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

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

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {e^2}{3 b d \left (a+b (c+d x)^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-e^2/(3*b*d*(a + b*(c + d*x)^3))

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]

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

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} -\frac {e^2}{3 b d \left (a+b (c+d x)^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*e^2/(b*d*(a + b*(c + d*x)^3))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c e+d e x)^2}{\left (a+b (c+d x)^3\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.20, size = 55, normalized size = 2.12 \begin {gather*} -\frac {e^{2}}{3 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + {\left (b^{2} c^{3} + a b\right )} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*e^2/(b^2*d^4*x^3 + 3*b^2*c*d^3*x^2 + 3*b^2*c^2*d^2*x + (b^2*c^3 + a*b)*d)

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giac [A]  time = 0.19, size = 45, normalized size = 1.73 \begin {gather*} -\frac {e^{2}}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*e^2/((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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maple [A]  time = 0.00, size = 47, normalized size = 1.81 \begin {gather*} -\frac {e^{2}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*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)

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maxima [B]  time = 0.55, size = 55, normalized size = 2.12 \begin {gather*} -\frac {e^{2}}{3 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + {\left (b^{2} c^{3} + a b\right )} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*e^2/(b^2*d^4*x^3 + 3*b^2*c*d^3*x^2 + 3*b^2*c^2*d^2*x + (b^2*c^3 + a*b)*d)

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mupad [B]  time = 1.25, size = 46, normalized size = 1.77 \begin {gather*} -\frac {e^2}{3\,b\,d\,\left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 1.37, size = 60, normalized size = 2.31 \begin {gather*} - \frac {e^{2}}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e**2/(3*a*b*d + 3*b**2*c**3*d + 9*b**2*c**2*d**2*x + 9*b**2*c*d**3*x**2 + 3*b**2*d**4*x**3)

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