3.434 \(\int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx\)

Optimal. Leaf size=95 \[ -\frac{c^3 d^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac{3 c^3 d^2 (d+e x)^{m+5}}{e^4 (m+5)}-\frac{3 c^3 d (d+e x)^{m+6}}{e^4 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^4 (m+7)} \]

[Out]

-((c^3*d^3*(d + e*x)^(4 + m))/(e^4*(4 + m))) + (3*c^3*d^2*(d + e*x)^(5 + m))/(e^
4*(5 + m)) - (3*c^3*d*(d + e*x)^(6 + m))/(e^4*(6 + m)) + (c^3*(d + e*x)^(7 + m))
/(e^4*(7 + m))

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Rubi [A]  time = 0.148869, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{c^3 d^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac{3 c^3 d^2 (d+e x)^{m+5}}{e^4 (m+5)}-\frac{3 c^3 d (d+e x)^{m+6}}{e^4 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^4 (m+7)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m*(c*d*x + c*e*x^2)^3,x]

[Out]

-((c^3*d^3*(d + e*x)^(4 + m))/(e^4*(4 + m))) + (3*c^3*d^2*(d + e*x)^(5 + m))/(e^
4*(5 + m)) - (3*c^3*d*(d + e*x)^(6 + m))/(e^4*(6 + m)) + (c^3*(d + e*x)^(7 + m))
/(e^4*(7 + m))

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Rubi in Sympy [A]  time = 30.9991, size = 85, normalized size = 0.89 \[ - \frac{c^{3} d^{3} \left (d + e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} + \frac{3 c^{3} d^{2} \left (d + e x\right )^{m + 5}}{e^{4} \left (m + 5\right )} - \frac{3 c^{3} d \left (d + e x\right )^{m + 6}}{e^{4} \left (m + 6\right )} + \frac{c^{3} \left (d + e x\right )^{m + 7}}{e^{4} \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(c*e*x**2+c*d*x)**3,x)

[Out]

-c**3*d**3*(d + e*x)**(m + 4)/(e**4*(m + 4)) + 3*c**3*d**2*(d + e*x)**(m + 5)/(e
**4*(m + 5)) - 3*c**3*d*(d + e*x)**(m + 6)/(e**4*(m + 6)) + c**3*(d + e*x)**(m +
7)/(e**4*(m + 7))

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Mathematica [A]  time = 0.0945339, size = 89, normalized size = 0.94 \[ \frac{c^3 (d+e x)^{m+4} \left (-6 d^3+6 d^2 e (m+4) x-3 d e^2 \left (m^2+9 m+20\right ) x^2+e^3 \left (m^3+15 m^2+74 m+120\right ) x^3\right )}{e^4 (m+4) (m+5) (m+6) (m+7)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^m*(c*d*x + c*e*x^2)^3,x]

[Out]

(c^3*(d + e*x)^(4 + m)*(-6*d^3 + 6*d^2*e*(4 + m)*x - 3*d*e^2*(20 + 9*m + m^2)*x^
2 + e^3*(120 + 74*m + 15*m^2 + m^3)*x^3))/(e^4*(4 + m)*(5 + m)*(6 + m)*(7 + m))

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Maple [A]  time = 0.008, size = 129, normalized size = 1.4 \[ -{\frac{{c}^{3} \left ( ex+d \right ) ^{4+m} \left ( -{e}^{3}{m}^{3}{x}^{3}-15\,{e}^{3}{m}^{2}{x}^{3}+3\,d{e}^{2}{m}^{2}{x}^{2}-74\,{e}^{3}m{x}^{3}+27\,d{e}^{2}m{x}^{2}-120\,{x}^{3}{e}^{3}-6\,{d}^{2}emx+60\,d{x}^{2}{e}^{2}-24\,{d}^{2}xe+6\,{d}^{3} \right ) }{{e}^{4} \left ({m}^{4}+22\,{m}^{3}+179\,{m}^{2}+638\,m+840 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*e*x^2+c*d*x)^3,x)

[Out]

-c^3*(e*x+d)^(4+m)*(-e^3*m^3*x^3-15*e^3*m^2*x^3+3*d*e^2*m^2*x^2-74*e^3*m*x^3+27*
d*e^2*m*x^2-120*e^3*x^3-6*d^2*e*m*x+60*d*e^2*x^2-24*d^2*e*x+6*d^3)/e^4/(m^4+22*m
^3+179*m^2+638*m+840)

