3.1087 \(\int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=43 \[ \frac{(d+e x)^{m+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (m+2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0523879, antiderivative size = 43, 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 \[ \frac{(d+e x)^{m+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (m+2 p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.9471, size = 39, normalized size = 0.91 \[ \frac{\left (d + e x\right )^{m + 1} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{e \left (m + 2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.029001, size = 32, normalized size = 0.74 \[ \frac{(d+e x)^{m+1} \left (c (d+e x)^2\right )^p}{e m+2 e p+e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 44, normalized size = 1. \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{e \left ( 1+m+2\,p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.691939, size = 58, normalized size = 1.35 \[ \frac{{\left (c^{p} e x + c^{p} d\right )} e^{\left (m \log \left (e x + d\right ) + 2 \, p \log \left (e x + d\right )\right )}}{e{\left (m + 2 \, p + 1\right )}} \]

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*(e*x + d)^m,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.227982, size = 53, normalized size = 1.23 \[ \frac{{\left (e x + d\right )}{\left (e x + d\right )}^{m} e^{\left (2 \, p \log \left (e x + d\right ) + p \log \left (c\right )\right )}}{e m + 2 \, e p + e} \]

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*(e*x + d)^m,x, algorithm="fricas")

[Out]

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

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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((e*x+d)**m*(c*e**2*x**2+2*c*d*e*x+c*d**2)**p,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22554, size = 96, normalized size = 2.23 \[ \frac{x e^{\left (m{\rm ln}\left (x e + d\right ) + 2 \, p{\rm ln}\left (x e + d\right ) + p{\rm ln}\left (c\right ) + 1\right )} + d e^{\left (m{\rm ln}\left (x e + d\right ) + 2 \, p{\rm ln}\left (x e + d\right ) + p{\rm ln}\left (c\right )\right )}}{m e + 2 \, p e + e} \]

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*(e*x + d)^m,x, algorithm="giac")

[Out]

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