3.11.11 \(\int \frac {(d+e x)^3}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx\) [1011]

Optimal. Leaf size=13 \[ \frac {\log (d+e x)}{c^2 e} \]

[Out]

ln(e*x+d)/c^2/e

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 31} \begin {gather*} \frac {\log (d+e x)}{c^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[d + e*x]/(c^2*e)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 31

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac {1}{c^2 (d+e x)} \, dx\\ &=\frac {\int \frac {1}{d+e x} \, dx}{c^2}\\ &=\frac {\log (d+e x)}{c^2 e}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {\log (d+e x)}{c^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[d + e*x]/(c^2*e)

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Maple [A]
time = 0.58, size = 14, normalized size = 1.08

method	result	size
default	\(\frac {\ln \left (e x +d \right )}{c^{2} e}\)	\(14\)
norman	\(\frac {\ln \left (e x +d \right )}{c^{2} e}\)	\(14\)
risch	\(\frac {\ln \left (e x +d \right )}{c^{2} e}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(e*x+d)/c^2/e

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Maxima [A]
time = 0.30, size = 13, normalized size = 1.00 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (x e + d\right )}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-1)*log(x*e + d)/c^2

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Fricas [A]
time = 2.43, size = 13, normalized size = 1.00 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (x e + d\right )}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-1)*log(x*e + d)/c^2

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Sympy [A]
time = 0.03, size = 17, normalized size = 1.31 \begin {gather*} \frac {\log {\left (c^{2} d + c^{2} e x \right )}}{c^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c**2*d + c**2*e*x)/(c**2*e)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (13) = 26\).
time = 0.71, size = 130, normalized size = 10.00 \begin {gather*} -\frac {d^{2} e^{\left (-1\right )}}{2 \, {\left (c d^{2} + {\left (x^{2} e + 2 \, d x\right )} c e\right )} c} + \frac {\frac {d^{2} e^{\left (-1\right )}}{c d^{2} + {\left (x^{2} e + 2 \, d x\right )} c e} - \frac {e^{\left (-1\right )} \log \left (\frac {{\left | c d^{2} + {\left (x^{2} e + 2 \, d x\right )} c e \right |} e^{\left (-1\right )}}{2 \, {\left (c d^{2} + {\left (x^{2} e + 2 \, d x\right )} c e\right )}^{2} {\left | c \right |}}\right )}{c}}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d^2*e^(-1)/((c*d^2 + (x^2*e + 2*d*x)*c*e)*c) + 1/2*(d^2*e^(-1)/(c*d^2 + (x^2*e + 2*d*x)*c*e) - e^(-1)*log
(1/2*abs(c*d^2 + (x^2*e + 2*d*x)*c*e)*e^(-1)/((c*d^2 + (x^2*e + 2*d*x)*c*e)^2*abs(c)))/c)/c

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Mupad [B]
time = 0.04, size = 13, normalized size = 1.00 \begin {gather*} \frac {\ln \left (d+e\,x\right )}{c^2\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(d + e*x)/(c^2*e)

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