\(\int \frac {(c e+d e x)^2}{a+b (c+d x)^3} \, dx\) [2888]

Optimal result

Integrand size = 24, antiderivative size = 25 \[ \int \frac {(c e+d e x)^2}{a+b (c+d x)^3} \, dx=\frac {e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {379, 266} \[ \int \frac {(c e+d e x)^2}{a+b (c+d x)^3} \, dx=\frac {e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(c e+d e x)^2}{a+b (c+d x)^3} \, dx=\frac {e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]

[In]

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

[Out]

(e^2*Log[a + b*(c + d*x)^3])/(3*b*d)

Maple [A] (verified)

Time = 4.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84

method	result	size
default	\(\frac {e^{2} \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\)	\(46\)
norman	\(\frac {e^{2} \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\)	\(46\)
risch	\(\frac {e^{2} \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\)	\(46\)
parallelrisch	\(\frac {e^{2} \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\)	\(46\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {(c e+d e x)^2}{a+b (c+d x)^3} \, dx=\frac {e^{2} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \]

[In]

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

[Out]

1/3*e^2*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. 46 vs. \(2 (19) = 38\).

Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {(c e+d e x)^2}{a+b (c+d x)^3} \, dx=\frac {e^{2} \log {\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}}{3 b d} \]

[In]

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

[Out]

e**2*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 = 45, normalized size of antiderivative = 1.80 \[ \int \frac {(c e+d e x)^2}{a+b (c+d x)^3} \, dx=\frac {e^{2} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \]

[In]

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

[Out]

1/3*e^2*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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {(c e+d e x)^2}{a+b (c+d x)^3} \, dx=\frac {e^{2} \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, b d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.54 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {(c e+d e x)^2}{a+b (c+d x)^3} \, dx=\frac {e^2\,\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,b\,d} \]

[In]

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

[Out]

(e^2*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)