\(\int \frac {a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^3} \, dx\) [1835]

Optimal result

Integrand size = 33, antiderivative size = 33 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=-\frac {a-\frac {c d^2}{e^2}}{d+e x}+\frac {c d \log (d+e x)}{e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {24, 45} \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d \log (d+e x)}{e^2}-\frac {a-\frac {c d^2}{e^2}}{d+e x} \]

[In]

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

[Out]

-((a - (c*d^2)/e^2)/(d + e*x)) + (c*d*Log[d + e*x])/e^2

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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {a e^3+c d e^2 x}{(d+e x)^2} \, dx}{e^2} \\ & = \frac {\int \left (\frac {-c d^2 e+a e^3}{(d+e x)^2}+\frac {c d e}{d+e x}\right ) \, dx}{e^2} \\ & = -\frac {a-\frac {c d^2}{e^2}}{d+e x}+\frac {c d \log (d+e x)}{e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d^2-a e^2}{e^2 (d+e x)}+\frac {c d \log (d+e x)}{e^2} \]

[In]

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

[Out]

(c*d^2 - a*e^2)/(e^2*(d + e*x)) + (c*d*Log[d + e*x])/e^2

Maple [A] (verified)

Time = 2.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15

method	result	size
default	\(-\frac {e^{2} a -c \,d^{2}}{e^{2} \left (e x +d \right )}+\frac {c d \ln \left (e x +d \right )}{e^{2}}\)	\(38\)
risch	\(-\frac {a}{e x +d}+\frac {c \,d^{2}}{e^{2} \left (e x +d \right )}+\frac {c d \ln \left (e x +d \right )}{e^{2}}\)	\(39\)
parallelrisch	\(\frac {\ln \left (e x +d \right ) x c d e +\ln \left (e x +d \right ) c \,d^{2}-e^{2} a +c \,d^{2}}{e^{2} \left (e x +d \right )}\)	\(46\)
norman	\(\frac {\frac {\left (a d \,e^{2}-d^{3} c \right ) x^{2}}{d^{2}}+\frac {\left (a d \,e^{2}-d^{3} c \right ) x}{d e}}{\left (e x +d \right )^{2}}+\frac {c d \ln \left (e x +d \right )}{e^{2}}\)	\(64\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d^{2} - a e^{2} + {\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{e^{3} x + d e^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d \log {\left (d + e x \right )}}{e^{2}} + \frac {- a e^{2} + c d^{2}}{d e^{2} + e^{3} x} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d \log \left (e x + d\right )}{e^{2}} + \frac {c d^{2} - a e^{2}}{e^{3} x + d e^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d \log \left ({\left | e x + d \right |}\right )}{e^{2}} + \frac {c d^{2} - a e^{2}}{{\left (e x + d\right )} e^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.81 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c\,d\,\ln \left (d+e\,x\right )}{e^2}-\frac {a\,e^2-c\,d^2}{e^2\,\left (d+e\,x\right )} \]

[In]

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

[Out]

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