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

Optimal. Leaf size=61 \[ \frac{B \left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac{\left (b x^2+c x^4\right )^{5/2} (2 b B-7 A c)}{35 c^2 x^5} \]

[Out]

-((2*b*B - 7*A*c)*(b*x^2 + c*x^4)^(5/2))/(35*c^2*x^5) + (B*(b*x^2 + c*x^4)^(5/2)
)/(7*c*x^3)

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Rubi [A]  time = 0.244465, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{B \left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac{\left (b x^2+c x^4\right )^{5/2} (2 b B-7 A c)}{35 c^2 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-((2*b*B - 7*A*c)*(b*x^2 + c*x^4)^(5/2))/(35*c^2*x^5) + (B*(b*x^2 + c*x^4)^(5/2)
)/(7*c*x^3)

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Rubi in Sympy [A]  time = 18.3493, size = 53, normalized size = 0.87 \[ \frac{B \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{7 c x^{3}} + \frac{\left (7 A c - 2 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{35 c^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(b*x**2 + c*x**4)**(5/2)/(7*c*x**3) + (7*A*c - 2*B*b)*(b*x**2 + c*x**4)**(5/2)
/(35*c**2*x**5)

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Mathematica [A]  time = 0.0533392, size = 48, normalized size = 0.79 \[ \frac{x \left (b+c x^2\right )^3 \left (7 A c-2 b B+5 B c x^2\right )}{35 c^2 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b + c*x^2)^3*(-2*b*B + 7*A*c + 5*B*c*x^2))/(35*c^2*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.006, size = 45, normalized size = 0.7 \[{\frac{ \left ( c{x}^{2}+b \right ) \left ( 5\,Bc{x}^{2}+7\,Ac-2\,Bb \right ) }{35\,{c}^{2}{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35*(c*x^2+b)*(5*B*c*x^2+7*A*c-2*B*b)*(c*x^4+b*x^2)^(3/2)/c^2/x^3

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Maxima [A]  time = 1.40318, size = 108, normalized size = 1.77 \[ \frac{{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt{c x^{2} + b} A}{5 \, c} + \frac{{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{2} + b} B}{35 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(c^2*x^4 + 2*b*c*x^2 + b^2)*sqrt(c*x^2 + b)*A/c + 1/35*(5*c^3*x^6 + 8*b*c^2*
x^4 + b^2*c*x^2 - 2*b^3)*sqrt(c*x^2 + b)*B/c^2

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Fricas [A]  time = 0.227219, size = 108, normalized size = 1.77 \[ \frac{{\left (5 \, B c^{3} x^{6} +{\left (8 \, B b c^{2} + 7 \, A c^{3}\right )} x^{4} - 2 \, B b^{3} + 7 \, A b^{2} c +{\left (B b^{2} c + 14 \, A b c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{35 \, c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35*(5*B*c^3*x^6 + (8*B*b*c^2 + 7*A*c^3)*x^4 - 2*B*b^3 + 7*A*b^2*c + (B*b^2*c +
14*A*b*c^2)*x^2)*sqrt(c*x^4 + b*x^2)/(c^2*x)

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

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**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214302, size = 203, normalized size = 3.33 \[ \frac{35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} A b{\rm sign}\left (x\right ) + 7 \,{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} A{\rm sign}\left (x\right ) + \frac{7 \,{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} B b{\rm sign}\left (x\right )}{c} + \frac{{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )} B{\rm sign}\left (x\right )}{c}}{105 \, c} + \frac{{\left (2 \, B b^{\frac{7}{2}} - 7 \, A b^{\frac{5}{2}} c\right )}{\rm sign}\left (x\right )}{35 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(35*(c*x^2 + b)^(3/2)*A*b*sign(x) + 7*(3*(c*x^2 + b)^(5/2) - 5*(c*x^2 + b)
^(3/2)*b)*A*sign(x) + 7*(3*(c*x^2 + b)^(5/2) - 5*(c*x^2 + b)^(3/2)*b)*B*b*sign(x
)/c + (15*(c*x^2 + b)^(7/2) - 42*(c*x^2 + b)^(5/2)*b + 35*(c*x^2 + b)^(3/2)*b^2)
*B*sign(x)/c)/c + 1/35*(2*B*b^(7/2) - 7*A*b^(5/2)*c)*sign(x)/c^2