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

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

[Out]

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

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Rubi [A]  time = 0.014199, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{13} b B x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x)*(A + B*x),x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0114178, size = 33, normalized size = 0.85 \[ \frac{2 x^{9/2} (13 a (11 A+9 B x)+9 b x (13 A+11 B x))}{1287} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x)*(A + B*x),x]

[Out]

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

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Maple [A]  time = 0.003, size = 28, normalized size = 0.7 \begin{align*}{\frac{198\,bB{x}^{2}+234\,Abx+234\,Bax+286\,Aa}{1287}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09306, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{9} \, A a x^{\frac{9}{2}} + \frac{2}{11} \,{\left (B a + A b\right )} x^{\frac{11}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.49648, size = 88, normalized size = 2.26 \begin{align*} \frac{2}{1287} \,{\left (99 \, B b x^{6} + 143 \, A a x^{4} + 117 \,{\left (B a + A b\right )} x^{5}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 7.87348, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A a x^{\frac{9}{2}}}{9} + \frac{2 A b x^{\frac{11}{2}}}{11} + \frac{2 B a x^{\frac{11}{2}}}{11} + \frac{2 B 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+a)*(B*x+A),x)

[Out]

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

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Giac [A]  time = 1.21406, size = 39, normalized size = 1. \begin{align*} \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{11} \, B a x^{\frac{11}{2}} + \frac{2}{11} \, A b x^{\frac{11}{2}} + \frac{2}{9} \, A a x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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