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

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

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a B+A b}{2 x^2}-\frac {a A}{5 x^5}+b B x \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 1.07 \begin {gather*} \frac {-a B-A b}{2 x^2}-\frac {a A}{5 x^5}+b B x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^3\right ) \left (A+B x^3\right )}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.74, size = 29, normalized size = 1.04 \begin {gather*} \frac {10 \, B b x^{6} - 5 \, {\left (B a + A b\right )} x^{3} - 2 \, A a}{10 \, x^{5}} \end {gather*}

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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giac [A]  time = 0.18, size = 29, normalized size = 1.04 \begin {gather*} B b x - \frac {5 \, B a x^{3} + 5 \, A b x^{3} + 2 \, A a}{10 \, x^{5}} \end {gather*}

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

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maple [A]  time = 0.04, size = 25, normalized size = 0.89 \begin {gather*} B b x -\frac {A b +B a}{2 x^{2}}-\frac {A a}{5 x^{5}} \end {gather*}

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

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maxima [A]  time = 0.49, size = 27, normalized size = 0.96 \begin {gather*} B b x - \frac {5 \, {\left (B a + A b\right )} x^{3} + 2 \, A a}{10 \, x^{5}} \end {gather*}

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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mupad [B]  time = 2.32, size = 28, normalized size = 1.00 \begin {gather*} B\,b\,x-\frac {\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x^3+\frac {A\,a}{5}}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3))/x^6,x)

[Out]

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

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sympy [A]  time = 0.29, size = 29, normalized size = 1.04 \begin {gather*} B b x + \frac {- 2 A a + x^{3} \left (- 5 A b - 5 B a\right )}{10 x^{5}} \end {gather*}

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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