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

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

[Out]

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

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Rubi [A]  time = 0.0173916, 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} \[ -\frac{a B+A b}{x}-\frac{a A}{4 x^4}+\frac{1}{2} b B x^2 \]

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*}

Mathematica [A]  time = 0.0118031, size = 32, normalized size = 1.03 \[ \frac{-a B-A b}{x}-\frac{a A}{4 x^4}+\frac{1}{2} b B x^2 \]

Antiderivative was successfully verified.

[In]

Integrate[((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

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Maple [A]  time = 0.006, size = 28, normalized size = 0.9 \begin{align*}{\frac{bB{x}^{2}}{2}}-{\frac{Aa}{4\,{x}^{4}}}-{\frac{Ab+Ba}{x}} \end{align*}

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 = 1.30636, size = 39, normalized size = 1.26 \begin{align*} \frac{1}{2} \, B b x^{2} - \frac{4 \,{\left (B a + A b\right )} x^{3} + A a}{4 \, x^{4}} \end{align*}

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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Fricas [A]  time = 1.37449, size = 63, normalized size = 2.03 \begin{align*} \frac{2 \, B b x^{6} - 4 \,{\left (B a + A b\right )} x^{3} - A a}{4 \, x^{4}} \end{align*}

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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Sympy [A]  time = 0.378447, size = 29, normalized size = 0.94 \begin{align*} \frac{B b x^{2}}{2} - \frac{A a + x^{3} \left (4 A b + 4 B a\right )}{4 x^{4}} \end{align*}

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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Giac [A]  time = 1.20865, size = 42, normalized size = 1.35 \begin{align*} \frac{1}{2} \, B b x^{2} - \frac{4 \, B a x^{3} + 4 \, A b x^{3} + A a}{4 \, x^{4}} \end{align*}

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