3.994 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^5} \, dx\)

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

[Out]

c^2*ln(e*x+d)/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} \[ \frac {c^2 \log (d+e x)}{e} \]

Antiderivative was successfully verified.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^5,x]

[Out]

(c^2*Log[d + e*x])/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 {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^5} \, dx &=\int \frac {c^2}{d+e x} \, dx\\ &=c^2 \int \frac {1}{d+e x} \, dx\\ &=\frac {c^2 \log (d+e x)}{e}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^5,x]

[Out]

(c^2*Log[d + e*x])/e

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fricas [A]  time = 1.09, size = 13, normalized size = 1.00 \[ \frac {c^{2} \log \left (e x + d\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^5,x, algorithm="fricas")

[Out]

c^2*log(e*x + d)/e

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giac [A]  time = 0.17, size = 26, normalized size = 2.00 \[ -c^{2} e^{\left (-1\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 14, normalized size = 1.08 \[ \frac {c^{2} \ln \left (e x +d \right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^5,x)

[Out]

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

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maxima [A]  time = 1.33, size = 13, normalized size = 1.00 \[ \frac {c^{2} \log \left (e x + d\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^5,x, algorithm="maxima")

[Out]

c^2*log(e*x + d)/e

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mupad [B]  time = 0.02, size = 13, normalized size = 1.00 \[ \frac {c^2\,\ln \left (d+e\,x\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^2/(d + e*x)^5,x)

[Out]

(c^2*log(d + e*x))/e

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sympy [A]  time = 0.12, size = 10, normalized size = 0.77 \[ \frac {c^{2} \log {\left (d + e x \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**2/(e*x+d)**5,x)

[Out]

c**2*log(d + e*x)/e

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