\(\int (d+e x)^p (c d^2+2 c d e x+c e^2 x^2)^{-p} \, dx\) [1099]

Optimal result

Integrand size = 32, antiderivative size = 44 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=\frac {(d+e x)^{1+p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {658, 32} \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=\frac {(d+e x)^{p+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 658

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.75 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=\frac {(d+e x)^{1+p} \left (c (d+e x)^2\right )^{-p}}{e (1-p)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.95 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00

method	result	size
gosper	\(-\frac {\left (e x +d \right )^{1+p} \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{-p}}{e \left (p -1\right )}\)	\(44\)
parallelrisch	\(\frac {\left (-x \left (e x +d \right )^{p} e -\left (e x +d \right )^{p} d \right ) {\left (c \left (x^{2} e^{2}+2 d e x +d^{2}\right )\right )}^{-p}}{e \left (p -1\right )}\)	\(54\)
norman	\(\left (-\frac {x \,{\mathrm e}^{p \ln \left (e x +d \right )}}{p -1}-\frac {d \,{\mathrm e}^{p \ln \left (e x +d \right )}}{e \left (p -1\right )}\right ) {\mathrm e}^{-p \ln \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )}\)	\(66\)
risch	\(-\frac {\left (e x +d \right ) \left (e x +d \right )^{p} \left (e x +d \right )^{-2 p} c^{-p} {\mathrm e}^{\frac {i \pi p \left (\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right )^{3}-2 \operatorname {csgn}\left (i \left (e x +d \right )^{2}\right )^{2} \operatorname {csgn}\left (i \left (e x +d \right )\right )+\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i \left (e x +d \right )\right )^{2}-\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right )^{2}+\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i c \right )+\operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right )^{3}-\operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right )^{2} \operatorname {csgn}\left (i c \right )\right )}{2}}}{e \left (p -1\right )}\)	\(197\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.68 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=-\frac {e x + d}{{\left (e p - e\right )} {\left (e x + d\right )}^{p} c^{p}} \]

[In]

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

[Out]

-(e*x + d)/((e*p - e)*(e*x + d)^p*c^p)

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> Invalid NaN comparison

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=-\frac {e x + d}{{\left (e x + d\right )}^{p} c^{p} e {\left (p - 1\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.43 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=-\frac {{\left (e x + d\right )}^{p} e x e^{\left (-2 \, p \log \left (e x + d\right ) - p \log \left (c\right )\right )} + {\left (e x + d\right )}^{p} d e^{\left (-2 \, p \log \left (e x + d\right ) - p \log \left (c\right )\right )}}{e p - e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.82 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx=-\frac {{\left (d+e\,x\right )}^{p+1}}{e\,\left (p-1\right )\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p} \]

[In]

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

[Out]

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