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

Optimal. Leaf size=31 \[ -\frac {a B+A b}{x}-\frac {a A}{4 x^4}+\frac {1}{2} b B x^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 31, 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}{x}-\frac {a A}{4 x^4}+\frac {1}{2} b B x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)*(A + B*x^3))/x^5,x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.03 \begin {gather*} \frac {-a B-A b}{x}-\frac {a A}{4 x^4}+\frac {1}{2} b B x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.29, size = 29, normalized size = 0.94 \begin {gather*} \frac {2 \, B b x^{6} - 4 \, {\left (B a + A b\right )} x^{3} - A a}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 31, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, B b x^{2} - \frac {4 \, B a x^{3} + 4 \, A b x^{3} + A a}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 28, normalized size = 0.90 \begin {gather*} \frac {B b \,x^{2}}{2}-\frac {A b +B a}{x}-\frac {A a}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 29, normalized size = 0.94 \begin {gather*} \frac {1}{2} \, B b x^{2} - \frac {4 \, {\left (B a + A b\right )} x^{3} + A a}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 29, normalized size = 0.94 \begin {gather*} \frac {B\,b\,x^2}{2}-\frac {\left (A\,b+B\,a\right )\,x^3+\frac {A\,a}{4}}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3))/x^5,x)

[Out]

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

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sympy [A]  time = 0.25, size = 31, normalized size = 1.00 \begin {gather*} \frac {B b x^{2}}{2} + \frac {- A a + x^{3} \left (- 4 A b - 4 B a\right )}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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