3.2095 \(\int (d+e x)^{-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\)

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 = 0.0511339, 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 \[ \frac{(d+e x)^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{c d (p+1)} \]

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
))

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Rubi in Sympy [A]  time = 18.2636, size = 44, normalized size = 0.85 \[ \frac{\left (d + e x\right )^{- p - 1} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{c d \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p/((e*x+d)**p),x)

[Out]

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

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Mathematica [A]  time = 0.0399096, 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 = 0.005, size = 56, normalized size = 1.1 \[{\frac{ \left ( cdx+ae \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{p}}{cd \left ( 1+p \right ) \left ( ex+d \right ) ^{p}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(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 = 0.822901, size = 41, normalized size = 0.79 \[ \frac{{\left (c d x + a e\right )}{\left (c d x + a e\right )}^{p}}{c d{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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 = 0.230833, size = 74, normalized size = 1.42 \[ \frac{{\left (c d x + a e\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (c d p + c d\right )}{\left (e x + d\right )}^{p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.214615, size = 62, normalized size = 1.19 \[ \frac{c d x e^{\left (p{\rm ln}\left (c d x + a e\right )\right )} + a e^{\left (p{\rm ln}\left (c d x + a e\right ) + 1\right )}}{c d p + c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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