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

Optimal. Leaf size=39 \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{17} b B x^{17/2} \]

[Out]

(2*a*A*x^(5/2))/5 + (2*(A*b + a*B)*x^(11/2))/11 + (2*b*B*x^(17/2))/17

________________________________________________________________________________________

Rubi [A]  time = 0.0153684, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{17} b B x^{17/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*A*x^(5/2))/5 + (2*(A*b + a*B)*x^(11/2))/11 + (2*b*B*x^(17/2))/17

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^{3/2} \left (a+b x^3\right ) \left (A+B x^3\right ) \, dx &=\int \left (a A x^{3/2}+(A b+a B) x^{9/2}+b B x^{15/2}\right ) \, dx\\ &=\frac{2}{5} a A x^{5/2}+\frac{2}{11} (A b+a B) x^{11/2}+\frac{2}{17} b B x^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0132942, size = 33, normalized size = 0.85 \[ \frac{2}{935} x^{5/2} \left (85 x^3 (a B+A b)+187 a A+55 b B x^6\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(187*a*A + 85*(A*b + a*B)*x^3 + 55*b*B*x^6))/935

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 32, normalized size = 0.8 \begin{align*}{\frac{110\,bB{x}^{6}+170\,A{x}^{3}b+170\,B{x}^{3}a+374\,Aa}{935}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/935*x^(5/2)*(55*B*b*x^6+85*A*b*x^3+85*B*a*x^3+187*A*a)

________________________________________________________________________________________

Maxima [A]  time = 0.934817, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{17} \, B b x^{\frac{17}{2}} + \frac{2}{11} \,{\left (B a + A b\right )} x^{\frac{11}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*b*x^(17/2) + 2/11*(B*a + A*b)*x^(11/2) + 2/5*A*a*x^(5/2)

________________________________________________________________________________________

Fricas [A]  time = 1.64699, size = 85, normalized size = 2.18 \begin{align*} \frac{2}{935} \,{\left (55 \, B b x^{8} + 85 \,{\left (B a + A b\right )} x^{5} + 187 \, A a x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/935*(55*B*b*x^8 + 85*(B*a + A*b)*x^5 + 187*A*a*x^2)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 5.73434, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 A b x^{\frac{11}{2}}}{11} + \frac{2 B a x^{\frac{11}{2}}}{11} + \frac{2 B b x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a*x**(5/2)/5 + 2*A*b*x**(11/2)/11 + 2*B*a*x**(11/2)/11 + 2*B*b*x**(17/2)/17

________________________________________________________________________________________

Giac [A]  time = 1.11139, size = 39, normalized size = 1. \begin{align*} \frac{2}{17} \, B b x^{\frac{17}{2}} + \frac{2}{11} \, B a x^{\frac{11}{2}} + \frac{2}{11} \, A b x^{\frac{11}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*b*x^(17/2) + 2/11*B*a*x^(11/2) + 2/11*A*b*x^(11/2) + 2/5*A*a*x^(5/2)