3.281 \(\int x^{5/2} (a+b x^2)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{6}{11} a^2 b x^{11/2}+\frac{2}{7} a^3 x^{7/2}+\frac{2}{5} a b^2 x^{15/2}+\frac{2}{19} b^3 x^{19/2} \]

[Out]

(2*a^3*x^(7/2))/7 + (6*a^2*b*x^(11/2))/11 + (2*a*b^2*x^(15/2))/5 + (2*b^3*x^(19/2))/19

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Rubi [A]  time = 0.012456, 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} \[ \frac{6}{11} a^2 b x^{11/2}+\frac{2}{7} a^3 x^{7/2}+\frac{2}{5} a b^2 x^{15/2}+\frac{2}{19} b^3 x^{19/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x^2)^3,x]

[Out]

(2*a^3*x^(7/2))/7 + (6*a^2*b*x^(11/2))/11 + (2*a*b^2*x^(15/2))/5 + (2*b^3*x^(19/2))/19

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^{5/2} \left (a+b x^2\right )^3 \, dx &=\int \left (a^3 x^{5/2}+3 a^2 b x^{9/2}+3 a b^2 x^{13/2}+b^3 x^{17/2}\right ) \, dx\\ &=\frac{2}{7} a^3 x^{7/2}+\frac{6}{11} a^2 b x^{11/2}+\frac{2}{5} a b^2 x^{15/2}+\frac{2}{19} b^3 x^{19/2}\\ \end{align*}

Mathematica [A]  time = 0.0094307, size = 41, normalized size = 0.8 \[ \frac{2 x^{7/2} \left (1995 a^2 b x^2+1045 a^3+1463 a b^2 x^4+385 b^3 x^6\right )}{7315} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x^2)^3,x]

[Out]

(2*x^(7/2)*(1045*a^3 + 1995*a^2*b*x^2 + 1463*a*b^2*x^4 + 385*b^3*x^6))/7315

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Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \begin{align*}{\frac{770\,{b}^{3}{x}^{6}+2926\,a{b}^{2}{x}^{4}+3990\,{a}^{2}b{x}^{2}+2090\,{a}^{3}}{7315}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7315*x^(7/2)*(385*b^3*x^6+1463*a*b^2*x^4+1995*a^2*b*x^2+1045*a^3)

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Maxima [A]  time = 2.38975, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{19} \, b^{3} x^{\frac{19}{2}} + \frac{2}{5} \, a b^{2} x^{\frac{15}{2}} + \frac{6}{11} \, a^{2} b x^{\frac{11}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a)^3,x, algorithm="maxima")

[Out]

2/19*b^3*x^(19/2) + 2/5*a*b^2*x^(15/2) + 6/11*a^2*b*x^(11/2) + 2/7*a^3*x^(7/2)

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Fricas [A]  time = 1.16818, size = 107, normalized size = 2.1 \begin{align*} \frac{2}{7315} \,{\left (385 \, b^{3} x^{9} + 1463 \, a b^{2} x^{7} + 1995 \, a^{2} b x^{5} + 1045 \, a^{3} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a)^3,x, algorithm="fricas")

[Out]

2/7315*(385*b^3*x^9 + 1463*a*b^2*x^7 + 1995*a^2*b*x^5 + 1045*a^3*x^3)*sqrt(x)

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Sympy [A]  time = 11.2705, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{7}{2}}}{7} + \frac{6 a^{2} b x^{\frac{11}{2}}}{11} + \frac{2 a b^{2} x^{\frac{15}{2}}}{5} + \frac{2 b^{3} x^{\frac{19}{2}}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(b*x**2+a)**3,x)

[Out]

2*a**3*x**(7/2)/7 + 6*a**2*b*x**(11/2)/11 + 2*a*b**2*x**(15/2)/5 + 2*b**3*x**(19/2)/19

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Giac [A]  time = 2.97581, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{19} \, b^{3} x^{\frac{19}{2}} + \frac{2}{5} \, a b^{2} x^{\frac{15}{2}} + \frac{6}{11} \, a^{2} b x^{\frac{11}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a)^3,x, algorithm="giac")

[Out]

2/19*b^3*x^(19/2) + 2/5*a*b^2*x^(15/2) + 6/11*a^2*b*x^(11/2) + 2/7*a^3*x^(7/2)