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]

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

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Rubi [A]  time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 1.00 \[ \frac {\log (c+d x)}{a d}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a d} \]

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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fricas [A]  time = 0.61, size = 51, normalized size = 1.42 \[ -\frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, \log \left (d x + c\right )}{3 \, a d} \]

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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giac [A]  time = 0.21, size = 58, normalized size = 1.61 \[ -\frac {\log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a d} + \frac {\log \left ({\left | d x + c \right |}\right )}{a d} \]

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)

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maple [A]  time = 0.01, size = 57, normalized size = 1.58 \[ \frac {\ln \left (d x +c \right )}{a d}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 a d} \]

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 = 0.55, size = 56, normalized size = 1.56 \[ -\frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a d} + \frac {\log \left (d x + c\right )}{a d} \]

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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mupad [B]  time = 1.28, size = 56, normalized size = 1.56 \[ \frac {\ln \left (c+d\,x\right )}{a\,d}-\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.47, size = 49, normalized size = 1.36 \[ \frac {\log {\left (\frac {c}{d} + x \right )}}{a d} - \frac {\log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a d} \]

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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