3.2864 \(\int \frac{1}{(c+d x) (a+b (c+d x)^3)} \, dx\)

Optimal. Leaf size=36 \[ \frac{\log (c+d x)}{a d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d} \]

[Out]

Log[c + d*x]/(a*d) - Log[a + b*(c + d*x)^3]/(3*a*d)

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Rubi [A]  time = 0.0277494, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {372, 266, 36, 29, 31} \[ \frac{\log (c+d x)}{a d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)*(a + b*(c + d*x)^3)),x]

[Out]

Log[c + d*x]/(a*d) - Log[a + b*(c + d*x)^3]/(3*a*d)

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 266

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0085122, size = 36, normalized size = 1. \[ \frac{\log (c+d x)}{a d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + d*x)*(a + b*(c + d*x)^3)),x]

[Out]

Log[c + d*x]/(a*d) - Log[a + b*(c + d*x)^3]/(3*a*d)

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Maple [A]  time = 0.005, size = 57, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{ad}}-{\frac{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,ad}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)/(a+b*(d*x+c)^3),x)

[Out]

ln(d*x+c)/a/d-1/3/a/d*ln(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)

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Maxima [A]  time = 1.30615, size = 76, normalized size = 2.11 \begin{align*} -\frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a d} + \frac{\log \left (d x + c\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+b*(d*x+c)^3),x, algorithm="maxima")

[Out]

-1/3*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)/(a*d) + log(d*x + c)/(a*d)

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Fricas [A]  time = 1.44372, size = 119, normalized size = 3.31 \begin{align*} -\frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, \log \left (d x + c\right )}{3 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+b*(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/3*(log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a) - 3*log(d*x + c))/(a*d)

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Sympy [A]  time = 0.441534, size = 49, normalized size = 1.36 \begin{align*} \frac{\log{\left (\frac{c}{d} + x \right )}}{a d} - \frac{\log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+b*(d*x+c)**3),x)

[Out]

log(c/d + x)/(a*d) - log(3*c**2*x/d**2 + 3*c*x**2/d + x**3 + (a + b*c**3)/(b*d**3))/(3*a*d)

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Giac [A]  time = 1.12358, size = 78, normalized size = 2.17 \begin{align*} -\frac{\log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a d} + \frac{\log \left ({\left | d x + c \right |}\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+b*(d*x+c)^3),x, algorithm="giac")

[Out]

-1/3*log(abs(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a))/(a*d) + log(abs(d*x + c))/(a*d)