3.19 \(\int \frac{(a+b x^3)^2 (A+B x^3)}{x^6} \, dx\)

Optimal. Leaf size=50 \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac{1}{4} b^2 B x^4 \]

[Out]

-(a^2*A)/(5*x^5) - (a*(2*A*b + a*B))/(2*x^2) + b*(A*b + 2*a*B)*x + (b^2*B*x^4)/4

________________________________________________________________________________________

Rubi [A]  time = 0.0283431, antiderivative size = 50, 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} \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac{1}{4} b^2 B x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^2*(A + B*x^3))/x^6,x]

[Out]

-(a^2*A)/(5*x^5) - (a*(2*A*b + a*B))/(2*x^2) + b*(A*b + 2*a*B)*x + (b^2*B*x^4)/4

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 )^2 \left (A+B x^3\right )}{x^6} \, dx &=\int \left (b (A b+2 a B)+\frac{a^2 A}{x^6}+\frac{a (2 A b+a B)}{x^3}+b^2 B x^3\right ) \, dx\\ &=-\frac{a^2 A}{5 x^5}-\frac{a (2 A b+a B)}{2 x^2}+b (A b+2 a B) x+\frac{1}{4} b^2 B x^4\\ \end{align*}

Mathematica [A]  time = 0.0182721, size = 50, normalized size = 1. \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac{1}{4} b^2 B x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^2*(A + B*x^3))/x^6,x]

[Out]

-(a^2*A)/(5*x^5) - (a*(2*A*b + a*B))/(2*x^2) + b*(A*b + 2*a*B)*x + (b^2*B*x^4)/4

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 46, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}B{x}^{4}}{4}}+{b}^{2}Ax+2\,abBx-{\frac{a \left ( 2\,Ab+Ba \right ) }{2\,{x}^{2}}}-{\frac{A{a}^{2}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*(B*x^3+A)/x^6,x)

[Out]

1/4*b^2*B*x^4+b^2*A*x+2*a*b*B*x-1/2*a*(2*A*b+B*a)/x^2-1/5*a^2*A/x^5

________________________________________________________________________________________

Maxima [A]  time = 1.28127, size = 69, normalized size = 1.38 \begin{align*} \frac{1}{4} \, B b^{2} x^{4} +{\left (2 \, B a b + A b^{2}\right )} x - \frac{5 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} + 2 \, A a^{2}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^6,x, algorithm="maxima")

[Out]

1/4*B*b^2*x^4 + (2*B*a*b + A*b^2)*x - 1/10*(5*(B*a^2 + 2*A*a*b)*x^3 + 2*A*a^2)/x^5

________________________________________________________________________________________

Fricas [A]  time = 1.69, size = 119, normalized size = 2.38 \begin{align*} \frac{5 \, B b^{2} x^{9} + 20 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} - 10 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} - 4 \, A a^{2}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^6,x, algorithm="fricas")

[Out]

1/20*(5*B*b^2*x^9 + 20*(2*B*a*b + A*b^2)*x^6 - 10*(B*a^2 + 2*A*a*b)*x^3 - 4*A*a^2)/x^5

________________________________________________________________________________________

Sympy [A]  time = 0.470905, size = 51, normalized size = 1.02 \begin{align*} \frac{B b^{2} x^{4}}{4} + x \left (A b^{2} + 2 B a b\right ) - \frac{2 A a^{2} + x^{3} \left (10 A a b + 5 B a^{2}\right )}{10 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*(B*x**3+A)/x**6,x)

[Out]

B*b**2*x**4/4 + x*(A*b**2 + 2*B*a*b) - (2*A*a**2 + x**3*(10*A*a*b + 5*B*a**2))/(10*x**5)

________________________________________________________________________________________

Giac [A]  time = 1.12228, size = 69, normalized size = 1.38 \begin{align*} \frac{1}{4} \, B b^{2} x^{4} + 2 \, B a b x + A b^{2} x - \frac{5 \, B a^{2} x^{3} + 10 \, A a b x^{3} + 2 \, A a^{2}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^6,x, algorithm="giac")

[Out]

1/4*B*b^2*x^4 + 2*B*a*b*x + A*b^2*x - 1/10*(5*B*a^2*x^3 + 10*A*a*b*x^3 + 2*A*a^2)/x^5