\(\int (d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^{-m} \, dx\) [2105]

Optimal result

Integrand size = 37, antiderivative size = 54 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\frac {(d+e x)^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (1-m)} \]

[Out]

(e*x+d)^(-1+m)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1-m)/c/d/(1-m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {662} \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\frac {(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (1-m)} \]

[In]

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

[Out]

((d + e*x)^(-1 + m)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c*d*(1 - m))

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (1-m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {(d+e x)^{-1+m} ((a e+c d x) (d+e x))^{1-m}}{c d (-1+m)} \]

[In]

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

[Out]

-(((d + e*x)^(-1 + m)*((a*e + c*d*x)*(d + e*x))^(1 - m))/(c*d*(-1 + m)))

Maple [A] (verified)

Time = 3.54 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06

method	result	size
gosper	\(-\frac {\left (c d x +a e \right ) \left (e x +d \right )^{m} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{c d \left (-1+m \right )}\)	\(57\)
parallelrisch	\(\frac {\left (-x \left (e x +d \right )^{m} c d e -\left (e x +d \right )^{m} a \,e^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{c d e \left (-1+m \right )}\)	\(71\)
norman	\(\left (-\frac {x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{-1+m}-\frac {a e \,{\mathrm e}^{m \ln \left (e x +d \right )}}{c d \left (-1+m \right )}\right ) {\mathrm e}^{-m \ln \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}\)	\(75\)
risch	\(-\frac {\left (c d x +a e \right ) \left (c d x +a e \right )^{-m} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i \left (c d x +a e \right ) \left (e x +d \right )\right ) m \left (-\operatorname {csgn}\left (i \left (c d x +a e \right ) \left (e x +d \right )\right )+\operatorname {csgn}\left (i \left (c d x +a e \right )\right )\right ) \left (-\operatorname {csgn}\left (i \left (c d x +a e \right ) \left (e x +d \right )\right )+\operatorname {csgn}\left (i \left (e x +d \right )\right )\right )}{2}}}{c d \left (-1+m \right )}\)	\(118\)

[In]

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

[Out]

-(c*d*x+a*e)/c/d/(-1+m)*(e*x+d)^m/((c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^m)

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {{\left (c d x + a e\right )} {\left (e x + d\right )}^{m}}{{\left (c d m - c d\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> Invalid NaN comparison

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.61 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {c d x + a e}{{\left (c d x + a e\right )}^{m} c d {\left (m - 1\right )}} \]

[In]

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

[Out]

-(c*d*x + a*e)/((c*d*x + a*e)^m*c*d*(m - 1))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.50 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {{\left (e x + d\right )}^{m} c d x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + {\left (e x + d\right )}^{m} a e e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )}}{c d m - c d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.92 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {\left (a\,e+c\,d\,x\right )\,{\left (d+e\,x\right )}^m}{c\,d\,\left (m-1\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \]

[In]

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

[Out]

-((a*e + c*d*x)*(d + e*x)^m)/(c*d*(m - 1)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^m)