\(\int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx\) [2861]

Optimal result

Integrand size = 21, antiderivative size = 22 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {379, 266} \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \]

[In]

Int[(c + d*x)^2/(a + b*(c + d*x)^3),x]

[Out]

Log[a + b*(c + d*x)^3]/(3*b*d)

Rule 266

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

Rule 379

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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \]

[In]

Integrate[(c + d*x)^2/(a + b*(c + d*x)^3),x]

[Out]

Log[a + b*(c + d*x)^3]/(3*b*d)

Maple [A] (verified)

Time = 6.42 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\ln \left (a +b \left (d x +c \right )^{3}\right )}{3 b d}\)	\(21\)
default	\(\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\)	\(43\)
norman	\(\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\)	\(43\)
risch	\(\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\)	\(43\)
parallelrisch	\(\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\)	\(43\)

[In]

int((d*x+c)^2/(a+b*(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \]

[In]

integrate((d*x+c)^2/(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)/(b*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).

Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log {\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}}{3 b d} \]

[In]

integrate((d*x+c)**2/(a+b*(d*x+c)**3),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left ({\left (d x + c\right )}^{3} b + a\right )}{3 \, b d} \]

[In]

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

[Out]

1/3*log((d*x + c)^3*b + a)/(b*d)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left ({\left | {\left (d x + c\right )}^{3} b + a \right |}\right )}{3 \, b d} \]

[In]

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

[Out]

1/3*log(abs((d*x + c)^3*b + a))/(b*d)

Mupad [B] (verification not implemented)

Time = 5.55 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,b\,d} \]

[In]

int((c + d*x)^2/(a + b*(c + d*x)^3),x)

[Out]

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