3.273 \(\int \frac{x^9}{\left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=109 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{7/2}}-\frac{15 b \sqrt{b x^2+c x^4}}{8 c^3}+\frac{5 x^2 \sqrt{b x^2+c x^4}}{4 c^2}-\frac{x^6}{c \sqrt{b x^2+c x^4}} \]

[Out]

-(x^6/(c*Sqrt[b*x^2 + c*x^4])) - (15*b*Sqrt[b*x^2 + c*x^4])/(8*c^3) + (5*x^2*Sqr
t[b*x^2 + c*x^4])/(4*c^2) + (15*b^2*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/
(8*c^(7/2))

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Rubi [A]  time = 0.257205, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{7/2}}-\frac{15 b \sqrt{b x^2+c x^4}}{8 c^3}+\frac{5 x^2 \sqrt{b x^2+c x^4}}{4 c^2}-\frac{x^6}{c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(x^6/(c*Sqrt[b*x^2 + c*x^4])) - (15*b*Sqrt[b*x^2 + c*x^4])/(8*c^3) + (5*x^2*Sqr
t[b*x^2 + c*x^4])/(4*c^2) + (15*b^2*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/
(8*c^(7/2))

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Rubi in Sympy [A]  time = 23.4583, size = 99, normalized size = 0.91 \[ \frac{15 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{8 c^{\frac{7}{2}}} - \frac{15 b \sqrt{b x^{2} + c x^{4}}}{8 c^{3}} - \frac{x^{6}}{c \sqrt{b x^{2} + c x^{4}}} + \frac{5 x^{2} \sqrt{b x^{2} + c x^{4}}}{4 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(c*x**4+b*x**2)**(3/2),x)

[Out]

15*b**2*atanh(sqrt(c)*x**2/sqrt(b*x**2 + c*x**4))/(8*c**(7/2)) - 15*b*sqrt(b*x**
2 + c*x**4)/(8*c**3) - x**6/(c*sqrt(b*x**2 + c*x**4)) + 5*x**2*sqrt(b*x**2 + c*x
**4)/(4*c**2)

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Mathematica [A]  time = 0.0758357, size = 92, normalized size = 0.84 \[ \frac{x \left (\sqrt{c} x \left (-15 b^2-5 b c x^2+2 c^2 x^4\right )+15 b^2 \sqrt{b+c x^2} \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )\right )}{8 c^{7/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(b*x^2 + c*x^4)^(3/2),x]

[Out]

(x*(Sqrt[c]*x*(-15*b^2 - 5*b*c*x^2 + 2*c^2*x^4) + 15*b^2*Sqrt[b + c*x^2]*Log[c*x
+ Sqrt[c]*Sqrt[b + c*x^2]]))/(8*c^(7/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.015, size = 87, normalized size = 0.8 \[{\frac{{x}^{3} \left ( c{x}^{2}+b \right ) }{8} \left ( 2\,{x}^{5}{c}^{7/2}-5\,{c}^{5/2}{x}^{3}b-15\,{c}^{3/2}x{b}^{2}+15\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) \sqrt{c{x}^{2}+b}{b}^{2}c \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(c*x^4+b*x^2)^(3/2),x)

[Out]

1/8*x^3*(c*x^2+b)*(2*x^5*c^(7/2)-5*c^(5/2)*x^3*b-15*c^(3/2)*x*b^2+15*ln(x*c^(1/2
)+(c*x^2+b)^(1/2))*(c*x^2+b)^(1/2)*b^2*c)/(c*x^4+b*x^2)^(3/2)/c^(9/2)

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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(x^9/(c*x^4 + b*x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281126, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} c x^{2} + b^{3}\right )} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) + 2 \,{\left (2 \, c^{3} x^{4} - 5 \, b c^{2} x^{2} - 15 \, b^{2} c\right )} \sqrt{c x^{4} + b x^{2}}}{16 \,{\left (c^{5} x^{2} + b c^{4}\right )}}, -\frac{15 \,{\left (b^{2} c x^{2} + b^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) -{\left (2 \, c^{3} x^{4} - 5 \, b c^{2} x^{2} - 15 \, b^{2} c\right )} \sqrt{c x^{4} + b x^{2}}}{8 \,{\left (c^{5} x^{2} + b c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + b*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(15*(b^2*c*x^2 + b^3)*sqrt(c)*log(-(2*c*x^2 + b)*sqrt(c) - 2*sqrt(c*x^4 +
b*x^2)*c) + 2*(2*c^3*x^4 - 5*b*c^2*x^2 - 15*b^2*c)*sqrt(c*x^4 + b*x^2))/(c^5*x^2
+ b*c^4), -1/8*(15*(b^2*c*x^2 + b^3)*sqrt(-c)*arctan(sqrt(-c)*x^2/sqrt(c*x^4 +
b*x^2)) - (2*c^3*x^4 - 5*b*c^2*x^2 - 15*b^2*c)*sqrt(c*x^4 + b*x^2))/(c^5*x^2 + b
*c^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral(x**9/(x**2*(b + c*x**2))**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + b*x^2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^9/(c*x^4 + b*x^2)^(3/2), x)