3.2888 \(\int \frac{(c e+d e x)^2}{a+b (c+d x)^3} \, dx\)

Optimal. Leaf size=25 \[ \frac{e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]

[Out]

(e^2*Log[a + b*(c + d*x)^3])/(3*b*d)

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Rubi [A]  time = 0.0201235, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {372, 260} \[ \frac{e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]

Antiderivative was successfully verified.

[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 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 260

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]

Rubi steps

\begin{align*} \int \frac{(c e+d e x)^2}{a+b (c+d x)^3} \, dx &=\frac{e^2 \operatorname{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]  time = 0.008426, size = 25, normalized size = 1. \[ \frac{e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.002, size = 46, normalized size = 1.8 \begin{align*}{\frac{{e}^{2}\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]  time = 0.972769, size = 61, normalized size = 2.44 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.56174, size = 97, normalized size = 3.88 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 0.447674, size = 46, normalized size = 1.84 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.1769, size = 61, normalized size = 2.44 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)