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

Optimal. Leaf size=121 \[ \frac{a^2 A (e x)^{m+1}}{e (m+1)}+\frac{a^2 B (e x)^{m+2}}{e^2 (m+2)}+\frac{2 a A c (e x)^{m+3}}{e^3 (m+3)}+\frac{2 a B c (e x)^{m+4}}{e^4 (m+4)}+\frac{A c^2 (e x)^{m+5}}{e^5 (m+5)}+\frac{B c^2 (e x)^{m+6}}{e^6 (m+6)} \]

[Out]

(a^2*A*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*B*(e*x)^(2 + m))/(e^2*(2 + m)) + (2*a*A
*c*(e*x)^(3 + m))/(e^3*(3 + m)) + (2*a*B*c*(e*x)^(4 + m))/(e^4*(4 + m)) + (A*c^2
*(e*x)^(5 + m))/(e^5*(5 + m)) + (B*c^2*(e*x)^(6 + m))/(e^6*(6 + m))

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Rubi [A]  time = 0.162847, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{a^2 A (e x)^{m+1}}{e (m+1)}+\frac{a^2 B (e x)^{m+2}}{e^2 (m+2)}+\frac{2 a A c (e x)^{m+3}}{e^3 (m+3)}+\frac{2 a B c (e x)^{m+4}}{e^4 (m+4)}+\frac{A c^2 (e x)^{m+5}}{e^5 (m+5)}+\frac{B c^2 (e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*A*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*B*(e*x)^(2 + m))/(e^2*(2 + m)) + (2*a*A
*c*(e*x)^(3 + m))/(e^3*(3 + m)) + (2*a*B*c*(e*x)^(4 + m))/(e^4*(4 + m)) + (A*c^2
*(e*x)^(5 + m))/(e^5*(5 + m)) + (B*c^2*(e*x)^(6 + m))/(e^6*(6 + m))

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Rubi in Sympy [A]  time = 31.1089, size = 112, normalized size = 0.93 \[ \frac{A a^{2} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{2 A a c \left (e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} + \frac{A c^{2} \left (e x\right )^{m + 5}}{e^{5} \left (m + 5\right )} + \frac{B a^{2} \left (e x\right )^{m + 2}}{e^{2} \left (m + 2\right )} + \frac{2 B a c \left (e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} + \frac{B c^{2} \left (e x\right )^{m + 6}}{e^{6} \left (m + 6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*(e*x)**(m + 1)/(e*(m + 1)) + 2*A*a*c*(e*x)**(m + 3)/(e**3*(m + 3)) + A*c*
*2*(e*x)**(m + 5)/(e**5*(m + 5)) + B*a**2*(e*x)**(m + 2)/(e**2*(m + 2)) + 2*B*a*
c*(e*x)**(m + 4)/(e**4*(m + 4)) + B*c**2*(e*x)**(m + 6)/(e**6*(m + 6))

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Mathematica [A]  time = 0.191701, size = 74, normalized size = 0.61 \[ (e x)^m \left (a^2 x \left (\frac{A}{m+1}+\frac{B x}{m+2}\right )+2 a c x^3 \left (\frac{A}{m+3}+\frac{B x}{m+4}\right )+c^2 x^5 \left (\frac{A}{m+5}+\frac{B x}{m+6}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(e*x)^m*(a^2*x*(A/(1 + m) + (B*x)/(2 + m)) + 2*a*c*x^3*(A/(3 + m) + (B*x)/(4 + m
)) + c^2*x^5*(A/(5 + m) + (B*x)/(6 + m)))

