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

Optimal. Leaf size=101 \[ \frac{4 c \left (b+2 c x^2\right ) (5 b B-6 A c)}{15 b^4 \sqrt{b x^2+c x^4}}-\frac{5 b B-6 A c}{15 b^2 x^2 \sqrt{b x^2+c x^4}}-\frac{A}{5 b x^4 \sqrt{b x^2+c x^4}} \]

[Out]

-A/(5*b*x^4*Sqrt[b*x^2 + c*x^4]) - (5*b*B - 6*A*c)/(15*b^2*x^2*Sqrt[b*x^2 + c*x^
4]) + (4*c*(5*b*B - 6*A*c)*(b + 2*c*x^2))/(15*b^4*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.417361, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{4 c \left (b+2 c x^2\right ) (5 b B-6 A c)}{15 b^4 \sqrt{b x^2+c x^4}}-\frac{5 b B-6 A c}{15 b^2 x^2 \sqrt{b x^2+c x^4}}-\frac{A}{5 b x^4 \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(5*b*x^4*Sqrt[b*x^2 + c*x^4]) - (5*b*B - 6*A*c)/(15*b^2*x^2*Sqrt[b*x^2 + c*x^
4]) + (4*c*(5*b*B - 6*A*c)*(b + 2*c*x^2))/(15*b^4*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 23.8695, size = 95, normalized size = 0.94 \[ - \frac{A}{5 b x^{4} \sqrt{b x^{2} + c x^{4}}} + \frac{6 A c - 5 B b}{15 b^{2} x^{2} \sqrt{b x^{2} + c x^{4}}} - \frac{2 c \left (2 b + 4 c x^{2}\right ) \left (6 A c - 5 B b\right )}{15 b^{4} \sqrt{b x^{2} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(5*b*x**4*sqrt(b*x**2 + c*x**4)) + (6*A*c - 5*B*b)/(15*b**2*x**2*sqrt(b*x**2
+ c*x**4)) - 2*c*(2*b + 4*c*x**2)*(6*A*c - 5*B*b)/(15*b**4*sqrt(b*x**2 + c*x**4)
)

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Mathematica [A]  time = 0.0991829, size = 85, normalized size = 0.84 \[ \frac{-3 A \left (b^3-2 b^2 c x^2+8 b c^2 x^4+16 c^3 x^6\right )-5 b B x^2 \left (b^2-4 b c x^2-8 c^2 x^4\right )}{15 b^4 x^4 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*b*B*x^2*(b^2 - 4*b*c*x^2 - 8*c^2*x^4) - 3*A*(b^3 - 2*b^2*c*x^2 + 8*b*c^2*x^4
+ 16*c^3*x^6))/(15*b^4*x^4*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.009, size = 94, normalized size = 0.9 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 48\,A{c}^{3}{x}^{6}-40\,B{x}^{6}b{c}^{2}+24\,Ab{c}^{2}{x}^{4}-20\,B{x}^{4}{b}^{2}c-6\,A{b}^{2}c{x}^{2}+5\,B{x}^{2}{b}^{3}+3\,A{b}^{3} \right ) }{15\,{x}^{2}{b}^{4}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(c*x^2+b)*(48*A*c^3*x^6-40*B*b*c^2*x^6+24*A*b*c^2*x^4-20*B*b^2*c*x^4-6*A*b
^2*c*x^2+5*B*b^3*x^2+3*A*b^3)/x^2/b^4/(c*x^4+b*x^2)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246311, size = 132, normalized size = 1.31 \[ \frac{{\left (8 \,{\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} x^{6} + 4 \,{\left (5 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{4} - 3 \, A b^{3} -{\left (5 \, B b^{3} - 6 \, A b^{2} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \,{\left (b^{4} c x^{8} + b^{5} x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(8*(5*B*b*c^2 - 6*A*c^3)*x^6 + 4*(5*B*b^2*c - 6*A*b*c^2)*x^4 - 3*A*b^3 - (5
*B*b^3 - 6*A*b^2*c)*x^2)*sqrt(c*x^4 + b*x^2)/(b^4*c*x^8 + b^5*x^6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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