3.335 \(\int x^{7/2} (a+b x)^3 (A+B x) \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{9} a^3 A x^{9/2}+\frac{2}{15} b^2 x^{15/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{17} b^3 B x^{17/2} \]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*
B)*x^(15/2))/15 + (2*b^3*B*x^(17/2))/17

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Rubi [A]  time = 0.0411008, antiderivative size = 85, 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} \[ \frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{9} a^3 A x^{9/2}+\frac{2}{15} b^2 x^{15/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{17} b^3 B x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x)^3*(A + B*x),x]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*
B)*x^(15/2))/15 + (2*b^3*B*x^(17/2))/17

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)^3 (A+B x) \, dx &=\int \left (a^3 A x^{7/2}+a^2 (3 A b+a B) x^{9/2}+3 a b (A b+a B) x^{11/2}+b^2 (A b+3 a B) x^{13/2}+b^3 B x^{15/2}\right ) \, dx\\ &=\frac{2}{9} a^3 A x^{9/2}+\frac{2}{11} a^2 (3 A b+a B) x^{11/2}+\frac{6}{13} a b (A b+a B) x^{13/2}+\frac{2}{15} b^2 (A b+3 a B) x^{15/2}+\frac{2}{17} b^3 B x^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0266091, size = 71, normalized size = 0.84 \[ \frac{2 x^{9/2} \left (2295 a^2 b x (13 A+11 B x)+1105 a^3 (11 A+9 B x)+1683 a b^2 x^2 (15 A+13 B x)+429 b^3 x^3 (17 A+15 B x)\right )}{109395} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x)^3*(A + B*x),x]

[Out]

(2*x^(9/2)*(1105*a^3*(11*A + 9*B*x) + 2295*a^2*b*x*(13*A + 11*B*x) + 1683*a*b^2*x^2*(15*A + 13*B*x) + 429*b^3*
x^3*(17*A + 15*B*x)))/109395

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Maple [A]  time = 0.004, size = 76, normalized size = 0.9 \begin{align*}{\frac{12870\,B{b}^{3}{x}^{4}+14586\,A{b}^{3}{x}^{3}+43758\,B{x}^{3}a{b}^{2}+50490\,aA{b}^{2}{x}^{2}+50490\,B{x}^{2}{a}^{2}b+59670\,{a}^{2}Abx+19890\,{a}^{3}Bx+24310\,{a}^{3}A}{109395}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x+a)^3*(B*x+A),x)

[Out]

2/109395*x^(9/2)*(6435*B*b^3*x^4+7293*A*b^3*x^3+21879*B*a*b^2*x^3+25245*A*a*b^2*x^2+25245*B*a^2*b*x^2+29835*A*
a^2*b*x+9945*B*a^3*x+12155*A*a^3)

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Maxima [A]  time = 1.07601, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{17} \, B b^{3} x^{\frac{17}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} + \frac{2}{15} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{15}{2}} + \frac{6}{13} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B a^{3} + 3 \, A a^{2} 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)^3*(B*x+A),x, algorithm="maxima")

[Out]

2/17*B*b^3*x^(17/2) + 2/9*A*a^3*x^(9/2) + 2/15*(3*B*a*b^2 + A*b^3)*x^(15/2) + 6/13*(B*a^2*b + A*a*b^2)*x^(13/2
) + 2/11*(B*a^3 + 3*A*a^2*b)*x^(11/2)

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Fricas [A]  time = 2.35894, size = 198, normalized size = 2.33 \begin{align*} \frac{2}{109395} \,{\left (6435 \, B b^{3} x^{8} + 12155 \, A a^{3} x^{4} + 7293 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + 25245 \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} + 9945 \,{\left (B a^{3} + 3 \, A a^{2} 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)^3*(B*x+A),x, algorithm="fricas")

[Out]

2/109395*(6435*B*b^3*x^8 + 12155*A*a^3*x^4 + 7293*(3*B*a*b^2 + A*b^3)*x^7 + 25245*(B*a^2*b + A*a*b^2)*x^6 + 99
45*(B*a^3 + 3*A*a^2*b)*x^5)*sqrt(x)

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Sympy [A]  time = 16.3243, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A a^{3} x^{\frac{9}{2}}}{9} + \frac{6 A a^{2} b x^{\frac{11}{2}}}{11} + \frac{6 A a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 A b^{3} x^{\frac{15}{2}}}{15} + \frac{2 B a^{3} x^{\frac{11}{2}}}{11} + \frac{6 B a^{2} b x^{\frac{13}{2}}}{13} + \frac{2 B a b^{2} x^{\frac{15}{2}}}{5} + \frac{2 B b^{3} x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x+a)**3*(B*x+A),x)

[Out]

2*A*a**3*x**(9/2)/9 + 6*A*a**2*b*x**(11/2)/11 + 6*A*a*b**2*x**(13/2)/13 + 2*A*b**3*x**(15/2)/15 + 2*B*a**3*x**
(11/2)/11 + 6*B*a**2*b*x**(13/2)/13 + 2*B*a*b**2*x**(15/2)/5 + 2*B*b**3*x**(17/2)/17

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Giac [A]  time = 1.22361, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{17} \, B b^{3} x^{\frac{17}{2}} + \frac{2}{5} \, B a b^{2} x^{\frac{15}{2}} + \frac{2}{15} \, A b^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B a^{2} b x^{\frac{13}{2}} + \frac{6}{13} \, A a b^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B a^{3} x^{\frac{11}{2}} + \frac{6}{11} \, A a^{2} b x^{\frac{11}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x+a)^3*(B*x+A),x, algorithm="giac")

[Out]

2/17*B*b^3*x^(17/2) + 2/5*B*a*b^2*x^(15/2) + 2/15*A*b^3*x^(15/2) + 6/13*B*a^2*b*x^(13/2) + 6/13*A*a*b^2*x^(13/
2) + 2/11*B*a^3*x^(11/2) + 6/11*A*a^2*b*x^(11/2) + 2/9*A*a^3*x^(9/2)