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

Optimal. Leaf size=63 \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]

[Out]

(2*a^2*A*x^(5/2))/5 + (2*a*(2*A*b + a*B)*x^(7/2))/7 + (2*b*(A*b + 2*a*B)*x^(9/2))/9 + (2*b^2*B*x^(11/2))/11

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Rubi [A]  time = 0.0269592, 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} \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x)^2*(A + B*x),x]

[Out]

(2*a^2*A*x^(5/2))/5 + (2*a*(2*A*b + a*B)*x^(7/2))/7 + (2*b*(A*b + 2*a*B)*x^(9/2))/9 + (2*b^2*B*x^(11/2))/11

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^{3/2} (a+b x)^2 (A+B x) \, dx &=\int \left (a^2 A x^{3/2}+a (2 A b+a B) x^{5/2}+b (A b+2 a B) x^{7/2}+b^2 B x^{9/2}\right ) \, dx\\ &=\frac{2}{5} a^2 A x^{5/2}+\frac{2}{7} a (2 A b+a B) x^{7/2}+\frac{2}{9} b (A b+2 a B) x^{9/2}+\frac{2}{11} b^2 B x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.01582, size = 52, normalized size = 0.83 \[ \frac{2 x^{5/2} \left (99 a^2 (7 A+5 B x)+110 a b x (9 A+7 B x)+35 b^2 x^2 (11 A+9 B x)\right )}{3465} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(99*a^2*(7*A + 5*B*x) + 110*a*b*x*(9*A + 7*B*x) + 35*b^2*x^2*(11*A + 9*B*x)))/3465

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Maple [A]  time = 0.006, size = 52, normalized size = 0.8 \begin{align*}{\frac{630\,B{b}^{2}{x}^{3}+770\,A{b}^{2}{x}^{2}+1540\,B{x}^{2}ab+1980\,aAbx+990\,{a}^{2}Bx+1386\,{a}^{2}A}{3465}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*x^(5/2)*(315*B*b^2*x^3+385*A*b^2*x^2+770*B*a*b*x^2+990*A*a*b*x+495*B*a^2*x+693*A*a^2)

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Maxima [A]  time = 1.6356, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*b^2*x^(11/2) + 2/5*A*a^2*x^(5/2) + 2/9*(2*B*a*b + A*b^2)*x^(9/2) + 2/7*(B*a^2 + 2*A*a*b)*x^(7/2)

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Fricas [A]  time = 2.37634, size = 140, normalized size = 2.22 \begin{align*} \frac{2}{3465} \,{\left (315 \, B b^{2} x^{5} + 693 \, A a^{2} x^{2} + 385 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 495 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*B*b^2*x^5 + 693*A*a^2*x^2 + 385*(2*B*a*b + A*b^2)*x^4 + 495*(B*a^2 + 2*A*a*b)*x^3)*sqrt(x)

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Sympy [A]  time = 2.2575, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A a^{2} x^{\frac{5}{2}}}{5} + \frac{4 A a b x^{\frac{7}{2}}}{7} + \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{2 B a^{2} x^{\frac{7}{2}}}{7} + \frac{4 B a b x^{\frac{9}{2}}}{9} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**2*x**(5/2)/5 + 4*A*a*b*x**(7/2)/7 + 2*A*b**2*x**(9/2)/9 + 2*B*a**2*x**(7/2)/7 + 4*B*a*b*x**(9/2)/9 + 2*
B*b**2*x**(11/2)/11

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Giac [A]  time = 1.14401, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{9} \, B a b x^{\frac{9}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{2}{7} \, B a^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A a b x^{\frac{7}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*b^2*x^(11/2) + 4/9*B*a*b*x^(9/2) + 2/9*A*b^2*x^(9/2) + 2/7*B*a^2*x^(7/2) + 4/7*A*a*b*x^(7/2) + 2/5*A*a^
2*x^(5/2)