3.2292 \(\int (d+e x)^{-3-2 p} (f+g x) (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2)^p \, dx\)

Optimal. Leaf size=64 \[ -\frac{(d+e x)^{-2 p-3} \left (e x (d g (2 p+3)+e f)+d (d g (p+1)+e f)+e^2 g (p+2) x^2\right )^{p+1}}{e^2 (p+2)} \]

[Out]

-(((d + e*x)^(-3 - 2*p)*(d*(e*f + d*g*(1 + p)) + e*(e*f + d*g*(3 + 2*p))*x + e^2*g*(2 + p)*x^2)^(1 + p))/(e^2*
(2 + p)))

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Rubi [A]  time = 0.035016, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 60, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.017, Rules used = {786} \[ -\frac{(d+e x)^{-2 p-3} \left (e x (d g (2 p+3)+e f)+d (d g (p+1)+e f)+e^2 g (p+2) x^2\right )^{p+1}}{e^2 (p+2)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(-3 - 2*p)*(f + g*x)*(d*(e*f + d*g + d*g*p) + e*(e*f + 3*d*g + 2*d*g*p)*x + e^2*g*(2 + p)*x^2)^p
,x]

[Out]

-(((d + e*x)^(-3 - 2*p)*(d*(e*f + d*g*(1 + p)) + e*(e*f + d*g*(3 + 2*p))*x + e^2*g*(2 + p)*x^2)^(1 + p))/(e^2*
(2 + p)))

Rule 786

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

Rubi steps

\begin{align*} \int (d+e x)^{-3-2 p} (f+g x) \left (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2\right )^p \, dx &=-\frac{(d+e x)^{-3-2 p} \left (d (e f+d g (1+p))+e (e f+d g (3+2 p)) x+e^2 g (2+p) x^2\right )^{1+p}}{e^2 (2+p)}\\ \end{align*}

Mathematica [A]  time = 0.1423, size = 48, normalized size = 0.75 \[ -\frac{(d+e x)^{-2 p-3} ((d+e x) (d g (p+1)+e (f+g (p+2) x)))^{p+1}}{e^2 (p+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(-3 - 2*p)*(f + g*x)*(d*(e*f + d*g + d*g*p) + e*(e*f + 3*d*g + 2*d*g*p)*x + e^2*g*(2 + p)*
x^2)^p,x]

[Out]

-(((d + e*x)^(-3 - 2*p)*((d + e*x)*(d*g*(1 + p) + e*(f + g*(2 + p)*x)))^(1 + p))/(e^2*(2 + p)))

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Maple [A]  time = 0.007, size = 98, normalized size = 1.5 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{-2-2\,p} \left ( egxp+dgp+2\,egx+dg+ef \right ) \left ({e}^{2}g{x}^{2}p+2\,degpx+2\,{e}^{2}g{x}^{2}+{d}^{2}gp+3\,degx+{e}^{2}fx+{d}^{2}g+def \right ) ^{p}}{ \left ( 2+p \right ){e}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(-3-2*p)*(g*x+f)*(d*(d*g*p+d*g+e*f)+e*(2*d*g*p+3*d*g+e*f)*x+e^2*g*(2+p)*x^2)^p,x)

[Out]

-(e*x+d)^(-2-2*p)*(e*g*p*x+d*g*p+2*e*g*x+d*g+e*f)/e^2/(2+p)*(e^2*g*p*x^2+2*d*e*g*p*x+2*e^2*g*x^2+d^2*g*p+3*d*e
*g*x+e^2*f*x+d^2*g+d*e*f)^p

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x + f\right )}{\left (e^{2} g{\left (p + 2\right )} x^{2} +{\left (2 \, d g p + e f + 3 \, d g\right )} e x +{\left (d g p + e f + d g\right )} d\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(g*x+f)*(d*(d*g*p+d*g+e*f)+e*(2*d*g*p+3*d*g+e*f)*x+e^2*g*(2+p)*x^2)^p,x, algorithm=
"maxima")

