3.385 \(\int x^{5/2} (A+B x) (a+c x^2) \, dx\)

Optimal. Leaf size=45 \[ \frac{2}{7} a A x^{7/2}+\frac{2}{9} a B x^{9/2}+\frac{2}{11} A c x^{11/2}+\frac{2}{13} B c x^{13/2} \]

[Out]

(2*a*A*x^(7/2))/7 + (2*a*B*x^(9/2))/9 + (2*A*c*x^(11/2))/11 + (2*B*c*x^(13/2))/13

________________________________________________________________________________________

Rubi [A]  time = 0.0123421, 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} \[ \frac{2}{7} a A x^{7/2}+\frac{2}{9} a B x^{9/2}+\frac{2}{11} A c x^{11/2}+\frac{2}{13} B c x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(A + B*x)*(a + c*x^2),x]

[Out]

(2*a*A*x^(7/2))/7 + (2*a*B*x^(9/2))/9 + (2*A*c*x^(11/2))/11 + (2*B*c*x^(13/2))/13

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^{5/2} (A+B x) \left (a+c x^2\right ) \, dx &=\int \left (a A x^{5/2}+a B x^{7/2}+A c x^{9/2}+B c x^{11/2}\right ) \, dx\\ &=\frac{2}{7} a A x^{7/2}+\frac{2}{9} a B x^{9/2}+\frac{2}{11} A c x^{11/2}+\frac{2}{13} B c x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0140148, size = 35, normalized size = 0.78 \[ \frac{2 x^{7/2} \left (143 a (9 A+7 B x)+63 c x^2 (13 A+11 B x)\right )}{9009} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(A + B*x)*(a + c*x^2),x]

[Out]

(2*x^(7/2)*(143*a*(9*A + 7*B*x) + 63*c*x^2*(13*A + 11*B*x)))/9009

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 30, normalized size = 0.7 \begin{align*}{\frac{1386\,Bc{x}^{3}+1638\,Ac{x}^{2}+2002\,aBx+2574\,aA}{9009}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x+A)*(c*x^2+a),x)

[Out]

2/9009*x^(7/2)*(693*B*c*x^3+819*A*c*x^2+1001*B*a*x+1287*A*a)

________________________________________________________________________________________

Maxima [A]  time = 0.98342, size = 39, normalized size = 0.87 \begin{align*} \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{11} \, A c x^{\frac{11}{2}} + \frac{2}{9} \, B a x^{\frac{9}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+a),x, algorithm="maxima")

[Out]

2/13*B*c*x^(13/2) + 2/11*A*c*x^(11/2) + 2/9*B*a*x^(9/2) + 2/7*A*a*x^(7/2)

________________________________________________________________________________________

Fricas [A]  time = 1.38773, size = 100, normalized size = 2.22 \begin{align*} \frac{2}{9009} \,{\left (693 \, B c x^{6} + 819 \, A c x^{5} + 1001 \, B a x^{4} + 1287 \, A a x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+a),x, algorithm="fricas")

[Out]

2/9009*(693*B*c*x^6 + 819*A*c*x^5 + 1001*B*a*x^4 + 1287*A*a*x^3)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 4.71777, size = 46, normalized size = 1.02 \begin{align*} \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 A c x^{\frac{11}{2}}}{11} + \frac{2 B a x^{\frac{9}{2}}}{9} + \frac{2 B c x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x+A)*(c*x**2+a),x)

[Out]

2*A*a*x**(7/2)/7 + 2*A*c*x**(11/2)/11 + 2*B*a*x**(9/2)/9 + 2*B*c*x**(13/2)/13

________________________________________________________________________________________

Giac [A]  time = 1.16391, size = 39, normalized size = 0.87 \begin{align*} \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{11} \, A c x^{\frac{11}{2}} + \frac{2}{9} \, B a x^{\frac{9}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+a),x, algorithm="giac")

[Out]

2/13*B*c*x^(13/2) + 2/11*A*c*x^(11/2) + 2/9*B*a*x^(9/2) + 2/7*A*a*x^(7/2)