3.140 \(\int x^{5/2} (a+b x^3)^2 (A+B x^3) \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{19} b x^{19/2} (2 a B+A b)+\frac{2}{13} a x^{13/2} (a B+2 A b)+\frac{2}{25} b^2 B x^{25/2} \]

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(13/2))/13 + (2*b*(A*b + 2*a*B)*x^(19/2))/19 + (2*b^2*B*x^(25/2))/2
5

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Rubi [A]  time = 0.0309277, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{19} b x^{19/2} (2 a B+A b)+\frac{2}{13} a x^{13/2} (a B+2 A b)+\frac{2}{25} b^2 B x^{25/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x^3)^2*(A + B*x^3),x]

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(13/2))/13 + (2*b*(A*b + 2*a*B)*x^(19/2))/19 + (2*b^2*B*x^(25/2))/2
5

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int x^{5/2} \left (a+b x^3\right )^2 \left (A+B x^3\right ) \, dx &=\int \left (a^2 A x^{5/2}+a (2 A b+a B) x^{11/2}+b (A b+2 a B) x^{17/2}+b^2 B x^{23/2}\right ) \, dx\\ &=\frac{2}{7} a^2 A x^{7/2}+\frac{2}{13} a (2 A b+a B) x^{13/2}+\frac{2}{19} b (A b+2 a B) x^{19/2}+\frac{2}{25} b^2 B x^{25/2}\\ \end{align*}

Mathematica [A]  time = 0.0280438, size = 63, normalized size = 1. \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{19} b x^{19/2} (2 a B+A b)+\frac{2}{13} a x^{13/2} (a B+2 A b)+\frac{2}{25} b^2 B x^{25/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x^3)^2*(A + B*x^3),x]

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(13/2))/13 + (2*b*(A*b + 2*a*B)*x^(19/2))/19 + (2*b^2*B*x^(25/2))/2
5

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Maple [A]  time = 0.009, size = 56, normalized size = 0.9 \begin{align*}{\frac{3458\,B{b}^{2}{x}^{9}+4550\,A{b}^{2}{x}^{6}+9100\,B{x}^{6}ab+13300\,aAb{x}^{3}+6650\,B{x}^{3}{a}^{2}+12350\,{a}^{2}A}{43225}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x^3+a)^2*(B*x^3+A),x)

[Out]

2/43225*x^(7/2)*(1729*B*b^2*x^9+2275*A*b^2*x^6+4550*B*a*b*x^6+6650*A*a*b*x^3+3325*B*a^2*x^3+6175*A*a^2)

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Maxima [A]  time = 0.959429, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{25} \, B b^{2} x^{\frac{25}{2}} + \frac{2}{19} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{19}{2}} + \frac{2}{13} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{13}{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^3+a)^2*(B*x^3+A),x, algorithm="maxima")

[Out]

2/25*B*b^2*x^(25/2) + 2/19*(2*B*a*b + A*b^2)*x^(19/2) + 2/13*(B*a^2 + 2*A*a*b)*x^(13/2) + 2/7*A*a^2*x^(7/2)

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Fricas [A]  time = 1.6829, size = 149, normalized size = 2.37 \begin{align*} \frac{2}{43225} \,{\left (1729 \, B b^{2} x^{12} + 2275 \,{\left (2 \, B a b + A b^{2}\right )} x^{9} + 3325 \,{\left (B a^{2} + 2 \, A a b\right )} x^{6} + 6175 \, A a^{2} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^3+a)^2*(B*x^3+A),x, algorithm="fricas")

[Out]

2/43225*(1729*B*b^2*x^12 + 2275*(2*B*a*b + A*b^2)*x^9 + 3325*(B*a^2 + 2*A*a*b)*x^6 + 6175*A*a^2*x^3)*sqrt(x)

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Sympy [A]  time = 46.9283, 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{13}{2}}}{13} + \frac{2 A b^{2} x^{\frac{19}{2}}}{19} + \frac{2 B a^{2} x^{\frac{13}{2}}}{13} + \frac{4 B a b x^{\frac{19}{2}}}{19} + \frac{2 B b^{2} x^{\frac{25}{2}}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(b*x**3+a)**2*(B*x**3+A),x)

[Out]

2*A*a**2*x**(7/2)/7 + 4*A*a*b*x**(13/2)/13 + 2*A*b**2*x**(19/2)/19 + 2*B*a**2*x**(13/2)/13 + 4*B*a*b*x**(19/2)
/19 + 2*B*b**2*x**(25/2)/25

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Giac [A]  time = 1.1303, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{25} \, B b^{2} x^{\frac{25}{2}} + \frac{4}{19} \, B a b x^{\frac{19}{2}} + \frac{2}{19} \, A b^{2} x^{\frac{19}{2}} + \frac{2}{13} \, B a^{2} x^{\frac{13}{2}} + \frac{4}{13} \, A a b x^{\frac{13}{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^3+a)^2*(B*x^3+A),x, algorithm="giac")

[Out]

2/25*B*b^2*x^(25/2) + 4/19*B*a*b*x^(19/2) + 2/19*A*b^2*x^(19/2) + 2/13*B*a^2*x^(13/2) + 4/13*A*a*b*x^(13/2) +
2/7*A*a^2*x^(7/2)