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

Optimal. Leaf size=80 \[ \frac{8 b^2 \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac{4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac{\left (b x^2+c x^4\right )^{5/2}}{9 c x} \]

[Out]

(8*b^2*(b*x^2 + c*x^4)^(5/2))/(315*c^3*x^5) - (4*b*(b*x^2 + c*x^4)^(5/2))/(63*c^
2*x^3) + (b*x^2 + c*x^4)^(5/2)/(9*c*x)

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Rubi [A]  time = 0.173749, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{8 b^2 \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac{4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac{\left (b x^2+c x^4\right )^{5/2}}{9 c x} \]

Antiderivative was successfully verified.

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

[Out]

(8*b^2*(b*x^2 + c*x^4)^(5/2))/(315*c^3*x^5) - (4*b*(b*x^2 + c*x^4)^(5/2))/(63*c^
2*x^3) + (b*x^2 + c*x^4)^(5/2)/(9*c*x)

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Rubi in Sympy [A]  time = 20.3437, size = 70, normalized size = 0.88 \[ \frac{8 b^{2} \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{315 c^{3} x^{5}} - \frac{4 b \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{63 c^{2} x^{3}} + \frac{\left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{9 c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*b**2*(b*x**2 + c*x**4)**(5/2)/(315*c**3*x**5) - 4*b*(b*x**2 + c*x**4)**(5/2)/(
63*c**2*x**3) + (b*x**2 + c*x**4)**(5/2)/(9*c*x)

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Mathematica [A]  time = 0.0399144, size = 53, normalized size = 0.66 \[ \frac{x \left (b+c x^2\right )^3 \left (8 b^2-20 b c x^2+35 c^2 x^4\right )}{315 c^3 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b + c*x^2)^3*(8*b^2 - 20*b*c*x^2 + 35*c^2*x^4))/(315*c^3*Sqrt[x^2*(b + c*x^2
)])

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Maple [A]  time = 0.008, size = 50, normalized size = 0.6 \[{\frac{ \left ( c{x}^{2}+b \right ) \left ( 35\,{c}^{2}{x}^{4}-20\,bc{x}^{2}+8\,{b}^{2} \right ) }{315\,{c}^{3}{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^4+b*x^2)^(3/2),x)

[Out]

1/315*(c*x^2+b)*(35*c^2*x^4-20*b*c*x^2+8*b^2)*(c*x^4+b*x^2)^(3/2)/c^3/x^3

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Maxima [A]  time = 0.695611, size = 77, normalized size = 0.96 \[ \frac{{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt{c x^{2} + b}}{315 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(35*c^4*x^8 + 50*b*c^3*x^6 + 3*b^2*c^2*x^4 - 4*b^3*c*x^2 + 8*b^4)*sqrt(c*x
^2 + b)/c^3

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Fricas [A]  time = 0.264651, size = 86, normalized size = 1.08 \[ \frac{{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(35*c^4*x^8 + 50*b*c^3*x^6 + 3*b^2*c^2*x^4 - 4*b^3*c*x^2 + 8*b^4)*sqrt(c*x
^4 + b*x^2)/(c^3*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272949, size = 163, normalized size = 2.04 \[ -\frac{8 \, b^{\frac{9}{2}}{\rm sign}\left (x\right )}{315 \, c^{3}} + \frac{\frac{3 \,{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )} b{\rm sign}\left (x\right )}{c^{2}} + \frac{{\left (35 \,{\left (c x^{2} + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{3}\right )}{\rm sign}\left (x\right )}{c^{2}}}{315 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/315*b^(9/2)*sign(x)/c^3 + 1/315*(3*(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^2 + (35*(c*x^2 + b)^(9/2) - 135*(c*
x^2 + b)^(7/2)*b + 189*(c*x^2 + b)^(5/2)*b^2 - 105*(c*x^2 + b)^(3/2)*b^3)*sign(x
)/c^2)/c