3.488 \(\int (e x)^m (A+B x) (a+c x^2) \, dx\)

Optimal. Leaf size=73 \[ \frac{a A (e x)^{m+1}}{e (m+1)}+\frac{a B (e x)^{m+2}}{e^2 (m+2)}+\frac{A c (e x)^{m+3}}{e^3 (m+3)}+\frac{B c (e x)^{m+4}}{e^4 (m+4)} \]

[Out]

(a*A*(e*x)^(1 + m))/(e*(1 + m)) + (a*B*(e*x)^(2 + m))/(e^2*(2 + m)) + (A*c*(e*x)^(3 + m))/(e^3*(3 + m)) + (B*c
*(e*x)^(4 + m))/(e^4*(4 + m))

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Rubi [A]  time = 0.0325873, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ \frac{a A (e x)^{m+1}}{e (m+1)}+\frac{a B (e x)^{m+2}}{e^2 (m+2)}+\frac{A c (e x)^{m+3}}{e^3 (m+3)}+\frac{B c (e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(A + B*x)*(a + c*x^2),x]

[Out]

(a*A*(e*x)^(1 + m))/(e*(1 + m)) + (a*B*(e*x)^(2 + m))/(e^2*(2 + m)) + (A*c*(e*x)^(3 + m))/(e^3*(3 + m)) + (B*c
*(e*x)^(4 + m))/(e^4*(4 + m))

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (e x)^m (A+B x) \left (a+c x^2\right ) \, dx &=\int \left (a A (e x)^m+\frac{a B (e x)^{1+m}}{e}+\frac{A c (e x)^{2+m}}{e^2}+\frac{B c (e x)^{3+m}}{e^3}\right ) \, dx\\ &=\frac{a A (e x)^{1+m}}{e (1+m)}+\frac{a B (e x)^{2+m}}{e^2 (2+m)}+\frac{A c (e x)^{3+m}}{e^3 (3+m)}+\frac{B c (e x)^{4+m}}{e^4 (4+m)}\\ \end{align*}

Mathematica [A]  time = 0.0428127, size = 47, normalized size = 0.64 \[ x (e x)^m \left (a \left (\frac{A}{m+1}+\frac{B x}{m+2}\right )+c x^2 \left (\frac{A}{m+3}+\frac{B x}{m+4}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(A + B*x)*(a + c*x^2),x]

[Out]

x*(e*x)^m*(a*(A/(1 + m) + (B*x)/(2 + m)) + c*x^2*(A/(3 + m) + (B*x)/(4 + m)))

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Maple [A]  time = 0.003, size = 145, normalized size = 2. \begin{align*}{\frac{ \left ( Bc{m}^{3}{x}^{3}+Ac{m}^{3}{x}^{2}+6\,Bc{m}^{2}{x}^{3}+7\,Ac{m}^{2}{x}^{2}+Ba{m}^{3}x+11\,Bcm{x}^{3}+Aa{m}^{3}+14\,Acm{x}^{2}+8\,Ba{m}^{2}x+6\,Bc{x}^{3}+9\,Aa{m}^{2}+8\,Ac{x}^{2}+19\,Bamx+26\,Aam+12\,aBx+24\,aA \right ) x \left ( ex \right ) ^{m}}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(B*x+A)*(c*x^2+a),x)

[Out]

x*(B*c*m^3*x^3+A*c*m^3*x^2+6*B*c*m^2*x^3+7*A*c*m^2*x^2+B*a*m^3*x+11*B*c*m*x^3+A*a*m^3+14*A*c*m*x^2+8*B*a*m^2*x
+6*B*c*x^3+9*A*a*m^2+8*A*c*x^2+19*B*a*m*x+26*A*a*m+12*B*a*x+24*A*a)*(e*x)^m/(4+m)/(3+m)/(2+m)/(1+m)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(B*x+A)*(c*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50643, size = 315, normalized size = 4.32 \begin{align*} \frac{{\left ({\left (B c m^{3} + 6 \, B c m^{2} + 11 \, B c m + 6 \, B c\right )} x^{4} +{\left (A c m^{3} + 7 \, A c m^{2} + 14 \, A c m + 8 \, A c\right )} x^{3} +{\left (B a m^{3} + 8 \, B a m^{2} + 19 \, B a m + 12 \, B a\right )} x^{2} +{\left (A a m^{3} + 9 \, A a m^{2} + 26 \, A a m + 24 \, A a\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(B*x+A)*(c*x^2+a),x, algorithm="fricas")

[Out]

((B*c*m^3 + 6*B*c*m^2 + 11*B*c*m + 6*B*c)*x^4 + (A*c*m^3 + 7*A*c*m^2 + 14*A*c*m + 8*A*c)*x^3 + (B*a*m^3 + 8*B*
a*m^2 + 19*B*a*m + 12*B*a)*x^2 + (A*a*m^3 + 9*A*a*m^2 + 26*A*a*m + 24*A*a)*x)*(e*x)^m/(m^4 + 10*m^3 + 35*m^2 +
50*m + 24)

