3.2936 \(\int x^5 \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=56 \[ \frac{2 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^2 c^3}-\frac{2 a \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^2 c^3} \]

[Out]

(-2*a*(a + b*(c*x^2)^(3/2))^(3/2))/(9*b^2*c^3) + (2*(a + b*(c*x^2)^(3/2))^(5/2))
/(15*b^2*c^3)

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Rubi [A]  time = 0.114753, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^2 c^3}-\frac{2 a \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^2 c^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[a + b*(c*x^2)^(3/2)],x]

[Out]

(-2*a*(a + b*(c*x^2)^(3/2))^(3/2))/(9*b^2*c^3) + (2*(a + b*(c*x^2)^(3/2))^(5/2))
/(15*b^2*c^3)

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Rubi in Sympy [A]  time = 11.9638, size = 51, normalized size = 0.91 \[ - \frac{2 a \left (a + b \left (c x^{2}\right )^{\frac{3}{2}}\right )^{\frac{3}{2}}}{9 b^{2} c^{3}} + \frac{2 \left (a + b \left (c x^{2}\right )^{\frac{3}{2}}\right )^{\frac{5}{2}}}{15 b^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(a+b*(c*x**2)**(3/2))**(1/2),x)

[Out]

-2*a*(a + b*(c*x**2)**(3/2))**(3/2)/(9*b**2*c**3) + 2*(a + b*(c*x**2)**(3/2))**(
5/2)/(15*b**2*c**3)

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Mathematica [A]  time = 0.0904102, size = 0, normalized size = 0. \[ \int x^5 \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[x^5*Sqrt[a + b*(c*x^2)^(3/2)],x]

[Out]

Integrate[x^5*Sqrt[a + b*(c*x^2)^(3/2)], x]

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{x}^{5}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(a+b*(c*x^2)^(3/2))^(1/2),x)

[Out]

int(x^5*(a+b*(c*x^2)^(3/2))^(1/2),x)

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Maxima [A]  time = 1.34099, size = 58, normalized size = 1.04 \[ \frac{2 \,{\left (\frac{3 \,{\left (\left (c x^{2}\right )^{\frac{3}{2}} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\left (c x^{2}\right )^{\frac{3}{2}} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{45 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)*x^5,x, algorithm="maxima")

[Out]

2/45*(3*((c*x^2)^(3/2)*b + a)^(5/2)/b^2 - 5*((c*x^2)^(3/2)*b + a)^(3/2)*a/b^2)/c
^3

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Fricas [A]  time = 0.210445, size = 76, normalized size = 1.36 \[ \frac{2 \,{\left (3 \, b^{2} c^{3} x^{6} + \sqrt{c x^{2}} a b c x^{2} - 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{45 \, b^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)*x^5,x, algorithm="fricas")

[Out]

2/45*(3*b^2*c^3*x^6 + sqrt(c*x^2)*a*b*c*x^2 - 2*a^2)*sqrt(sqrt(c*x^2)*b*c*x^2 +
a)/(b^2*c^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{5} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(a+b*(c*x**2)**(3/2))**(1/2),x)

[Out]

Integral(x**5*sqrt(a + b*(c*x**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.218228, size = 51, normalized size = 0.91 \[ \frac{2 \,{\left (3 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{3}{2}} a\right )}}{45 \, b^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)*x^5,x, algorithm="giac")

[Out]

2/45*(3*(b*c^(3/2)*x^3 + a)^(5/2) - 5*(b*c^(3/2)*x^3 + a)^(3/2)*a)/(b^2*c^3)