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

Optimal. Leaf size=28 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{5 x^5}+b B x \]

[Out]

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

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Rubi [A]  time = 0.0164566, antiderivative size = 28, 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} \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{5 x^5}+b B x \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)*(A + B*x^3))/x^6,x]

[Out]

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

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

Mathematica [A]  time = 0.0111974, size = 30, normalized size = 1.07 \[ \frac{-a B-A b}{2 x^2}-\frac{a A}{5 x^5}+b B x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 25, normalized size = 0.9 \begin{align*} bBx-{\frac{Ab+Ba}{2\,{x}^{2}}}-{\frac{Aa}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.33122, size = 36, normalized size = 1.29 \begin{align*} B b x - \frac{5 \,{\left (B a + A b\right )} x^{3} + 2 \, A a}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.40485, size = 69, normalized size = 2.46 \begin{align*} \frac{10 \, B b x^{6} - 5 \,{\left (B a + A b\right )} x^{3} - 2 \, A a}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.408755, size = 27, normalized size = 0.96 \begin{align*} B b x - \frac{2 A a + x^{3} \left (5 A b + 5 B a\right )}{10 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13208, size = 39, normalized size = 1.39 \begin{align*} B b x - \frac{5 \, B a x^{3} + 5 \, A b x^{3} + 2 \, A a}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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