3.1092 \(\int \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=38 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0340363, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.6634, size = 39, normalized size = 1.03 \[ \frac{\left (2 d + 2 e x\right ) \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e \left (2 p + 1\right )} \]

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,x)

[Out]

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

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Mathematica [A]  time = 0.0152926, size = 25, normalized size = 0.66 \[ \frac{(d+e x) \left (c (d+e x)^2\right )^p}{2 e p+e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 39, normalized size = 1. \[{\frac{ \left ( ex+d \right ) \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{e \left ( 1+2\,p \right ) }} \]

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,x)

[Out]

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

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Maxima [A]  time = 0.688277, size = 43, normalized size = 1.13 \[ \frac{{\left (c^{p} e x + c^{p} d\right )}{\left (e x + d\right )}^{2 \, p}}{e{\left (2 \, p + 1\right )}} \]

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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.233868, size = 49, normalized size = 1.29 \[ \frac{{\left (e x + d\right )}{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, e p + e} \]

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,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.218607, size = 89, normalized size = 2.34 \[ \frac{x e^{\left (p{\rm ln}\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right ) + 1\right )} + d e^{\left (p{\rm ln}\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )\right )}}{2 \, p e + e} \]

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,x, algorithm="giac")

[Out]

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