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Maple [B]  time = 0.01, size = 395, normalized size = 3.3 \[{\frac{ \left ( B{c}^{2}{m}^{5}{x}^{5}+A{c}^{2}{m}^{5}{x}^{4}+15\,B{c}^{2}{m}^{4}{x}^{5}+16\,A{c}^{2}{m}^{4}{x}^{4}+2\,Bac{m}^{5}{x}^{3}+85\,B{c}^{2}{m}^{3}{x}^{5}+2\,Aac{m}^{5}{x}^{2}+95\,A{c}^{2}{m}^{3}{x}^{4}+34\,Bac{m}^{4}{x}^{3}+225\,B{c}^{2}{m}^{2}{x}^{5}+36\,Aac{m}^{4}{x}^{2}+260\,A{c}^{2}{m}^{2}{x}^{4}+B{a}^{2}{m}^{5}x+214\,Bac{m}^{3}{x}^{3}+274\,B{c}^{2}m{x}^{5}+A{a}^{2}{m}^{5}+242\,Aac{m}^{3}{x}^{2}+324\,A{c}^{2}m{x}^{4}+19\,B{a}^{2}{m}^{4}x+614\,Bac{m}^{2}{x}^{3}+120\,B{c}^{2}{x}^{5}+20\,A{a}^{2}{m}^{4}+744\,Aac{m}^{2}{x}^{2}+144\,A{c}^{2}{x}^{4}+137\,B{a}^{2}{m}^{3}x+792\,Bacm{x}^{3}+155\,A{a}^{2}{m}^{3}+1016\,Aacm{x}^{2}+461\,B{a}^{2}{m}^{2}x+360\,aBc{x}^{3}+580\,A{a}^{2}{m}^{2}+480\,aAc{x}^{2}+702\,B{a}^{2}mx+1044\,A{a}^{2}m+360\,{a}^{2}Bx+720\,A{a}^{2} \right ) x \left ( ex \right ) ^{m}}{ \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(B*c^2*m^5*x^5+A*c^2*m^5*x^4+15*B*c^2*m^4*x^5+16*A*c^2*m^4*x^4+2*B*a*c*m^5*x^3
+85*B*c^2*m^3*x^5+2*A*a*c*m^5*x^2+95*A*c^2*m^3*x^4+34*B*a*c*m^4*x^3+225*B*c^2*m^
2*x^5+36*A*a*c*m^4*x^2+260*A*c^2*m^2*x^4+B*a^2*m^5*x+214*B*a*c*m^3*x^3+274*B*c^2
*m*x^5+A*a^2*m^5+242*A*a*c*m^3*x^2+324*A*c^2*m*x^4+19*B*a^2*m^4*x+614*B*a*c*m^2*
x^3+120*B*c^2*x^5+20*A*a^2*m^4+744*A*a*c*m^2*x^2+144*A*c^2*x^4+137*B*a^2*m^3*x+7
92*B*a*c*m*x^3+155*A*a^2*m^3+1016*A*a*c*m*x^2+461*B*a^2*m^2*x+360*B*a*c*x^3+580*
A*a^2*m^2+480*A*a*c*x^2+702*B*a^2*m*x+1044*A*a^2*m+360*B*a^2*x+720*A*a^2)*(e*x)^
m/(6+m)/(5+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)^2*(B*x + A)*(e*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295769, size = 463, normalized size = 3.83 \[ \frac{{\left ({\left (B c^{2} m^{5} + 15 \, B c^{2} m^{4} + 85 \, B c^{2} m^{3} + 225 \, B c^{2} m^{2} + 274 \, B c^{2} m + 120 \, B c^{2}\right )} x^{6} +{\left (A c^{2} m^{5} + 16 \, A c^{2} m^{4} + 95 \, A c^{2} m^{3} + 260 \, A c^{2} m^{2} + 324 \, A c^{2} m + 144 \, A c^{2}\right )} x^{5} + 2 \,{\left (B a c m^{5} + 17 \, B a c m^{4} + 107 \, B a c m^{3} + 307 \, B a c m^{2} + 396 \, B a c m + 180 \, B a c\right )} x^{4} + 2 \,{\left (A a c m^{5} + 18 \, A a c m^{4} + 121 \, A a c m^{3} + 372 \, A a c m^{2} + 508 \, A a c m + 240 \, A a c\right )} x^{3} +{\left (B a^{2} m^{5} + 19 \, B a^{2} m^{4} + 137 \, B a^{2} m^{3} + 461 \, B a^{2} m^{2} + 702 \, B a^{2} m + 360 \, B a^{2}\right )} x^{2} +{\left (A a^{2} m^{5} + 20 \, A a^{2} m^{4} + 155 \, A a^{2} m^{3} + 580 \, A a^{2} m^{2} + 1044 \, A a^{2} m + 720 \, A a^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((B*c^2*m^5 + 15*B*c^2*m^4 + 85*B*c^2*m^3 + 225*B*c^2*m^2 + 274*B*c^2*m + 120*B*
c^2)*x^6 + (A*c^2*m^5 + 16*A*c^2*m^4 + 95*A*c^2*m^3 + 260*A*c^2*m^2 + 324*A*c^2*
m + 144*A*c^2)*x^5 + 2*(B*a*c*m^5 + 17*B*a*c*m^4 + 107*B*a*c*m^3 + 307*B*a*c*m^2
+ 396*B*a*c*m + 180*B*a*c)*x^4 + 2*(A*a*c*m^5 + 18*A*a*c*m^4 + 121*A*a*c*m^3 +
372*A*a*c*m^2 + 508*A*a*c*m + 240*A*a*c)*x^3 + (B*a^2*m^5 + 19*B*a^2*m^4 + 137*B
*a^2*m^3 + 461*B*a^2*m^2 + 702*B*a^2*m + 360*B*a^2)*x^2 + (A*a^2*m^5 + 20*A*a^2*
m^4 + 155*A*a^2*m^3 + 580*A*a^2*m^2 + 1044*A*a^2*m + 720*A*a^2)*x)*(e*x)^m/(m^6
+ 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

