3.2106 \(\int (d+e x)^{-p} (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\)

Optimal. Leaf size=52 \[ \frac{(d+e x)^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{c d (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0160988, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {648} \[ \frac{(d+e x)^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{c d (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 648

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

Rubi steps

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

Mathematica [A]  time = 0.0206808, size = 41, normalized size = 0.79 \[ \frac{(d+e x)^{-p-1} ((d+e x) (a e+c d x))^{p+1}}{c d (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 56, normalized size = 1.1 \begin{align*}{\frac{ \left ( cdx+ae \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{p}}{cd \left ( 1+p \right ) \left ( ex+d \right ) ^{p}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p/((e*x+d)^p),x)

[Out]

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

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Maxima [A]  time = 1.02852, size = 41, normalized size = 0.79 \begin{align*} \frac{{\left (c d x + a e\right )}{\left (c d x + a e\right )}^{p}}{c d{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.13223, size = 115, normalized size = 2.21 \begin{align*} \frac{{\left (c d x + a e\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (c d p + c d\right )}{\left (e x + d\right )}^{p}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p/((e*x+d)**p),x)

[Out]

Timed out

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Giac [A]  time = 1.14888, size = 115, normalized size = 2.21 \begin{align*} \frac{\frac{c d x e^{\left (p \log \left (c d x + a e\right ) + p \log \left (x e + d\right )\right )}}{{\left (x e + d\right )}^{p}} + \frac{a e^{\left (p \log \left (c d x + a e\right ) + p \log \left (x e + d\right ) + 1\right )}}{{\left (x e + d\right )}^{p}}}{c d p + c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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