\(\int \frac {(d+e x)^2}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx\) [1868]

Optimal result

Integrand size = 35, antiderivative size = 38 \[ \int \frac {(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {e x}{c d}+\frac {\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int \frac {(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac {e x}{c d} \]

[In]

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

[Out]

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

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])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {c d e x+\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.39 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {e x}{c d}+\frac {\left (-e^{2} a +c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{2} d^{2}}\)	\(39\)
norman	\(\frac {e x}{c d}-\frac {\left (e^{2} a -c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{2} d^{2}}\)	\(40\)
risch	\(\frac {e x}{c d}-\frac {\ln \left (c d x +a e \right ) e^{2} a}{c^{2} d^{2}}+\frac {\ln \left (c d x +a e \right )}{c}\)	\(45\)
parallelrisch	\(-\frac {\ln \left (c d x +a e \right ) a \,e^{2}-\ln \left (c d x +a e \right ) c \,d^{2}-x c d e}{c^{2} d^{2}}\)	\(45\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {c d e x + {\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {e x}{c d} - \frac {\left (a e^{2} - c d^{2}\right ) \log {\left (a e + c d x \right )}}{c^{2} d^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {e x}{c d} + \frac {{\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {e x}{c d} + \frac {{\left (c d^{2} - a e^{2}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{2} d^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {e\,x}{c\,d}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (a\,e^2-c\,d^2\right )}{c^2\,d^2} \]

[In]

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

[Out]

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