### 3.1099 $$\int (d+e x)^p (c d^2+2 c d e x+c e^2 x^2)^{-p} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0173963, antiderivative size = 44, normalized size of antiderivative = 1., 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} $\frac{(d+e x)^{p+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)}$

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 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]

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]

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 = 0.0160117, size = 31, normalized size = 0.7 $\frac{(d+e x)^{p+1} \left (c (d+e x)^2\right )^{-p}}{e-e p}$

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 = 0.04, size = 44, normalized size = 1. \begin{align*} -{\frac{ \left ( ex+d \right ) ^{1+p}}{e \left ( p-1 \right ) \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}} \end{align*}

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

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Maxima [A]  time = 1.15625, size = 39, normalized size = 0.89 \begin{align*} -\frac{e x + d}{{\left (e x + d\right )}^{p} c^{p} e{\left (p - 1\right )}} \end{align*}

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))

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Fricas [A]  time = 2.44774, size = 54, normalized size = 1.23 \begin{align*} -\frac{e x + d}{{\left (e p - e\right )}{\left (e x + d\right )}^{p} c^{p}} \end{align*}

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)

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

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

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Giac [A]  time = 1.2373, size = 93, normalized size = 2.11 \begin{align*} -\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{align*}

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)