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

Optimal. Leaf size=31 \[ \frac{1}{2} x^2 (a B+A b)-\frac{a A}{x}+\frac{1}{5} b B x^5 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0153374, antiderivative size = 31, 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 = {448} \[ \frac{1}{2} x^2 (a B+A b)-\frac{a A}{x}+\frac{1}{5} b B x^5 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)*(A + B*x^3))/x^2,x]

[Out]

-((a*A)/x) + ((A*b + a*B)*x^2)/2 + (b*B*x^5)/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 \frac{\left (a+b x^3\right ) \left (A+B x^3\right )}{x^2} \, dx &=\int \left (\frac{a A}{x^2}+(A b+a B) x+b B x^4\right ) \, dx\\ &=-\frac{a A}{x}+\frac{1}{2} (A b+a B) x^2+\frac{1}{5} b B x^5\\ \end{align*}

Mathematica [A]  time = 0.0099321, size = 31, normalized size = 1. \[ \frac{1}{2} x^2 (a B+A b)-\frac{a A}{x}+\frac{1}{5} b B x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 30, normalized size = 1. \begin{align*}{\frac{bB{x}^{5}}{5}}+{\frac{A{x}^{2}b}{2}}+{\frac{B{x}^{2}a}{2}}-{\frac{Aa}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*B*x^5+1/2*A*x^2*b+1/2*B*x^2*a-a*A/x

________________________________________________________________________________________

Maxima [A]  time = 1.15168, size = 36, normalized size = 1.16 \begin{align*} \frac{1}{5} \, B b x^{5} + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} - \frac{A a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*b*x^5 + 1/2*(B*a + A*b)*x^2 - A*a/x

________________________________________________________________________________________

Fricas [A]  time = 1.38473, size = 66, normalized size = 2.13 \begin{align*} \frac{2 \, B b x^{6} + 5 \,{\left (B a + A b\right )} x^{3} - 10 \, A a}{10 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(2*B*b*x^6 + 5*(B*a + A*b)*x^3 - 10*A*a)/x

________________________________________________________________________________________

Sympy [A]  time = 0.250824, size = 26, normalized size = 0.84 \begin{align*} - \frac{A a}{x} + \frac{B b x^{5}}{5} + x^{2} \left (\frac{A b}{2} + \frac{B a}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.20218, size = 39, normalized size = 1.26 \begin{align*} \frac{1}{5} \, B b x^{5} + \frac{1}{2} \, B a x^{2} + \frac{1}{2} \, A b x^{2} - \frac{A a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*b*x^5 + 1/2*B*a*x^2 + 1/2*A*b*x^2 - A*a/x