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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, 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 = {644, 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 644

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*}

________________________________________________________________________________________

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 30, normalized size = 0.68 \begin {gather*} -\frac {e x + d}{{\left (e p - e\right )} {\left (e x + d\right )}^{p} c^{p}} \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]

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

________________________________________________________________________________________

giac [A]  time = 0.21, 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 \relax (c) + 1\right )} + {\left (x e + d\right )}^{p} d e^{\left (-2 \, p \log \left (x e + d\right ) - p \log \relax (c)\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)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 44, normalized size = 1.00 \begin {gather*} -\frac {\left (e x +d \right )^{p +1} \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{-p}}{\left (p -1\right ) e} \end {gather*}

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)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.47, size = 29, normalized size = 0.66 \begin {gather*} -\frac {e x + d}{{\left (e x + d\right )}^{p} c^{p} e {\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]

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

________________________________________________________________________________________

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)

________________________________________________________________________________________

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

________________________________________________________________________________________