3.1439 \(\int \left (\frac{b e}{2 c}+e x\right )^m \left (\frac{b^2}{4 c}+b x+c x^2\right )^n \, dx\)

Optimal. Leaf size=50 \[ \frac{\left (\frac{b^2}{4 c}+b x+c x^2\right )^n \left (\frac{b e}{2 c}+e x\right )^{m+1}}{e (m+2 n+1)} \]

[Out]

(((b*e)/(2*c) + e*x)^(1 + m)*(b^2/(4*c) + b*x + c*x^2)^n)/(e*(1 + m + 2*n))

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Rubi [A]  time = 0.054894, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{\left (\frac{b^2}{4 c}+b x+c x^2\right )^n \left (\frac{b e}{2 c}+e x\right )^{m+1}}{e (m+2 n+1)} \]

Antiderivative was successfully verified.

[In]  Int[((b*e)/(2*c) + e*x)^m*(b^2/(4*c) + b*x + c*x^2)^n,x]

[Out]

(((b*e)/(2*c) + e*x)^(1 + m)*(b^2/(4*c) + b*x + c*x^2)^n)/(e*(1 + m + 2*n))

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Rubi in Sympy [A]  time = 16.2715, size = 37, normalized size = 0.74 \[ \frac{\left (\frac{b e}{2 c} + e x\right )^{m + 1} \left (\frac{b^{2}}{4 c} + b x + c x^{2}\right )^{n}}{e \left (m + 2 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1/2*b*e/c+e*x)**m*(1/4/c*b**2+b*x+c*x**2)**n,x)

[Out]

(b*e/(2*c) + e*x)**(m + 1)*(b**2/(4*c) + b*x + c*x**2)**n/(e*(m + 2*n + 1))

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Mathematica [A]  time = 0.0448261, size = 54, normalized size = 1.08 \[ \frac{2^{-2 n-1} (b+2 c x) \left (\frac{(b+2 c x)^2}{c}\right )^n \left (\frac{b e}{2 c}+e x\right )^m}{c (m+2 n+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[((b*e)/(2*c) + e*x)^m*(b^2/(4*c) + b*x + c*x^2)^n,x]

[Out]

(2^(-1 - 2*n)*(b + 2*c*x)*((b + 2*c*x)^2/c)^n*((b*e)/(2*c) + e*x)^m)/(c*(1 + m +
2*n))

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Maple [A]  time = 0.005, size = 58, normalized size = 1.2 \[{\frac{2\,cx+b}{2\,c \left ( 1+m+2\,n \right ) } \left ({\frac{e \left ( 2\,cx+b \right ) }{2\,c}} \right ) ^{m} \left ({\frac{4\,{c}^{2}{x}^{2}+4\,bxc+{b}^{2}}{4\,c}} \right ) ^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1/2*b*e/c+e*x)^m*(1/4*b^2/c+b*x+c*x^2)^n,x)

[Out]

1/2*(2*c*x+b)/c/(1+m+2*n)*(1/2*e*(2*c*x+b)/c)^m*(1/4*(4*c^2*x^2+4*b*c*x+b^2)/c)^
n

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Maxima [A]  time = 0.77516, size = 107, normalized size = 2.14 \[ \frac{{\left (2 \, c e^{m} x + b e^{m}\right )} c^{-m - n - 1} e^{\left (m \log \left (2 \, c x + b\right ) + 2 \, n \log \left (2 \, c x + b\right )\right )}}{{\left (2^{2 \, n + 2} n + 2^{2 \, n + 1}\right )} 2^{m} + 2^{m + 2 \, n + 1} m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + 1/4*b^2/c)^n*(e*x + 1/2*b*e/c)^m,x, algorithm="maxima")

[Out]

(2*c*e^m*x + b*e^m)*c^(-m - n - 1)*e^(m*log(2*c*x + b) + 2*n*log(2*c*x + b))/((2
^(2*n + 2)*n + 2^(2*n + 1))*2^m + 2^(m + 2*n + 1)*m)

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Fricas [A]  time = 0.225417, size = 85, normalized size = 1.7 \[ \frac{{\left (2 \, c x + b\right )} \left (\frac{2 \, c e x + b e}{2 \, c}\right )^{m} e^{\left (2 \, n \log \left (\frac{2 \, c e x + b e}{2 \, c}\right ) + n \log \left (\frac{c}{e^{2}}\right )\right )}}{2 \,{\left (c m + 2 \, c n + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + 1/4*b^2/c)^n*(e*x + 1/2*b*e/c)^m,x, algorithm="fricas")

[Out]

1/2*(2*c*x + b)*(1/2*(2*c*e*x + b*e)/c)^m*e^(2*n*log(1/2*(2*c*e*x + b*e)/c) + n*
log(c/e^2))/(c*m + 2*c*n + c)

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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((1/2*b*e/c+e*x)**m*(1/4/c*b**2+b*x+c*x**2)**n,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221909, size = 140, normalized size = 2.8 \[ \frac{2 \, c x e^{\left (-m{\rm ln}\left (2\right ) - 2 \, n{\rm ln}\left (2\right ) + m{\rm ln}\left (2 \, c x + b\right ) + 2 \, n{\rm ln}\left (2 \, c x + b\right ) - m{\rm ln}\left (c\right ) - n{\rm ln}\left (c\right ) + m\right )} + b e^{\left (-m{\rm ln}\left (2\right ) - 2 \, n{\rm ln}\left (2\right ) + m{\rm ln}\left (2 \, c x + b\right ) + 2 \, n{\rm ln}\left (2 \, c x + b\right ) - m{\rm ln}\left (c\right ) - n{\rm ln}\left (c\right ) + m\right )}}{2 \,{\left (c m + 2 \, c n + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + 1/4*b^2/c)^n*(e*x + 1/2*b*e/c)^m,x, algorithm="giac")

[Out]

1/2*(2*c*x*e^(-m*ln(2) - 2*n*ln(2) + m*ln(2*c*x + b) + 2*n*ln(2*c*x + b) - m*ln(
c) - n*ln(c) + m) + b*e^(-m*ln(2) - 2*n*ln(2) + m*ln(2*c*x + b) + 2*n*ln(2*c*x +
b) - m*ln(c) - n*ln(c) + m))/(c*m + 2*c*n + c)