3.1093 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{d+e x} \, dx\)

Optimal. Leaf size=32 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p} \]

[Out]

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

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Rubi [A]  time = 0.0550035, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.8986, size = 27, normalized size = 0.84 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00854067, size = 21, normalized size = 0.66 \[ \frac{\left (c (d+e x)^2\right )^p}{2 e p} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)^2)^p/(2*e*p)

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Maple [A]  time = 0.002, size = 31, normalized size = 1. \[{\frac{ \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{2\,ep}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*e^2*x^2+2*c*d*e*x+c*d^2)^p/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 0.685717, size = 27, normalized size = 0.84 \[ \frac{{\left (e x + d\right )}^{2 \, p} c^{p}}{2 \, e p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(e*x + d)^(2*p)*c^p/(e*p)

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Fricas [A]  time = 0.236119, size = 41, normalized size = 1.28 \[ \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, e p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.987994, size = 48, normalized size = 1.5 \[ \begin{cases} \frac{x}{d} & \text{for}\: e = 0 \wedge p = 0 \\\frac{x \left (c d^{2}\right )^{p}}{d} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{e} & \text{for}\: p = 0 \\\frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((x/d, Eq(e, 0) & Eq(p, 0)), (x*(c*d**2)**p/d, Eq(e, 0)), (log(d/e + x)
/e, Eq(p, 0)), ((c*d**2 + 2*c*d*e*x + c*e**2*x**2)**p/(2*e*p), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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