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

Optimal. Leaf size=11 \[ \frac {c \log (d+e x)}{e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {24, 21, 31} \[ \frac {c \log (d+e x)}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Log[d + e*x])/e

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ \frac {c \log (d+e x)}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Log[d + e*x])/e

________________________________________________________________________________________

fricas [A]  time = 1.15, size = 11, normalized size = 1.00 \[ \frac {c \log \left (e x + d\right )}{e} \]

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)/(e*x+d)^3,x, algorithm="fricas")

[Out]

c*log(e*x + d)/e

________________________________________________________________________________________

giac [A]  time = 0.18, size = 12, normalized size = 1.09 \[ c e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right ) \]

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)/(e*x+d)^3,x, algorithm="giac")

[Out]

c*e^(-1)*log(abs(x*e + d))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 12, normalized size = 1.09 \[ \frac {c \ln \left (e x +d \right )}{e} \]

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)/(e*x+d)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 11, normalized size = 1.00 \[ \frac {c \log \left (e x + d\right )}{e} \]

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)/(e*x+d)^3,x, algorithm="maxima")

[Out]

c*log(e*x + d)/e

________________________________________________________________________________________

mupad [B]  time = 0.43, size = 11, normalized size = 1.00 \[ \frac {c\,\ln \left (d+e\,x\right )}{e} \]

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)/(d + e*x)^3,x)

[Out]

(c*log(d + e*x))/e

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 8, normalized size = 0.73 \[ \frac {c \log {\left (d + e x \right )}}{e} \]

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)/(e*x+d)**3,x)

[Out]

c*log(d + e*x)/e

________________________________________________________________________________________