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

Optimal. Leaf size=39 \[ \frac{2}{11} x^{11/2} (A c+b B)+\frac{2}{9} A b x^{9/2}+\frac{2}{13} B c x^{13/2} \]

[Out]

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

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Rubi [A]  time = 0.0165978, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{2}{11} x^{11/2} (A c+b B)+\frac{2}{9} A b x^{9/2}+\frac{2}{13} B c x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(A + B*x)*(b*x + c*x^2),x]

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0118826, size = 33, normalized size = 0.85 \[ \frac{2 x^{9/2} (13 A (11 b+9 c x)+9 B x (13 b+11 c x))}{1287} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(13*A*(11*b + 9*c*x) + 9*B*x*(13*b + 11*c*x)))/1287

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Maple [A]  time = 0.005, size = 28, normalized size = 0.7 \begin{align*}{\frac{198\,Bc{x}^{2}+234\,Acx+234\,bBx+286\,Ab}{1287}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1287*x^(9/2)*(99*B*c*x^2+117*A*c*x+117*B*b*x+143*A*b)

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Maxima [A]  time = 1.08308, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{9} \, A b x^{\frac{9}{2}} + \frac{2}{11} \,{\left (B b + A c\right )} x^{\frac{11}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.77324, size = 88, normalized size = 2.26 \begin{align*} \frac{2}{1287} \,{\left (99 \, B c x^{6} + 143 \, A b x^{4} + 117 \,{\left (B b + A c\right )} x^{5}\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+b*x),x, algorithm="fricas")

[Out]

2/1287*(99*B*c*x^6 + 143*A*b*x^4 + 117*(B*b + A*c)*x^5)*sqrt(x)

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Sympy [A]  time = 4.35817, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A b x^{\frac{9}{2}}}{9} + \frac{2 A c x^{\frac{11}{2}}}{11} + \frac{2 B b x^{\frac{11}{2}}}{11} + \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+b*x),x)

[Out]

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

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Giac [A]  time = 1.10685, size = 39, normalized size = 1. \begin{align*} \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{11} \, B b x^{\frac{11}{2}} + \frac{2}{11} \, A c x^{\frac{11}{2}} + \frac{2}{9} \, A b x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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