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

Optimal. Leaf size=52 \[ \frac{\left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac{2 b \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5} \]

[Out]

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

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Rubi [A]  time = 0.0896004, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac{2 b \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.5922, size = 44, normalized size = 0.85 \[ - \frac{2 b \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{35 c^{2} x^{5}} + \frac{\left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{7 c x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0345354, size = 42, normalized size = 0.81 \[ \frac{x \left (b+c x^2\right )^3 \left (5 c x^2-2 b\right )}{35 c^2 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 39, normalized size = 0.8 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -5\,c{x}^{2}+2\,b \right ) }{35\,{c}^{2}{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.705537, size = 61, normalized size = 1.17 \[ \frac{{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{2} + b}}{35 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)/c^2

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Fricas [A]  time = 0.262872, size = 70, normalized size = 1.35 \[ \frac{{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{35 \, c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((b*x**2 + c*x**4)**(3/2), x)

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GIAC/XCAS [A]  time = 0.273338, size = 126, normalized size = 2.42 \[ \frac{2 \, b^{\frac{7}{2}}{\rm sign}\left (x\right )}{35 \, c^{2}} + \frac{\frac{7 \,{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} b{\rm sign}\left (x\right )}{c} + \frac{{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )}{\rm sign}\left (x\right )}{c}}{105 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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