3.264 \(\int x^{7/2} (a+b x^2) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{9} a x^{9/2}+\frac{2}{13} b x^{13/2} \]

[Out]

(2*a*x^(9/2))/9 + (2*b*x^(13/2))/13

________________________________________________________________________________________

Rubi [A]  time = 0.0043362, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{2}{9} a x^{9/2}+\frac{2}{13} b x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x^2),x]

[Out]

(2*a*x^(9/2))/9 + (2*b*x^(13/2))/13

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0049709, size = 21, normalized size = 1. \[ \frac{2}{9} a x^{9/2}+\frac{2}{13} b x^{13/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x^2),x]

[Out]

(2*a*x^(9/2))/9 + (2*b*x^(13/2))/13

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \begin{align*}{\frac{18\,b{x}^{2}+26\,a}{117}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2+a),x)

[Out]

2/117*x^(9/2)*(9*b*x^2+13*a)

________________________________________________________________________________________

Maxima [A]  time = 2.0432, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{13} \, b x^{\frac{13}{2}} + \frac{2}{9} \, a x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a),x, algorithm="maxima")

[Out]

2/13*b*x^(13/2) + 2/9*a*x^(9/2)

________________________________________________________________________________________

Fricas [A]  time = 1.46267, size = 49, normalized size = 2.33 \begin{align*} \frac{2}{117} \,{\left (9 \, b x^{6} + 13 \, a x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a),x, algorithm="fricas")

[Out]

2/117*(9*b*x^6 + 13*a*x^4)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 5.79703, size = 19, normalized size = 0.9 \begin{align*} \frac{2 a x^{\frac{9}{2}}}{9} + \frac{2 b x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x**2+a),x)

[Out]

2*a*x**(9/2)/9 + 2*b*x**(13/2)/13

________________________________________________________________________________________

Giac [A]  time = 2.72801, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{13} \, b x^{\frac{13}{2}} + \frac{2}{9} \, a x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a),x, algorithm="giac")

[Out]

2/13*b*x^(13/2) + 2/9*a*x^(9/2)