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

Optimal. Leaf size=83 \[ \frac {(e x)^{m+2} (a B+A b)}{e^2 (m+2)}+\frac {a A (e x)^{m+1}}{e (m+1)}+\frac {(e x)^{m+3} (A c+b B)}{e^3 (m+3)}+\frac {B c (e x)^{m+4}}{e^4 (m+4)} \]

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Rubi [A]  time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {765} \begin {gather*} \frac {(e x)^{m+2} (a B+A b)}{e^2 (m+2)}+\frac {a A (e x)^{m+1}}{e (m+1)}+\frac {(e x)^{m+3} (A c+b B)}{e^3 (m+3)}+\frac {B c (e x)^{m+4}}{e^4 (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 91, normalized size = 1.10 \begin {gather*} \frac {x (e x)^m \left (a \left (m^2+7 m+12\right ) (A (m+2)+B (m+1) x)+(m+1) x (A (m+4) (b (m+3)+c (m+2) x)+B (m+2) x (b (m+4)+c (m+3) x))\right )}{(m+1) (m+2) (m+3) (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(a*(12 + 7*m + m^2)*(A*(2 + m) + B*(1 + m)*x) + (1 + m)*x*(A*(4 + m)*(b*(3 + m) + c*(2 + m)*x) + B*
(2 + m)*x*(b*(4 + m) + c*(3 + m)*x))))/((1 + m)*(2 + m)*(3 + m)*(4 + m))

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IntegrateAlgebraic [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^m (A+B x) \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.42, size = 171, normalized size = 2.06 \begin {gather*} \frac {{\left ({\left (B c m^{3} + 6 \, B c m^{2} + 11 \, B c m + 6 \, B c\right )} x^{4} + {\left ({\left (B b + A c\right )} m^{3} + 7 \, {\left (B b + A c\right )} m^{2} + 8 \, B b + 8 \, A c + 14 \, {\left (B b + A c\right )} m\right )} x^{3} + {\left ({\left (B a + A b\right )} m^{3} + 8 \, {\left (B a + A b\right )} m^{2} + 12 \, B a + 12 \, A b + 19 \, {\left (B a + A b\right )} m\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*c*m^3 + 6*B*c*m^2 + 11*B*c*m + 6*B*c)*x^4 + ((B*b + A*c)*m^3 + 7*(B*b + A*c)*m^2 + 8*B*b + 8*A*c + 14*(B*b
+ A*c)*m)*x^3 + ((B*a + A*b)*m^3 + 8*(B*a + A*b)*m^2 + 12*B*a + 12*A*b + 19*(B*a + A*b)*m)*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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giac [B]  time = 0.17, size = 338, normalized size = 4.07 \begin {gather*} \frac {B c m^{3} x^{4} x^{m} e^{m} + B b m^{3} x^{3} 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} + A b m^{3} x^{2} x^{m} e^{m} + 7 \, B b m^{2} x^{3} 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} + 8 \, A b m^{2} x^{2} x^{m} e^{m} + 14 \, B b m x^{3} 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} + 19 \, A b m x^{2} x^{m} e^{m} + 8 \, B b x^{3} 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} + 12 \, A b 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*c*m^3*x^4*x^m*e^m + B*b*m^3*x^3*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
+ A*b*m^3*x^2*x^m*e^m + 7*B*b*m^2*x^3*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 + 8*A*b*m^2*x^2*x^m*e^m + 14*B*b*m*x^3*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 + 19*A*b*m*x^2*x^m*e^m + 8*B*b*x^3*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 + 12*A*b*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)

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maple [B]  time = 0.04, size = 205, normalized size = 2.47 \begin {gather*} \frac {\left (B c \,m^{3} x^{3}+A c \,m^{3} x^{2}+B b \,m^{3} x^{2}+6 B c \,m^{2} x^{3}+A b \,m^{3} x +7 A c \,m^{2} x^{2}+B a \,m^{3} x +7 B b \,m^{2} x^{2}+11 B c m \,x^{3}+A a \,m^{3}+8 A b \,m^{2} x +14 A c m \,x^{2}+8 B a \,m^{2} x +14 B b m \,x^{2}+6 B c \,x^{3}+9 A a \,m^{2}+19 A b m x +8 A c \,x^{2}+19 B a m x +8 B b \,x^{2}+26 A a m +12 A b x +12 B a x +24 A a \right ) x \left (e x \right )^{m}}{\left (m +4\right ) \left (m +3\right ) \left (m +2\right ) \left (m +1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.69, size = 104, normalized size = 1.25 \begin {gather*} \frac {B c e^{m} x^{4} x^{m}}{m + 4} + \frac {B b e^{m} x^{3} x^{m}}{m + 3} + \frac {A c e^{m} x^{3} x^{m}}{m + 3} + \frac {B a e^{m} x^{2} x^{m}}{m + 2} + \frac {A b e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} A a}{e {\left (m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.44, size = 171, normalized size = 2.06 \begin {gather*} {\left (e\,x\right )}^m\,\left (\frac {x^2\,\left (A\,b+B\,a\right )\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {x^3\,\left (A\,c+B\,b\right )\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {A\,a\,x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {B\,c\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x)^m*((x^2*(A*b + B*a)*(19*m + 8*m^2 + m^3 + 12))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (x^3*(A*c + B*b)*(1
4*m + 7*m^2 + m^3 + 8))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (A*a*x*(26*m + 9*m^2 + m^3 + 24))/(50*m + 35*m^2
+ 10*m^3 + m^4 + 24) + (B*c*x^4*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24))

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sympy [A]  time = 1.02, size = 1022, normalized size = 12.31 \begin {gather*} \begin {cases} \frac {- \frac {A a}{3 x^{3}} - \frac {A b}{2 x^{2}} - \frac {A c}{x} - \frac {B a}{2 x^{2}} - \frac {B b}{x} + B c \log {\relax (x )}}{e^{4}} & \text {for}\: m = -4 \\\frac {- \frac {A a}{2 x^{2}} - \frac {A b}{x} + A c \log {\relax (x )} - \frac {B a}{x} + B b \log {\relax (x )} + B c x}{e^{3}} & \text {for}\: m = -3 \\\frac {- \frac {A a}{x} + A b \log {\relax (x )} + A c x + B a \log {\relax (x )} + B b x + \frac {B c x^{2}}{2}}{e^{2}} & \text {for}\: m = -2 \\\frac {A a \log {\relax (x )} + A b x + \frac {A c x^{2}}{2} + B a x + \frac {B b x^{2}}{2} + \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 b e^{m} m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 A b e^{m} m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {19 A b e^{m} m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {12 A b e^{m} x^{2} 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 b e^{m} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {7 B b e^{m} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {14 B b e^{m} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 B b e^{m} x^{3} 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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