3.386 \(\int x^{3/2} (A+B x) (a+c x^2) \, dx\)

Optimal. Leaf size=45 \[ \frac{2}{5} a A x^{5/2}+\frac{2}{7} a B x^{7/2}+\frac{2}{9} A c x^{9/2}+\frac{2}{11} B c x^{11/2} \]

[Out]

(2*a*A*x^(5/2))/5 + (2*a*B*x^(7/2))/7 + (2*A*c*x^(9/2))/9 + (2*B*c*x^(11/2))/11

________________________________________________________________________________________

Rubi [A]  time = 0.0125184, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ \frac{2}{5} a A x^{5/2}+\frac{2}{7} a B x^{7/2}+\frac{2}{9} A c x^{9/2}+\frac{2}{11} B c x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(A + B*x)*(a + c*x^2),x]

[Out]

(2*a*A*x^(5/2))/5 + (2*a*B*x^(7/2))/7 + (2*A*c*x^(9/2))/9 + (2*B*c*x^(11/2))/11

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^{3/2} (A+B x) \left (a+c x^2\right ) \, dx &=\int \left (a A x^{3/2}+a B x^{5/2}+A c x^{7/2}+B c x^{9/2}\right ) \, dx\\ &=\frac{2}{5} a A x^{5/2}+\frac{2}{7} a B x^{7/2}+\frac{2}{9} A c x^{9/2}+\frac{2}{11} B c x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0169846, size = 37, normalized size = 0.82 \[ \frac{2}{35} a x^{5/2} (7 A+5 B x)+\frac{2}{99} c x^{9/2} (11 A+9 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(A + B*x)*(a + c*x^2),x]

[Out]

(2*a*x^(5/2)*(7*A + 5*B*x))/35 + (2*c*x^(9/2)*(11*A + 9*B*x))/99

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 30, normalized size = 0.7 \begin{align*}{\frac{630\,Bc{x}^{3}+770\,Ac{x}^{2}+990\,aBx+1386\,aA}{3465}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(B*x+A)*(c*x^2+a),x)

[Out]

2/3465*x^(5/2)*(315*B*c*x^3+385*A*c*x^2+495*B*a*x+693*A*a)

________________________________________________________________________________________

Maxima [A]  time = 0.991566, size = 39, normalized size = 0.87 \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \, A c x^{\frac{9}{2}} + \frac{2}{7} \, B a x^{\frac{7}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*c*x^(11/2) + 2/9*A*c*x^(9/2) + 2/7*B*a*x^(7/2) + 2/5*A*a*x^(5/2)

________________________________________________________________________________________

Fricas [A]  time = 1.28202, size = 97, normalized size = 2.16 \begin{align*} \frac{2}{3465} \,{\left (315 \, B c x^{5} + 385 \, A c x^{4} + 495 \, B a x^{3} + 693 \, A a x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*B*c*x^5 + 385*A*c*x^4 + 495*B*a*x^3 + 693*A*a*x^2)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 2.02586, size = 46, normalized size = 1.02 \begin{align*} \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 A c x^{\frac{9}{2}}}{9} + \frac{2 B a x^{\frac{7}{2}}}{7} + \frac{2 B c x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(B*x+A)*(c*x**2+a),x)

[Out]

2*A*a*x**(5/2)/5 + 2*A*c*x**(9/2)/9 + 2*B*a*x**(7/2)/7 + 2*B*c*x**(11/2)/11

________________________________________________________________________________________

Giac [A]  time = 1.14921, size = 39, normalized size = 0.87 \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \, A c x^{\frac{9}{2}} + \frac{2}{7} \, B a x^{\frac{7}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*c*x^(11/2) + 2/9*A*c*x^(9/2) + 2/7*B*a*x^(7/2) + 2/5*A*a*x^(5/2)