3.2085 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{d+e x} \, dx\)

Optimal. Leaf size=93 \[ \frac{\left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p,p;p+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]

[Out]

((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p*Hypergeometric2F1[-p, p, 1 + p, (c*d*
(d + e*x))/(c*d^2 - a*e^2)])/(e*p*(-((e*(a*e + c*d*x))/(c*d^2 - a*e^2)))^p)

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Rubi [A]  time = 0.175214, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{\left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p,p;p+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]

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),x]

[Out]

((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p*Hypergeometric2F1[-p, p, 1 + p, (c*d*
(d + e*x))/(c*d^2 - a*e^2)])/(e*p*(-((e*(a*e + c*d*x))/(c*d^2 - a*e^2)))^p)

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Rubi in Sympy [A]  time = 23.1475, size = 76, normalized size = 0.82 \[ \frac{\left (\frac{e \left (a e + c d x\right )}{a e^{2} - c d^{2}}\right )^{- p} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p \\ p + 1 \end{matrix}\middle |{\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}} \right )}}{e p} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.102829, size = 81, normalized size = 0.87 \[ \frac{\left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (-p,p;p+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]

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),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[-p, p, 1 + p, (c*d*(d + e*x))/(c*
d^2 - a*e^2)])/(e*p*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^p)

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Maple [F]  time = 0.137, size = 0, normalized size = 0. \[ \int{\frac{ \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}}{ex+d}}\, dx \]

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),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{e x + d}\,{d x} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{e x + d}, x\right ) \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{p}}{d + e x}\, dx \]

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),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{e x + d}\,{d x} \]

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),x, algorithm="giac")

[Out]

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