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Sympy [A]  time = 0.88246, size = 685, normalized size = 9.38 \begin{align*} \begin{cases} \frac{- \frac{A a}{3 x^{3}} - \frac{A c}{x} - \frac{B a}{2 x^{2}} + B c \log{\left (x \right )}}{e^{4}} & \text{for}\: m = -4 \\\frac{- \frac{A a}{2 x^{2}} + A c \log{\left (x \right )} - \frac{B a}{x} + B c x}{e^{3}} & \text{for}\: m = -3 \\\frac{- \frac{A a}{x} + A c x + B a \log{\left (x \right )} + \frac{B c x^{2}}{2}}{e^{2}} & \text{for}\: m = -2 \\\frac{A a \log{\left (x \right )} + \frac{A c x^{2}}{2} + B a x + \frac{B c x^{3}}{3}}{e} & \text{for}\: m = -1 \\\frac{A a e^{m} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{9 A a e^{m} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{26 A a e^{m} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 A a e^{m} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{A c e^{m} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{7 A c e^{m} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{14 A c e^{m} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{8 A c e^{m} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{B a e^{m} m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{8 B a e^{m} m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{19 B a e^{m} m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{12 B a e^{m} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{B c e^{m} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{6 B c e^{m} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{11 B c e^{m} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{6 B c e^{m} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(B*x+A)*(c*x**2+a),x)

[Out]

Piecewise(((-A*a/(3*x**3) - A*c/x - B*a/(2*x**2) + B*c*log(x))/e**4, Eq(m, -4)), ((-A*a/(2*x**2) + A*c*log(x)
- B*a/x + B*c*x)/e**3, Eq(m, -3)), ((-A*a/x + A*c*x + B*a*log(x) + B*c*x**2/2)/e**2, Eq(m, -2)), ((A*a*log(x)
+ A*c*x**2/2 + B*a*x + B*c*x**3/3)/e, Eq(m, -1)), (A*a*e**m*m**3*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24)
+ 9*A*a*e**m*m**2*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 26*A*a*e**m*m*x*x**m/(m**4 + 10*m**3 + 35*m
**2 + 50*m + 24) + 24*A*a*e**m*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + A*c*e**m*m**3*x**3*x**m/(m**4 +
10*m**3 + 35*m**2 + 50*m + 24) + 7*A*c*e**m*m**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 14*A*c*e*
*m*m*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8*A*c*e**m*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m
+ 24) + B*a*e**m*m**3*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8*B*a*e**m*m**2*x**2*x**m/(m**4 + 10*
m**3 + 35*m**2 + 50*m + 24) + 19*B*a*e**m*m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 12*B*a*e**m*x**
2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + B*c*e**m*m**3*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24)
+ 6*B*c*e**m*m**2*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 11*B*c*e**m*m*x**4*x**m/(m**4 + 10*m**3
+ 35*m**2 + 50*m + 24) + 6*B*c*e**m*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24), True))

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Giac [B]  time = 1.13268, size = 311, normalized size = 4.26 \begin{align*} \frac{B c m^{3} x^{4} x^{m} e^{m} + A c m^{3} x^{3} x^{m} e^{m} + 6 \, B c m^{2} x^{4} x^{m} e^{m} + B a m^{3} x^{2} x^{m} e^{m} + 7 \, A c m^{2} x^{3} x^{m} e^{m} + 11 \, B c m x^{4} x^{m} e^{m} + A a m^{3} x x^{m} e^{m} + 8 \, B a m^{2} x^{2} x^{m} e^{m} + 14 \, A c m x^{3} x^{m} e^{m} + 6 \, B c x^{4} x^{m} e^{m} + 9 \, A a m^{2} x x^{m} e^{m} + 19 \, B a m x^{2} x^{m} e^{m} + 8 \, A c x^{3} x^{m} e^{m} + 26 \, A a m x x^{m} e^{m} + 12 \, B a x^{2} x^{m} e^{m} + 24 \, A a x x^{m} e^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(B*x+A)*(c*x^2+a),x, algorithm="giac")

[Out]

(B*c*m^3*x^4*x^m*e^m + A*c*m^3*x^3*x^m*e^m + 6*B*c*m^2*x^4*x^m*e^m + B*a*m^3*x^2*x^m*e^m + 7*A*c*m^2*x^3*x^m*e
^m + 11*B*c*m*x^4*x^m*e^m + A*a*m^3*x*x^m*e^m + 8*B*a*m^2*x^2*x^m*e^m + 14*A*c*m*x^3*x^m*e^m + 6*B*c*x^4*x^m*e
^m + 9*A*a*m^2*x*x^m*e^m + 19*B*a*m*x^2*x^m*e^m + 8*A*c*x^3*x^m*e^m + 26*A*a*m*x*x^m*e^m + 12*B*a*x^2*x^m*e^m
+ 24*A*a*x*x^m*e^m)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)