3.147 \(\int \frac{x^5 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=112 \[ -\frac{(3 b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{5/2}}+\frac{\sqrt{b x^2+c x^4} (3 b B-2 A c)}{2 b c^2}-\frac{x^4 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

[Out]

-(((b*B - A*c)*x^4)/(b*c*Sqrt[b*x^2 + c*x^4])) + ((3*b*B - 2*A*c)*Sqrt[b*x^2 + c
*x^4])/(2*b*c^2) - ((3*b*B - 2*A*c)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/
(2*c^(5/2))

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Rubi [A]  time = 0.453996, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{(3 b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{5/2}}+\frac{\sqrt{b x^2+c x^4} (3 b B-2 A c)}{2 b c^2}-\frac{x^4 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(((b*B - A*c)*x^4)/(b*c*Sqrt[b*x^2 + c*x^4])) + ((3*b*B - 2*A*c)*Sqrt[b*x^2 + c
*x^4])/(2*b*c^2) - ((3*b*B - 2*A*c)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/
(2*c^(5/2))

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Rubi in Sympy [A]  time = 28.4644, size = 94, normalized size = 0.84 \[ \frac{\left (A c - \frac{3 B b}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{c^{\frac{5}{2}}} + \frac{x^{4} \left (A c - B b\right )}{b c \sqrt{b x^{2} + c x^{4}}} - \frac{\left (A c - \frac{3 B b}{2}\right ) \sqrt{b x^{2} + c x^{4}}}{b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(B*x**2+A)/(c*x**4+b*x**2)**(3/2),x)

[Out]

(A*c - 3*B*b/2)*atanh(sqrt(c)*x**2/sqrt(b*x**2 + c*x**4))/c**(5/2) + x**4*(A*c -
B*b)/(b*c*sqrt(b*x**2 + c*x**4)) - (A*c - 3*B*b/2)*sqrt(b*x**2 + c*x**4)/(b*c**
2)

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Mathematica [A]  time = 0.105996, size = 91, normalized size = 0.81 \[ \frac{x \left (\sqrt{c} x \left (-2 A c+3 b B+B c x^2\right )+\sqrt{b+c x^2} (2 A c-3 b B) \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )\right )}{2 c^{5/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]

[Out]

(x*(Sqrt[c]*x*(3*b*B - 2*A*c + B*c*x^2) + (-3*b*B + 2*A*c)*Sqrt[b + c*x^2]*Log[c
*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(2*c^(5/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.015, size = 117, normalized size = 1. \[ -{\frac{ \left ( c{x}^{2}+b \right ){x}^{3}}{2} \left ( -B{x}^{3}{c}^{{\frac{7}{2}}}+2\,Ax{c}^{7/2}-3\,xBb{c}^{5/2}-2\,A\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){c}^{3}\sqrt{c{x}^{2}+b}+3\,Bb\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){c}^{2}\sqrt{c{x}^{2}+b} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(B*x^2+A)/(c*x^4+b*x^2)^(3/2),x)

[Out]

-1/2*x^3*(c*x^2+b)*(-B*x^3*c^(7/2)+2*A*x*c^(7/2)-3*x*B*b*c^(5/2)-2*A*ln(c^(1/2)*
x+(c*x^2+b)^(1/2))*c^3*(c*x^2+b)^(1/2)+3*B*b*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*c^2*(
c*x^2+b)^(1/2))/(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((B*x^2 + A)*x^5/(c*x^4 + b*x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255283, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, B b^{2} - 2 \, A b c +{\left (3 \, B b c - 2 \, A c^{2}\right )} x^{2}\right )} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) - 2 \,{\left (B c^{2} x^{2} + 3 \, B b c - 2 \, A c^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{4 \,{\left (c^{4} x^{2} + b c^{3}\right )}}, \frac{{\left (3 \, B b^{2} - 2 \, A b c +{\left (3 \, B b c - 2 \, A c^{2}\right )} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (B c^{2} x^{2} + 3 \, B b c - 2 \, A c^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{2 \,{\left (c^{4} x^{2} + b c^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^5/(c*x^4 + b*x^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/4*((3*B*b^2 - 2*A*b*c + (3*B*b*c - 2*A*c^2)*x^2)*sqrt(c)*log(-(2*c*x^2 + b)*
sqrt(c) - 2*sqrt(c*x^4 + b*x^2)*c) - 2*(B*c^2*x^2 + 3*B*b*c - 2*A*c^2)*sqrt(c*x^
4 + b*x^2))/(c^4*x^2 + b*c^3), 1/2*((3*B*b^2 - 2*A*b*c + (3*B*b*c - 2*A*c^2)*x^2
)*sqrt(-c)*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + b*x^2)) + (B*c^2*x^2 + 3*B*b*c - 2*A
*c^2)*sqrt(c*x^4 + b*x^2))/(c^4*x^2 + b*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5} \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(B*x**2+A)/(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral(x**5*(A + B*x**2)/(x**2*(b + c*x**2))**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{5}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^5/(c*x^4 + b*x^2)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^5/(c*x^4 + b*x^2)^(3/2), x)