrt{2} d^{13/4} (9 a d-17 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{13/4} (b c-a d)^3}+\frac{10 \sqrt{2} d^{13/4} (17 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{13/4} (b c-a d)^3}+\frac{40 d^4 x^{3/2}}{c^3 \left (c+d x^2\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

(-32/(a^2*c^2*x^(5/2)) + (320*(b*c + a*d))/(a^3*c^3*Sqrt[x]) + (40*b^4*x^(3/2))/
(a^3*(b*c - a*d)^2*(a + b*x^2)) + (40*d^4*x^(3/2))/(c^3*(b*c - a*d)^2*(c + d*x^2
)) + (10*Sqrt[2]*b^(13/4)*(-9*b*c + 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])
/a^(1/4)])/(a^(13/4)*(b*c - a*d)^3) + (10*Sqrt[2]*b^(13/4)*(-9*b*c + 17*a*d)*Arc
Tan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(13/4)*(-(b*c) + a*d)^3) + (10*Sq
rt[2]*d^(13/4)*(-17*b*c + 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/
(c^(13/4)*(b*c - a*d)^3) + (10*Sqrt[2]*d^(13/4)*(17*b*c - 9*a*d)*ArcTan[1 + (Sqr
t[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(13/4)*(b*c - a*d)^3) + (5*Sqrt[2]*b^(13/4)*(
-9*b*c + 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^
(13/4)*(-(b*c) + a*d)^3) + (5*Sqrt[2]*b^(13/4)*(-9*b*c + 17*a*d)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(13/4)*(b*c - a*d)^3) + (5*Sqrt[
2]*d^(13/4)*(17*b*c - 9*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr
t[d]*x])/(c^(13/4)*(b*c - a*d)^3) + (5*Sqrt[2]*d^(13/4)*(17*b*c - 9*a*d)*Log[Sqr
t[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(13/4)*(-(b*c) + a*d)^3)
)/80

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Maple [A]  time = 0.041, size = 849, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

1/2*d^5/c^3/(a*d-b*c)^3*x^(3/2)/(d*x^2+c)*a-1/2*d^4/c^2/(a*d-b*c)^3*x^(3/2)/(d*x
^2+c)*b+9/16*d^4/c^3/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*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)))+9/8*d^4/c^3/(
a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+9/8*d^4/c
^3/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-17/16
*d^3/c^2/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*b*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)))-17/8*d^3/c^2/(a*d-b*c)^3/
(c/d)^(1/4)*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-17/8*d^3/c^2/(a*d-b*
c)^3/(c/d)^(1/4)*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/5/a^2/c^2/x^(
5/2)+4/x^(1/2)/a^2/c^3*d+4/x^(1/2)/a^3/c^2*b+1/2*b^4/a^2/(a*d-b*c)^3*x^(3/2)/(b*
x^2+a)*d-1/2*b^5/a^3/(a*d-b*c)^3*x^(3/2)/(b*x^2+a)*c+17/16*b^3/a^2/(a*d-b*c)^3/(
a/b)^(1/4)*2^(1/2)*d*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)))+17/8*b^3/a^2/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*d*
arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+17/8*b^3/a^2/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/
2)*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-9/16*b^4/a^3/(a*d-b*c)^3/(a/b)^(1/4)*
2^(1/2)*c*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)))-9/8*b^4/a^3/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*c*arctan(2^(1/
2)/(a/b)^(1/4)*x^(1/2)+1)-9/8*b^4/a^3/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*c*arctan(2
^(1/2)/(a/b)^(1/4)*x^(1/2)-1)

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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^(7/2)),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^(7/2)),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**(7/2)/(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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{2} x^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2*x^(7/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2*x^(7/2)), x)