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

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 = 0.0870757, 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 \[ \frac{a A (e x)^{m+1}}{e (m+1)}+\frac{a B (e x)^{m+2}}{e^2 (m+2)}+\frac{A c (e x)^{m+3}}{e^3 (m+3)}+\frac{B c (e x)^{m+4}}{e^4 (m+4)} \]

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))

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Rubi in Sympy [A]  time = 16.6219, size = 65, normalized size = 0.89 \[ \frac{A a \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{A c \left (e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} + \frac{B a \left (e x\right )^{m + 2}}{e^{2} \left (m + 2\right )} + \frac{B c \left (e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(B*x+A)*(c*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.108368, size = 49, normalized size = 0.67 \[ (e x)^m \left (\frac{a A x}{m+1}+\frac{a B x^2}{m+2}+\frac{A c x^3}{m+3}+\frac{B c x^4}{m+4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 145, normalized size = 2. \[{\frac{ \left ( Bc{m}^{3}{x}^{3}+Ac{m}^{3}{x}^{2}+6\,Bc{m}^{2}{x}^{3}+7\,Ac{m}^{2}{x}^{2}+Ba{m}^{3}x+11\,Bcm{x}^{3}+Aa{m}^{3}+14\,Acm{x}^{2}+8\,Ba{m}^{2}x+6\,Bc{x}^{3}+9\,Aa{m}^{2}+8\,Ac{x}^{2}+19\,Bamx+26\,Aam+12\,aBx+24\,aA \right ) x \left ( ex \right ) ^{m}}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310358, size = 180, normalized size = 2.47 \[ \frac{{\left ({\left (B c m^{3} + 6 \, B c m^{2} + 11 \, B c m + 6 \, B c\right )} x^{4} +{\left (A c m^{3} + 7 \, A c m^{2} + 14 \, A c m + 8 \, A c\right )} x^{3} +{\left (B a m^{3} + 8 \, B a m^{2} + 19 \, B a m + 12 \, B a\right )} x^{2} +{\left (A a m^{3} + 9 \, A a m^{2} + 26 \, A a m + 24 \, A a\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)*(e*x)^m,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 = 2.51115, size = 685, normalized size = 9.38 \[ \begin{cases} \frac{- \frac{A a}{3 x^{3}} - \frac{A c}{x} - \frac{B a}{2 x^{2}} + B c \log{\left (x \right )}}{e^{4}} & \text{for}\: m = -4 \\\frac{- \frac{A a}{2 x^{2}} + A c \log{\left (x \right )} - \frac{B a}{x} + B c x}{e^{3}} & \text{for}\: m = -3 \\\frac{- \frac{A a}{x} + A c x + B a \log{\left (x \right )} + \frac{B c x^{2}}{2}}{e^{2}} & \text{for}\: m = -2 \\\frac{A a \log{\left (x \right )} + \frac{A c x^{2}}{2} + B a x + \frac{B c x^{3}}{3}}{e} & \text{for}\: m = -1 \\\frac{A a e^{m} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{9 A a e^{m} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{26 A a e^{m} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 A a e^{m} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{A c e^{m} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{7 A c e^{m} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{14 A c e^{m} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{8 A c e^{m} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{B a e^{m} m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{8 B a e^{m} m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{19 B a e^{m} m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{12 B a e^{m} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{B c e^{m} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{6 B c e^{m} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{11 B c e^{m} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{6 B c e^{m} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.273072, size = 354, normalized size = 4.85 \[ \frac{B c m^{3} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + A c m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 6 \, B c m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + B a m^{3} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 7 \, A c m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 11 \, B c m x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + A a m^{3} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 8 \, B a m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 14 \, A c m x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 6 \, B c x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 9 \, A a m^{2} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 19 \, B a m x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 8 \, A c x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 26 \, A a m x e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 12 \, B a x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 24 \, A a x e^{\left (m{\rm ln}\left (x\right ) + m\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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