3.1450 \(\int (\frac {b e}{2 c}+e x)^m (\frac {b^2}{4 c}+b x+c x^2)^n \, dx\)

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]

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

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Rubi [A]  time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {644, 32} \[ \frac {\left (\frac {b^2}{4 c}+b x+c x^2\right )^n \left (\frac {b e}{2 c}+e x\right )^{m+1}}{e (m+2 n+1)} \]

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))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 644

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^p/(d
+ e*x)^(2*p), Int[(d + e*x)^(m + 2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !
IntegerQ[p] && EqQ[2*c*d - b*e, 0] &&  !IntegerQ[m]

Rubi steps

\begin {align*} \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 &=\left (\left (\frac {b e}{2 c}+e x\right )^{-2 n} \left (\frac {b^2}{4 c}+b x+c x^2\right )^n\right ) \int \left (\frac {b e}{2 c}+e x\right )^{m+2 n} \, dx\\ &=\frac {\left (\frac {b e}{2 c}+e x\right )^{1+m} \left (\frac {b^2}{4 c}+b x+c x^2\right )^n}{e (1+m+2 n)}\\ \end {align*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.99, size = 63, normalized size = 1.26 \[ \frac {{\left (2 \, c x + b\right )} \left (\frac {2 \, c e x + b e}{2 \, c}\right )^{m} e^{\left (2 \, n \log \left (\frac {2 \, c e x + b e}{2 \, c}\right ) + n \log \left (\frac {c}{e^{2}}\right )\right )}}{2 \, {\left (c m + 2 \, c n + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/2*b*e/c+e*x)^m*(1/4/c*b^2+b*x+c*x^2)^n,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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giac [B]  time = 0.24, size = 104, normalized size = 2.08 \[ \frac {2 \, c x e^{\left (-m \log \relax (2) - 2 \, n \log \relax (2) + m \log \left (2 \, c x + b\right ) + 2 \, n \log \left (2 \, c x + b\right ) - m \log \relax (c) - n \log \relax (c) + m\right )} + b e^{\left (-m \log \relax (2) - 2 \, n \log \relax (2) + m \log \left (2 \, c x + b\right ) + 2 \, n \log \left (2 \, c x + b\right ) - m \log \relax (c) - n \log \relax (c) + m\right )}}{2 \, {\left (c m + 2 \, c n + c\right )}} \]

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, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 58, normalized size = 1.16 \[ \frac {\left (2 c x +b \right ) \left (\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{4 c}\right )^{n} \left (\frac {\left (2 c x +b \right ) e}{2 c}\right )^{m}}{2 \left (m +2 n +1\right ) c} \]

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/(m+2*n+1)*(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 = 3.16, size = 79, normalized size = 1.58 \[ \frac {{\left (2 \, c e^{m} x + b e^{m}\right )} c^{-m - n - 1} e^{\left (m \log \left (2 \, c x + b\right ) + 2 \, n \log \left (2 \, c x + b\right )\right )}}{{\left (2^{2 \, n + 2} n + 2^{2 \, n + 1}\right )} 2^{m} + 2^{m + 2 \, n + 1} m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/2*b*e/c+e*x)^m*(1/4/c*b^2+b*x+c*x^2)^n,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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mupad [B]  time = 0.67, size = 51, normalized size = 1.02 \[ \frac {{\left (e\,x+\frac {b\,e}{2\,c}\right )}^m\,\left (b+2\,c\,x\right )\,{\left (b\,x+c\,x^2+\frac {b^2}{4\,c}\right )}^n}{2\,c\,\left (m+2\,n+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {b \left (\frac {b e}{2 c} + e x\right )^{m} \left (\frac {b^{2}}{4 c} + b x + c x^{2}\right )^{n}}{2 c m + 4 c n + 2 c} + \frac {2 c x \left (\frac {b e}{2 c} + e x\right )^{m} \left (\frac {b^{2}}{4 c} + b x + c x^{2}\right )^{n}}{2 c m + 4 c n + 2 c} & \text {for}\: m \neq - 2 n - 1 \\2^{2 n + 1} \cdot 4^{- n} \int \frac {\left (\frac {b^{2}}{c} + 4 b x + 4 c x^{2}\right )^{n}}{\frac {b e \left (\frac {b e}{c} + 2 e x\right )^{2 n}}{c} + 2 e x \left (\frac {b e}{c} + 2 e x\right )^{2 n}}\, dx & \text {otherwise} \end {cases} \]

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]

Piecewise((b*(b*e/(2*c) + e*x)**m*(b**2/(4*c) + b*x + c*x**2)**n/(2*c*m + 4*c*n + 2*c) + 2*c*x*(b*e/(2*c) + e*
x)**m*(b**2/(4*c) + b*x + c*x**2)**n/(2*c*m + 4*c*n + 2*c), Ne(m, -2*n - 1)), (2**(2*n + 1)*4**(-n)*Integral((
b**2/c + 4*b*x + 4*c*x**2)**n/(b*e*(b*e/c + 2*e*x)**(2*n)/c + 2*e*x*(b*e/c + 2*e*x)**(2*n)), x), True))

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