3.2904 \(\int \frac{(c e+d e x)^2}{(a+b (c+d x)^3)^3} \, dx\)

Optimal. Leaf size=26 \[ -\frac{e^2}{6 b d \left (a+b (c+d x)^3\right )^2} \]

[Out]

-e^2/(6*b*d*(a + b*(c + d*x)^3)^2)

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Rubi [A]  time = 0.0205789, antiderivative size = 26, 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, 261} \[ -\frac{e^2}{6 b d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-e^2/(6*b*d*(a + b*(c + d*x)^3)^2)

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 261

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

Rubi steps

\begin{align*} \int \frac{(c e+d e x)^2}{\left (a+b (c+d x)^3\right )^3} \, dx &=\frac{e^2 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^2}{6 b d \left (a+b (c+d x)^3\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0129327, size = 26, normalized size = 1. \[ -\frac{e^2}{6 b d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-e^2/(6*b*d*(a + b*(c + d*x)^3)^2)

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Maple [A]  time = 0.003, size = 47, normalized size = 1.8 \begin{align*} -{\frac{{e}^{2}}{6\,bd \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}} \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)^3,x)

[Out]

-1/6*e^2/b/d/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2

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Maxima [B]  time = 1.00036, size = 184, normalized size = 7.08 \begin{align*} -\frac{e^{2}}{6 \,{\left (b^{3} d^{7} x^{6} + 6 \, b^{3} c d^{6} x^{5} + 15 \, b^{3} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, b^{3} c^{3} + a b^{2}\right )} d^{4} x^{3} + 3 \,{\left (5 \, b^{3} c^{4} + 2 \, a b^{2} c\right )} d^{3} x^{2} + 6 \,{\left (b^{3} c^{5} + a b^{2} c^{2}\right )} d^{2} x +{\left (b^{3} c^{6} + 2 \, a b^{2} c^{3} + a^{2} b\right )} d\right )}} \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)^3,x, algorithm="maxima")

[Out]

-1/6*e^2/(b^3*d^7*x^6 + 6*b^3*c*d^6*x^5 + 15*b^3*c^2*d^5*x^4 + 2*(10*b^3*c^3 + a*b^2)*d^4*x^3 + 3*(5*b^3*c^4 +
2*a*b^2*c)*d^3*x^2 + 6*(b^3*c^5 + a*b^2*c^2)*d^2*x + (b^3*c^6 + 2*a*b^2*c^3 + a^2*b)*d)

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Fricas [B]  time = 1.62632, size = 270, normalized size = 10.38 \begin{align*} -\frac{e^{2}}{6 \,{\left (b^{3} d^{7} x^{6} + 6 \, b^{3} c d^{6} x^{5} + 15 \, b^{3} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, b^{3} c^{3} + a b^{2}\right )} d^{4} x^{3} + 3 \,{\left (5 \, b^{3} c^{4} + 2 \, a b^{2} c\right )} d^{3} x^{2} + 6 \,{\left (b^{3} c^{5} + a b^{2} c^{2}\right )} d^{2} x +{\left (b^{3} c^{6} + 2 \, a b^{2} c^{3} + a^{2} b\right )} d\right )}} \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)^3,x, algorithm="fricas")

[Out]

-1/6*e^2/(b^3*d^7*x^6 + 6*b^3*c*d^6*x^5 + 15*b^3*c^2*d^5*x^4 + 2*(10*b^3*c^3 + a*b^2)*d^4*x^3 + 3*(5*b^3*c^4 +
2*a*b^2*c)*d^3*x^2 + 6*(b^3*c^5 + a*b^2*c^2)*d^2*x + (b^3*c^6 + 2*a*b^2*c^3 + a^2*b)*d)

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Sympy [B]  time = 20.058, size = 155, normalized size = 5.96 \begin{align*} - \frac{e^{2}}{6 a^{2} b d + 12 a b^{2} c^{3} d + 6 b^{3} c^{6} d + 90 b^{3} c^{2} d^{5} x^{4} + 36 b^{3} c d^{6} x^{5} + 6 b^{3} d^{7} x^{6} + x^{3} \left (12 a b^{2} d^{4} + 120 b^{3} c^{3} d^{4}\right ) + x^{2} \left (36 a b^{2} c d^{3} + 90 b^{3} c^{4} d^{3}\right ) + x \left (36 a b^{2} c^{2} d^{2} + 36 b^{3} c^{5} d^{2}\right )} \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)**3,x)

[Out]

-e**2/(6*a**2*b*d + 12*a*b**2*c**3*d + 6*b**3*c**6*d + 90*b**3*c**2*d**5*x**4 + 36*b**3*c*d**6*x**5 + 6*b**3*d
**7*x**6 + x**3*(12*a*b**2*d**4 + 120*b**3*c**3*d**4) + x**2*(36*a*b**2*c*d**3 + 90*b**3*c**4*d**3) + x*(36*a*
b**2*c**2*d**2 + 36*b**3*c**5*d**2))

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Giac [A]  time = 1.25637, size = 61, normalized size = 2.35 \begin{align*} -\frac{e^{2}}{6 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} 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)^3,x, algorithm="giac")

[Out]

-1/6*e^2/((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)^2*b*d)