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

Optimal. Leaf size=51 \[ -\frac{A \log \left (a+b x^3\right )}{3 a^2}+\frac{A \log (x)}{a^2}+\frac{A b-a B}{3 a b \left (a+b x^3\right )} \]

[Out]

(A*b - a*B)/(3*a*b*(a + b*x^3)) + (A*Log[x])/a^2 - (A*Log[a + b*x^3])/(3*a^2)

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Rubi [A]  time = 0.0458923, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ -\frac{A \log \left (a+b x^3\right )}{3 a^2}+\frac{A \log (x)}{a^2}+\frac{A b-a B}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^3)/(x*(a + b*x^3)^2),x]

[Out]

(A*b - a*B)/(3*a*b*(a + b*x^3)) + (A*Log[x])/a^2 - (A*Log[a + b*x^3])/(3*a^2)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{A+B x^3}{x \left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A}{a^2 x}+\frac{-A b+a B}{a (a+b x)^2}-\frac{A b}{a^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{A b-a B}{3 a b \left (a+b x^3\right )}+\frac{A \log (x)}{a^2}-\frac{A \log \left (a+b x^3\right )}{3 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0302738, size = 46, normalized size = 0.9 \[ \frac{\frac{a (A b-a B)}{b \left (a+b x^3\right )}-A \log \left (a+b x^3\right )+3 A \log (x)}{3 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(A*b - a*B))/(b*(a + b*x^3)) + 3*A*Log[x] - A*Log[a + b*x^3])/(3*a^2)

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Maple [A]  time = 0.014, size = 53, normalized size = 1. \begin{align*}{\frac{A}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{B}{3\,b \left ( b{x}^{3}+a \right ) }}-{\frac{A\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2}}}+{\frac{A\ln \left ( x \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03537, size = 69, normalized size = 1.35 \begin{align*} -\frac{B a - A b}{3 \,{\left (a b^{2} x^{3} + a^{2} b\right )}} - \frac{A \log \left (b x^{3} + a\right )}{3 \, a^{2}} + \frac{A \log \left (x^{3}\right )}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.712, size = 151, normalized size = 2.96 \begin{align*} -\frac{B a^{2} - A a b +{\left (A b^{2} x^{3} + A a b\right )} \log \left (b x^{3} + a\right ) - 3 \,{\left (A b^{2} x^{3} + A a b\right )} \log \left (x\right )}{3 \,{\left (a^{2} b^{2} x^{3} + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.15689, size = 46, normalized size = 0.9 \begin{align*} \frac{A \log{\left (x \right )}}{a^{2}} - \frac{A \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2}} - \frac{- A b + B a}{3 a^{2} b + 3 a b^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11872, size = 82, normalized size = 1.61 \begin{align*} -\frac{A \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{A \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{A b^{2} x^{3} - B a^{2} + 2 \, A a b}{3 \,{\left (b x^{3} + a\right )} a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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