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

Optimal. Leaf size=63 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(9/2))/9 + (2*b*(A*b + 2*a*B)*x^(11/2))/11 + (2*b^2*B*x^(13/2))/13

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Rubi [A]  time = 0.0284251, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(9/2))/9 + (2*b*(A*b + 2*a*B)*x^(11/2))/11 + (2*b^2*B*x^(13/2))/13

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

Mathematica [A]  time = 0.017825, size = 52, normalized size = 0.83 \[ \frac{2 x^{7/2} \left (143 a^2 (9 A+7 B x)+182 a b x (11 A+9 B x)+63 b^2 x^2 (13 A+11 B x)\right )}{9009} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(7/2)*(143*a^2*(9*A + 7*B*x) + 182*a*b*x*(11*A + 9*B*x) + 63*b^2*x^2*(13*A + 11*B*x)))/9009

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Maple [A]  time = 0.006, size = 52, normalized size = 0.8 \begin{align*}{\frac{1386\,{b}^{2}B{x}^{3}+1638\,A{b}^{2}{x}^{2}+3276\,B{x}^{2}ab+4004\,aAbx+2002\,{a}^{2}Bx+2574\,A{a}^{2}}{9009}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*x^(7/2)*(693*B*b^2*x^3+819*A*b^2*x^2+1638*B*a*b*x^2+2002*A*a*b*x+1001*B*a^2*x+1287*A*a^2)

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Maxima [A]  time = 1.03974, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*B*b^2*x^(13/2) + 2/7*A*a^2*x^(7/2) + 2/11*(2*B*a*b + A*b^2)*x^(11/2) + 2/9*(B*a^2 + 2*A*a*b)*x^(9/2)

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Fricas [A]  time = 1.57553, size = 143, normalized size = 2.27 \begin{align*} \frac{2}{9009} \,{\left (693 \, B b^{2} x^{6} + 1287 \, A a^{2} x^{3} + 819 \,{\left (2 \, B a b + A b^{2}\right )} x^{5} + 1001 \,{\left (B a^{2} + 2 \, A a b\right )} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(693*B*b^2*x^6 + 1287*A*a^2*x^3 + 819*(2*B*a*b + A*b^2)*x^5 + 1001*(B*a^2 + 2*A*a*b)*x^4)*sqrt(x)

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Sympy [A]  time = 4.52832, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{4 A a b x^{\frac{9}{2}}}{9} + \frac{2 A b^{2} x^{\frac{11}{2}}}{11} + \frac{2 B a^{2} x^{\frac{9}{2}}}{9} + \frac{4 B a b x^{\frac{11}{2}}}{11} + \frac{2 B b^{2} 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)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

2*A*a**2*x**(7/2)/7 + 4*A*a*b*x**(9/2)/9 + 2*A*b**2*x**(11/2)/11 + 2*B*a**2*x**(9/2)/9 + 4*B*a*b*x**(11/2)/11
+ 2*B*b**2*x**(13/2)/13

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Giac [A]  time = 1.15679, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{11} \, B a b x^{\frac{11}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} + \frac{2}{9} \, B a^{2} x^{\frac{9}{2}} + \frac{4}{9} \, A a b x^{\frac{9}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*B*b^2*x^(13/2) + 4/11*B*a*b*x^(11/2) + 2/11*A*b^2*x^(11/2) + 2/9*B*a^2*x^(9/2) + 4/9*A*a*b*x^(9/2) + 2/7*
A*a^2*x^(7/2)