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

Optimal. Leaf size=39 \[ \frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{9} b B x^{9/2} \]

[Out]

(-2*a*A)/(3*x^(3/2)) + (2*(A*b + a*B)*x^(3/2))/3 + (2*b*B*x^(9/2))/9

________________________________________________________________________________________

Rubi [A]  time = 0.0150596, 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} \[ \frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{9} b B x^{9/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*A)/(3*x^(3/2)) + (2*(A*b + a*B)*x^(3/2))/3 + (2*b*B*x^(9/2))/9

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

Mathematica [A]  time = 0.0111206, size = 34, normalized size = 0.87 \[ \frac{2 \left (-3 a A+3 a B x^3+3 A b x^3+b B x^6\right )}{9 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-3*a*A + 3*A*b*x^3 + 3*a*B*x^3 + b*B*x^6))/(9*x^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \begin{align*} -{\frac{-2\,bB{x}^{6}-6\,A{x}^{3}b-6\,B{x}^{3}a+6\,Aa}{9}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*(-B*b*x^6-3*A*b*x^3-3*B*a*x^3+3*A*a)/x^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 0.936978, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{3} \,{\left (B a + A b\right )} x^{\frac{3}{2}} - \frac{2 \, A a}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*B*b*x^(9/2) + 2/3*(B*a + A*b)*x^(3/2) - 2/3*A*a/x^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.71192, size = 69, normalized size = 1.77 \begin{align*} \frac{2 \,{\left (B b x^{6} + 3 \,{\left (B a + A b\right )} x^{3} - 3 \, A a\right )}}{9 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(B*b*x^6 + 3*(B*a + A*b)*x^3 - 3*A*a)/x^(3/2)

________________________________________________________________________________________

Sympy [A]  time = 3.10627, size = 46, normalized size = 1.18 \begin{align*} - \frac{2 A a}{3 x^{\frac{3}{2}}} + \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 B a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a/(3*x**(3/2)) + 2*A*b*x**(3/2)/3 + 2*B*a*x**(3/2)/3 + 2*B*b*x**(9/2)/9

________________________________________________________________________________________

Giac [A]  time = 1.12935, size = 39, normalized size = 1. \begin{align*} \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} - \frac{2 \, A a}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*B*b*x^(9/2) + 2/3*B*a*x^(3/2) + 2/3*A*b*x^(3/2) - 2/3*A*a/x^(3/2)