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Sympy [A]  time = 5.13837, size = 2076, normalized size = 17.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise(((-A*a**2/(5*x**5) - 2*A*a*c/(3*x**3) - A*c**2/x - B*a**2/(4*x**4) - B
*a*c/x**2 + B*c**2*log(x))/e**6, Eq(m, -6)), ((-A*a**2/(4*x**4) - A*a*c/x**2 + A
*c**2*log(x) - B*a**2/(3*x**3) - 2*B*a*c/x + B*c**2*x)/e**5, Eq(m, -5)), ((-A*a*
*2/(3*x**3) - 2*A*a*c/x + A*c**2*x - B*a**2/(2*x**2) + 2*B*a*c*log(x) + B*c**2*x
**2/2)/e**4, Eq(m, -4)), ((-A*a**2/(2*x**2) + 2*A*a*c*log(x) + A*c**2*x**2/2 - B
*a**2/x + 2*B*a*c*x + B*c**2*x**3/3)/e**3, Eq(m, -3)), ((-A*a**2/x + 2*A*a*c*x +
A*c**2*x**3/3 + B*a**2*log(x) + B*a*c*x**2 + B*c**2*x**4/4)/e**2, Eq(m, -2)), (
(A*a**2*log(x) + A*a*c*x**2 + A*c**2*x**4/4 + B*a**2*x + 2*B*a*c*x**3/3 + B*c**2
*x**5/5)/e, Eq(m, -1)), (A*a**2*e**m*m**5*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 73
5*m**3 + 1624*m**2 + 1764*m + 720) + 20*A*a**2*e**m*m**4*x*x**m/(m**6 + 21*m**5
+ 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 155*A*a**2*e**m*m**3*x*x**m/
(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*A*a**2*e
**m*m**2*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720
) + 1044*A*a**2*e**m*m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) + 720*A*a**2*e**m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 +
1624*m**2 + 1764*m + 720) + 2*A*a*c*e**m*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m
**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 36*A*a*c*e**m*m**4*x**3*x**m/(m**6
+ 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 242*A*a*c*e**m*m**
3*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) +
744*A*a*c*e**m*m**2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) + 1016*A*a*c*e**m*m*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m
**3 + 1624*m**2 + 1764*m + 720) + 480*A*a*c*e**m*x**3*x**m/(m**6 + 21*m**5 + 175
*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + A*c**2*e**m*m**5*x**5*x**m/(m**6
+ 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 16*A*c**2*e**m*m**
4*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) +
95*A*c**2*e**m*m**3*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) + 260*A*c**2*e**m*m**2*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 73
5*m**3 + 1624*m**2 + 1764*m + 720) + 324*A*c**2*e**m*m*x**5*x**m/(m**6 + 21*m**5
+ 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 144*A*c**2*e**m*x**5*x**m/(
m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + B*a**2*e**m*m
**5*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720)
+ 19*B*a**2*e**m*m**4*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**
2 + 1764*m + 720) + 137*B*a**2*e**m*m**3*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) + 461*B*a**2*e**m*m**2*x**2*x**m/(m**6 + 21
*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 702*B*a**2*e**m*m*x**2
*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 360*B*
a**2*e**m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m +
720) + 2*B*a*c*e**m*m**5*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624
*m**2 + 1764*m + 720) + 34*B*a*c*e**m*m**4*x**4*x**m/(m**6 + 21*m**5 + 175*m**4
+ 735*m**3 + 1624*m**2 + 1764*m + 720) + 214*B*a*c*e**m*m**3*x**4*x**m/(m**6 + 2
1*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 614*B*a*c*e**m*m**2*x
**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 792
*B*a*c*e**m*m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764
*m + 720) + 360*B*a*c*e**m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 162
4*m**2 + 1764*m + 720) + B*c**2*e**m*m**5*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) + 15*B*c**2*e**m*m**4*x**6*x**m/(m**6 + 21
*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 85*B*c**2*e**m*m**3*x*
*6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 225*
B*c**2*e**m*m**2*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1
764*m + 720) + 274*B*c**2*e**m*m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3
+ 1624*m**2 + 1764*m + 720) + 120*B*c**2*e**m*x**6*x**m/(m**6 + 21*m**5 + 175*m
**4 + 735*m**3 + 1624*m**2 + 1764*m + 720), True))

