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

Optimal. Leaf size=63 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2} \]

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(11/2))/11 + (2*b*(A*b + 2*a*B)*x^(13/2))/13 + (2*b^2*B*x^(15/2))/1
5

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Rubi [A]  time = 0.0287165, antiderivative size = 63, 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 = {76} \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x)^2*(A + B*x),x]

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(11/2))/11 + (2*b*(A*b + 2*a*B)*x^(13/2))/13 + (2*b^2*B*x^(15/2))/1
5

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)^2 (A+B x) \, dx &=\int \left (a^2 A x^{7/2}+a (2 A b+a B) x^{9/2}+b (A b+2 a B) x^{11/2}+b^2 B x^{13/2}\right ) \, dx\\ &=\frac{2}{9} a^2 A x^{9/2}+\frac{2}{11} a (2 A b+a B) x^{11/2}+\frac{2}{13} b (A b+2 a B) x^{13/2}+\frac{2}{15} b^2 B x^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0168972, size = 52, normalized size = 0.83 \[ \frac{2 x^{9/2} \left (65 a^2 (11 A+9 B x)+90 a b x (13 A+11 B x)+33 b^2 x^2 (15 A+13 B x)\right )}{6435} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(65*a^2*(11*A + 9*B*x) + 90*a*b*x*(13*A + 11*B*x) + 33*b^2*x^2*(15*A + 13*B*x)))/6435

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Maple [A]  time = 0.004, size = 52, normalized size = 0.8 \begin{align*}{\frac{858\,B{b}^{2}{x}^{3}+990\,A{b}^{2}{x}^{2}+1980\,B{x}^{2}ab+2340\,aAbx+1170\,{a}^{2}Bx+1430\,{a}^{2}A}{6435}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/6435*x^(9/2)*(429*B*b^2*x^3+495*A*b^2*x^2+990*B*a*b*x^2+1170*A*a*b*x+585*B*a^2*x+715*A*a^2)

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Maxima [A]  time = 1.10033, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B a^{2} + 2 \, 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)^2*(B*x+A),x, algorithm="maxima")

[Out]

2/15*B*b^2*x^(15/2) + 2/9*A*a^2*x^(9/2) + 2/13*(2*B*a*b + A*b^2)*x^(13/2) + 2/11*(B*a^2 + 2*A*a*b)*x^(11/2)

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Fricas [A]  time = 2.22922, size = 140, normalized size = 2.22 \begin{align*} \frac{2}{6435} \,{\left (429 \, B b^{2} x^{7} + 715 \, A a^{2} x^{4} + 495 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + 585 \,{\left (B a^{2} + 2 \, 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)^2*(B*x+A),x, algorithm="fricas")

[Out]

2/6435*(429*B*b^2*x^7 + 715*A*a^2*x^4 + 495*(2*B*a*b + A*b^2)*x^6 + 585*(B*a^2 + 2*A*a*b)*x^5)*sqrt(x)

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Sympy [A]  time = 13.3671, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a b x^{\frac{11}{2}}}{11} + \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{2 B a^{2} x^{\frac{11}{2}}}{11} + \frac{4 B a b x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**2*x**(9/2)/9 + 4*A*a*b*x**(11/2)/11 + 2*A*b**2*x**(13/2)/13 + 2*B*a**2*x**(11/2)/11 + 4*B*a*b*x**(13/2)
/13 + 2*B*b**2*x**(15/2)/15

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Giac [A]  time = 1.22717, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B a b x^{\frac{13}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A a b x^{\frac{11}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*B*b^2*x^(15/2) + 4/13*B*a*b*x^(13/2) + 2/13*A*b^2*x^(13/2) + 2/11*B*a^2*x^(11/2) + 4/11*A*a*b*x^(11/2) +
2/9*A*a^2*x^(9/2)