3.1091 \(\int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=39 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{p+1}}{2 c e (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0303593, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{p+1}}{2 c e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.6129, size = 32, normalized size = 0.82 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p + 1}}{2 c e \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0224545, size = 28, normalized size = 0.72 \[ \frac{\left (c (d+e x)^2\right )^{p+1}}{2 c e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 40, normalized size = 1. \[{\frac{ \left ( ex+d \right ) ^{2} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{2\,e \left ( 1+p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228948, size = 63, normalized size = 1.62 \[ \frac{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \,{\left (e p + e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.35228, size = 139, normalized size = 3.56 \[ \begin{cases} \frac{x}{c d} & \text{for}\: e = 0 \wedge p = -1 \\d x \left (c d^{2}\right )^{p} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{c e} & \text{for}\: p = -1 \\\frac{d^{2} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} + \frac{2 d e x \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} + \frac{e^{2} x^{2} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((x/(c*d), Eq(e, 0) & Eq(p, -1)), (d*x*(c*d**2)**p, Eq(e, 0)), (log(d/e
+ x)/(c*e), Eq(p, -1)), (d**2*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**p/(2*e*p + 2*
e) + 2*d*e*x*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**p/(2*e*p + 2*e) + e**2*x**2*(c*
d**2 + 2*c*d*e*x + c*e**2*x**2)**p/(2*e*p + 2*e), True))

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GIAC/XCAS [A]  time = 0.225532, size = 135, normalized size = 3.46 \[ \frac{x^{2} e^{\left (p{\rm ln}\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right ) + 2\right )} + 2 \, d x e^{\left (p{\rm ln}\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right ) + 1\right )} + d^{2} e^{\left (p{\rm ln}\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )\right )}}{2 \,{\left (p e + e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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