[Out]

integrate((g*x + f)*(e^2*g*(p + 2)*x^2 + (2*d*g*p + e*f + 3*d*g)*e*x + (d*g*p + e*f + d*g)*d)^p*(e*x + d)^(-2*
p - 3), x)

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Fricas [B]  time = 1.39722, size = 288, normalized size = 4.5 \begin{align*} -\frac{{\left (d^{2} g p + d e f + d^{2} g +{\left (e^{2} g p + 2 \, e^{2} g\right )} x^{2} +{\left (2 \, d e g p + e^{2} f + 3 \, d e g\right )} x\right )}{\left (d^{2} g p + d e f + d^{2} g +{\left (e^{2} g p + 2 \, e^{2} g\right )} x^{2} +{\left (2 \, d e g p + e^{2} f + 3 \, d e g\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}}{e^{2} p + 2 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(g*x+f)*(d*(d*g*p+d*g+e*f)+e*(2*d*g*p+3*d*g+e*f)*x+e^2*g*(2+p)*x^2)^p,x, algorithm=
"fricas")

[Out]

-(d^2*g*p + d*e*f + d^2*g + (e^2*g*p + 2*e^2*g)*x^2 + (2*d*e*g*p + e^2*f + 3*d*e*g)*x)*(d^2*g*p + d*e*f + d^2*
g + (e^2*g*p + 2*e^2*g)*x^2 + (2*d*e*g*p + e^2*f + 3*d*e*g)*x)^p*(e*x + d)^(-2*p - 3)/(e^2*p + 2*e^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(-3-2*p)*(g*x+f)*(d*(d*g*p+d*g+e*f)+e*(2*d*g*p+3*d*g+e*f)*x+e**2*g*(2+p)*x**2)**p,x)

[Out]

Timed out

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Giac [B]  time = 1.2156, size = 599, normalized size = 9.36 \begin{align*} -\frac{g p x^{2} e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 2\right )} + 2 \, d g p x e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 1\right )} + d^{2} g p e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right )\right )} + 2 \, g x^{2} e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 2\right )} + 3 \, d g x e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 1\right )} + d^{2} g e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right )\right )} + f x e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 2\right )} + d f e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 1\right )}}{p e^{2} + 2 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(g*x+f)*(d*(d*g*p+d*g+e*f)+e*(2*d*g*p+3*d*g+e*f)*x+e^2*g*(2+p)*x^2)^p,x, algorithm=
"giac")

[Out]

-(g*p*x^2*e^(p*log(g*p*x*e + d*g*p + 2*g*x*e + d*g + f*e) - p*log(x*e + d) - 3*log(x*e + d) + 2) + 2*d*g*p*x*e
^(p*log(g*p*x*e + d*g*p + 2*g*x*e + d*g + f*e) - p*log(x*e + d) - 3*log(x*e + d) + 1) + d^2*g*p*e^(p*log(g*p*x
*e + d*g*p + 2*g*x*e + d*g + f*e) - p*log(x*e + d) - 3*log(x*e + d)) + 2*g*x^2*e^(p*log(g*p*x*e + d*g*p + 2*g*
x*e + d*g + f*e) - p*log(x*e + d) - 3*log(x*e + d) + 2) + 3*d*g*x*e^(p*log(g*p*x*e + d*g*p + 2*g*x*e + d*g + f
*e) - p*log(x*e + d) - 3*log(x*e + d) + 1) + d^2*g*e^(p*log(g*p*x*e + d*g*p + 2*g*x*e + d*g + f*e) - p*log(x*e
+ d) - 3*log(x*e + d)) + f*x*e^(p*log(g*p*x*e + d*g*p + 2*g*x*e + d*g + f*e) - p*log(x*e + d) - 3*log(x*e + d
) + 2) + d*f*e^(p*log(g*p*x*e + d*g*p + 2*g*x*e + d*g + f*e) - p*log(x*e + d) - 3*log(x*e + d) + 1))/(p*e^2 +
2*e^2)