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GIAC/XCAS [A]  time = 0.276291, size = 888, normalized size = 7.34 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*c^2*m^5*x^6*e^(m*ln(x) + m) + A*c^2*m^5*x^5*e^(m*ln(x) + m) + 15*B*c^2*m^4*x^
6*e^(m*ln(x) + m) + 2*B*a*c*m^5*x^4*e^(m*ln(x) + m) + 16*A*c^2*m^4*x^5*e^(m*ln(x
) + m) + 85*B*c^2*m^3*x^6*e^(m*ln(x) + m) + 2*A*a*c*m^5*x^3*e^(m*ln(x) + m) + 34
*B*a*c*m^4*x^4*e^(m*ln(x) + m) + 95*A*c^2*m^3*x^5*e^(m*ln(x) + m) + 225*B*c^2*m^
2*x^6*e^(m*ln(x) + m) + B*a^2*m^5*x^2*e^(m*ln(x) + m) + 36*A*a*c*m^4*x^3*e^(m*ln
(x) + m) + 214*B*a*c*m^3*x^4*e^(m*ln(x) + m) + 260*A*c^2*m^2*x^5*e^(m*ln(x) + m)
+ 274*B*c^2*m*x^6*e^(m*ln(x) + m) + A*a^2*m^5*x*e^(m*ln(x) + m) + 19*B*a^2*m^4*
x^2*e^(m*ln(x) + m) + 242*A*a*c*m^3*x^3*e^(m*ln(x) + m) + 614*B*a*c*m^2*x^4*e^(m
*ln(x) + m) + 324*A*c^2*m*x^5*e^(m*ln(x) + m) + 120*B*c^2*x^6*e^(m*ln(x) + m) +
20*A*a^2*m^4*x*e^(m*ln(x) + m) + 137*B*a^2*m^3*x^2*e^(m*ln(x) + m) + 744*A*a*c*m
^2*x^3*e^(m*ln(x) + m) + 792*B*a*c*m*x^4*e^(m*ln(x) + m) + 144*A*c^2*x^5*e^(m*ln
(x) + m) + 155*A*a^2*m^3*x*e^(m*ln(x) + m) + 461*B*a^2*m^2*x^2*e^(m*ln(x) + m) +
1016*A*a*c*m*x^3*e^(m*ln(x) + m) + 360*B*a*c*x^4*e^(m*ln(x) + m) + 580*A*a^2*m^
2*x*e^(m*ln(x) + m) + 702*B*a^2*m*x^2*e^(m*ln(x) + m) + 480*A*a*c*x^3*e^(m*ln(x)
+ m) + 1044*A*a^2*m*x*e^(m*ln(x) + m) + 360*B*a^2*x^2*e^(m*ln(x) + m) + 720*A*a
^2*x*e^(m*ln(x) + m))/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 72
0)