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

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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)} \end {gather*}

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 = 0.13, size = 48, normalized size = 0.75 \begin {gather*} -\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)} \end {gather*}

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)))

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.53, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(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]

Defer[IntegrateAlgebraic][(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]

________________________________________________________________________________________

fricas [B]  time = 0.44, size = 132, normalized size = 2.06 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

giac [B]  time = 0.51, size = 444, normalized size = 6.94 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 98, normalized size = 1.53 \begin {gather*} -\frac {\left (e g x p +d g p +2 e g x +d g +e f \right ) \left (e x +d \right )^{-2 p -2} \left (e^{2} g \,x^{2} p +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 \right )^{p}}{\left (p +2\right ) e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 2.63, size = 138, normalized size = 2.16 \begin {gather*} -{\left (d\,\left (d\,g+e\,f+d\,g\,p\right )+e\,x\,\left (3\,d\,g+e\,f+2\,d\,g\,p\right )+e^2\,g\,x^2\,\left (p+2\right )\right )}^p\,\left (\frac {g\,x^2}{{\left (d+e\,x\right )}^{2\,p+3}}+\frac {d^2\,g+d\,e\,f+d^2\,g\,p}{e^2\,\left (p+2\right )\,{\left (d+e\,x\right )}^{2\,p+3}}+\frac {x\,\left (3\,d\,g+e\,f+2\,d\,g\,p\right )}{e\,\left (p+2\right )\,{\left (d+e\,x\right )}^{2\,p+3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________