3.231 \(\int x^4 \sqrt{b x^2+c x^4} \, dx\)

Optimal. Leaf size=78 \[ \frac{8 b^2 \left (b x^2+c x^4\right )^{3/2}}{105 c^3 x^3}-\frac{4 b \left (b x^2+c x^4\right )^{3/2}}{35 c^2 x}+\frac{x \left (b x^2+c x^4\right )^{3/2}}{7 c} \]

[Out]

(8*b^2*(b*x^2 + c*x^4)^(3/2))/(105*c^3*x^3) - (4*b*(b*x^2 + c*x^4)^(3/2))/(35*c^
2*x) + (x*(b*x^2 + c*x^4)^(3/2))/(7*c)

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Rubi [A]  time = 0.168645, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{8 b^2 \left (b x^2+c x^4\right )^{3/2}}{105 c^3 x^3}-\frac{4 b \left (b x^2+c x^4\right )^{3/2}}{35 c^2 x}+\frac{x \left (b x^2+c x^4\right )^{3/2}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[x^4*Sqrt[b*x^2 + c*x^4],x]

[Out]

(8*b^2*(b*x^2 + c*x^4)^(3/2))/(105*c^3*x^3) - (4*b*(b*x^2 + c*x^4)^(3/2))/(35*c^
2*x) + (x*(b*x^2 + c*x^4)^(3/2))/(7*c)

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Rubi in Sympy [A]  time = 20.3978, size = 68, normalized size = 0.87 \[ \frac{8 b^{2} \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{105 c^{3} x^{3}} - \frac{4 b \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{35 c^{2} x} + \frac{x \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(c*x**4+b*x**2)**(1/2),x)

[Out]

8*b**2*(b*x**2 + c*x**4)**(3/2)/(105*c**3*x**3) - 4*b*(b*x**2 + c*x**4)**(3/2)/(
35*c**2*x) + x*(b*x**2 + c*x**4)**(3/2)/(7*c)

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Mathematica [A]  time = 0.028546, size = 57, normalized size = 0.73 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (8 b^3-4 b^2 c x^2+3 b c^2 x^4+15 c^3 x^6\right )}{105 c^3 x} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*Sqrt[b*x^2 + c*x^4],x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*(8*b^3 - 4*b^2*c*x^2 + 3*b*c^2*x^4 + 15*c^3*x^6))/(105*c^
3*x)

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Maple [A]  time = 0.007, size = 50, normalized size = 0.6 \[{\frac{ \left ( c{x}^{2}+b \right ) \left ( 15\,{c}^{2}{x}^{4}-12\,bc{x}^{2}+8\,{b}^{2} \right ) }{105\,{c}^{3}x}\sqrt{c{x}^{4}+b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(c*x^4+b*x^2)^(1/2),x)

[Out]

1/105*(c*x^2+b)*(15*c^2*x^4-12*b*c*x^2+8*b^2)*(c*x^4+b*x^2)^(1/2)/c^3/x

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Maxima [A]  time = 0.734401, size = 62, normalized size = 0.79 \[ \frac{{\left (15 \, c^{3} x^{6} + 3 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt{c x^{2} + b}}{105 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(15*c^3*x^6 + 3*b*c^2*x^4 - 4*b^2*c*x^2 + 8*b^3)*sqrt(c*x^2 + b)/c^3

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Fricas [A]  time = 0.270043, size = 72, normalized size = 0.92 \[ \frac{{\left (15 \, c^{3} x^{6} + 3 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{105 \, c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(15*c^3*x^6 + 3*b*c^2*x^4 - 4*b^2*c*x^2 + 8*b^3)*sqrt(c*x^4 + b*x^2)/(c^3*
x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(x**4*sqrt(x**2*(b + c*x**2)), x)

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GIAC/XCAS [A]  time = 0.271534, size = 76, normalized size = 0.97 \[ -\frac{8 \, b^{\frac{7}{2}}{\rm sign}\left (x\right )}{105 \, c^{3}} + \frac{{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )}{\rm sign}\left (x\right )}{105 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/105*b^(7/2)*sign(x)/c^3 + 1/105*(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^3