3.282 \(\int x^{3/2} (a+b x^2)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{2}{3} a^2 b x^{9/2}+\frac{2}{5} a^3 x^{5/2}+\frac{6}{13} a b^2 x^{13/2}+\frac{2}{17} b^3 x^{17/2} \]

[Out]

(2*a^3*x^(5/2))/5 + (2*a^2*b*x^(9/2))/3 + (6*a*b^2*x^(13/2))/13 + (2*b^3*x^(17/2))/17

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Rubi [A]  time = 0.0125412, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ \frac{2}{3} a^2 b x^{9/2}+\frac{2}{5} a^3 x^{5/2}+\frac{6}{13} a b^2 x^{13/2}+\frac{2}{17} b^3 x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x^2)^3,x]

[Out]

(2*a^3*x^(5/2))/5 + (2*a^2*b*x^(9/2))/3 + (6*a*b^2*x^(13/2))/13 + (2*b^3*x^(17/2))/17

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^{3/2} \left (a+b x^2\right )^3 \, dx &=\int \left (a^3 x^{3/2}+3 a^2 b x^{7/2}+3 a b^2 x^{11/2}+b^3 x^{15/2}\right ) \, dx\\ &=\frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{6}{13} a b^2 x^{13/2}+\frac{2}{17} b^3 x^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0096036, size = 41, normalized size = 0.8 \[ \frac{2 x^{5/2} \left (1105 a^2 b x^2+663 a^3+765 a b^2 x^4+195 b^3 x^6\right )}{3315} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + b*x^2)^3,x]

[Out]

(2*x^(5/2)*(663*a^3 + 1105*a^2*b*x^2 + 765*a*b^2*x^4 + 195*b^3*x^6))/3315

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Maple [A]  time = 0.003, size = 38, normalized size = 0.8 \begin{align*}{\frac{390\,{b}^{3}{x}^{6}+1530\,a{b}^{2}{x}^{4}+2210\,{a}^{2}b{x}^{2}+1326\,{a}^{3}}{3315}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3315*x^(5/2)*(195*b^3*x^6+765*a*b^2*x^4+1105*a^2*b*x^2+663*a^3)

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Maxima [A]  time = 2.17352, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{17} \, b^{3} x^{\frac{17}{2}} + \frac{6}{13} \, a b^{2} x^{\frac{13}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^3,x, algorithm="maxima")

[Out]

2/17*b^3*x^(17/2) + 6/13*a*b^2*x^(13/2) + 2/3*a^2*b*x^(9/2) + 2/5*a^3*x^(5/2)

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Fricas [A]  time = 1.30844, size = 104, normalized size = 2.04 \begin{align*} \frac{2}{3315} \,{\left (195 \, b^{3} x^{8} + 765 \, a b^{2} x^{6} + 1105 \, a^{2} b x^{4} + 663 \, a^{3} x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^3,x, algorithm="fricas")

[Out]

2/3315*(195*b^3*x^8 + 765*a*b^2*x^6 + 1105*a^2*b*x^4 + 663*a^3*x^2)*sqrt(x)

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Sympy [A]  time = 5.72263, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{5}{2}}}{5} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 b^{3} x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(b*x**2+a)**3,x)

[Out]

2*a**3*x**(5/2)/5 + 2*a**2*b*x**(9/2)/3 + 6*a*b**2*x**(13/2)/13 + 2*b**3*x**(17/2)/17

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Giac [A]  time = 2.25685, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{17} \, b^{3} x^{\frac{17}{2}} + \frac{6}{13} \, a b^{2} x^{\frac{13}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^3,x, algorithm="giac")

[Out]

2/17*b^3*x^(17/2) + 6/13*a*b^2*x^(13/2) + 2/3*a^2*b*x^(9/2) + 2/5*a^3*x^(5/2)