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

Optimal. Leaf size=44 \[ \frac {(d+e x)^{1+p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {658, 32} \begin {gather*} \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)} \end {gather*}

Antiderivative was successfully verified.

[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*} \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx &=\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*}

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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.70 \begin {gather*} \frac {(d+e x)^{1+p} \left (c (d+e x)^2\right )^{-p}}{e-e p} \end {gather*}

Antiderivative was successfully verified.

[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 - e*p)*(c*(d + e*x)^2)^p)

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Maple [A]
time = 0.67, size = 44, normalized size = 1.00

method	result	size
gosper	\(-\frac {\left (e x +d \right )^{1+p} \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{-p}}{e \left (-1+p \right )}\)	\(44\)
norman	\(\left (-\frac {x \,{\mathrm e}^{p \ln \left (e x +d \right )}}{-1+p}-\frac {d \,{\mathrm e}^{p \ln \left (e x +d \right )}}{e \left (-1+p \right )}\right ) {\mathrm e}^{-p \ln \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )}\)	\(66\)
risch	\(-\frac {\left (e x +d \right ) \left (e x +d \right )^{p} {\mathrm e}^{-\frac {p \left (-i \mathrm {csgn}\left (i \left (e x +d \right )^{2}\right )^{3} \pi +2 i \mathrm {csgn}\left (i \left (e x +d \right )^{2}\right )^{2} \mathrm {csgn}\left (i \left (e x +d \right )\right ) \pi -i \mathrm {csgn}\left (i \left (e x +d \right )^{2}\right ) \mathrm {csgn}\left (i \left (e x +d \right )\right )^{2} \pi +i \mathrm {csgn}\left (i \left (e x +d \right )^{2}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{2}\right )^{2} \pi -i \mathrm {csgn}\left (i \left (e x +d \right )^{2}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{2}\right ) \mathrm {csgn}\left (i c \right ) \pi -i \mathrm {csgn}\left (i c \left (e x +d \right )^{2}\right )^{3} \pi +i \mathrm {csgn}\left (i c \left (e x +d \right )^{2}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi +4 \ln \left (e x +d \right )+2 \ln \left (c \right )\right )}{2}}}{e \left (-1+p \right )}\)	\(211\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 30, normalized size = 0.68 \begin {gather*} -\frac {{\left (x e + d\right )} e^{\left (-1\right )}}{{\left (x e + d\right )}^{p} c^{p} {\left (p - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 4.13, size = 30, normalized size = 0.68 \begin {gather*} -\frac {{\left (x e + d\right )} e^{\left (-1\right )}}{{\left (x e + d\right )}^{p} c^{p} {\left (p - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]
time = 1.04, size = 69, normalized size = 1.57 \begin {gather*} -\frac {{\left (x e + d\right )}^{p} x e^{\left (-2 \, p \log \left (x e + d\right ) - p \log \left (c\right ) + 1\right )} + {\left (x e + d\right )}^{p} d e^{\left (-2 \, p \log \left (x e + d\right ) - p \log \left (c\right )\right )}}{p e - e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.47, size = 43, normalized size = 0.98 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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