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Maxima [A]  time = 0.780694, size = 910, normalized size = 9.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e*x^2 + c*d*x)^3*(e*x + d)^m,x, algorithm="maxima")

[Out]

((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*
d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*c^3*d^3/((m^4 + 10*m^3 + 35*m^2 +
50*m + 24)*e^4) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3
+ 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^
3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*c^3*d^2/((m^5 + 15*m^4 + 85*m^3 +
225*m^2 + 274*m + 120)*e^4) + 3*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120
)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^4 + 6*m^3 +
11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d
^4*e^2*x^2 + 120*d^5*e*m*x - 120*d^6)*(e*x + d)^m*c^3*d/((m^6 + 21*m^5 + 175*m^4
+ 735*m^3 + 1624*m^2 + 1764*m + 720)*e^4) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3
+ 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2
+ 120*m)*d*e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*
(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4*x^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*e^3*x^3 +
360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6*e*m*x + 720*d^7)*(e*x + d)^m*c^3/((m^7 + 28
*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*e^4)

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Fricas [A]  time = 0.233696, size = 463, normalized size = 4.87 \[ \frac{{\left (6 \, c^{3} d^{6} e m x - 6 \, c^{3} d^{7} +{\left (c^{3} e^{7} m^{3} + 15 \, c^{3} e^{7} m^{2} + 74 \, c^{3} e^{7} m + 120 \, c^{3} e^{7}\right )} x^{7} +{\left (4 \, c^{3} d e^{6} m^{3} + 57 \, c^{3} d e^{6} m^{2} + 269 \, c^{3} d e^{6} m + 420 \, c^{3} d e^{6}\right )} x^{6} + 6 \,{\left (c^{3} d^{2} e^{5} m^{3} + 13 \, c^{3} d^{2} e^{5} m^{2} + 57 \, c^{3} d^{2} e^{5} m + 84 \, c^{3} d^{2} e^{5}\right )} x^{5} + 2 \,{\left (2 \, c^{3} d^{3} e^{4} m^{3} + 21 \, c^{3} d^{3} e^{4} m^{2} + 79 \, c^{3} d^{3} e^{4} m + 105 \, c^{3} d^{3} e^{4}\right )} x^{4} +{\left (c^{3} d^{4} e^{3} m^{3} + 3 \, c^{3} d^{4} e^{3} m^{2} + 2 \, c^{3} d^{4} e^{3} m\right )} x^{3} - 3 \,{\left (c^{3} d^{5} e^{2} m^{2} + c^{3} d^{5} e^{2} m\right )} x^{2}\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 22 \, e^{4} m^{3} + 179 \, e^{4} m^{2} + 638 \, e^{4} m + 840 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e*x^2 + c*d*x)^3*(e*x + d)^m,x, algorithm="fricas")

[Out]

(6*c^3*d^6*e*m*x - 6*c^3*d^7 + (c^3*e^7*m^3 + 15*c^3*e^7*m^2 + 74*c^3*e^7*m + 12
0*c^3*e^7)*x^7 + (4*c^3*d*e^6*m^3 + 57*c^3*d*e^6*m^2 + 269*c^3*d*e^6*m + 420*c^3
*d*e^6)*x^6 + 6*(c^3*d^2*e^5*m^3 + 13*c^3*d^2*e^5*m^2 + 57*c^3*d^2*e^5*m + 84*c^
3*d^2*e^5)*x^5 + 2*(2*c^3*d^3*e^4*m^3 + 21*c^3*d^3*e^4*m^2 + 79*c^3*d^3*e^4*m +
105*c^3*d^3*e^4)*x^4 + (c^3*d^4*e^3*m^3 + 3*c^3*d^4*e^3*m^2 + 2*c^3*d^4*e^3*m)*x
^3 - 3*(c^3*d^5*e^2*m^2 + c^3*d^5*e^2*m)*x^2)*(e*x + d)^m/(e^4*m^4 + 22*e^4*m^3
+ 179*e^4*m^2 + 638*e^4*m + 840*e^4)

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Sympy [A]  time = 16.6263, size = 2218, normalized size = 23.35 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(c*e*x**2+c*d*x)**3,x)

[Out]

Piecewise((c**3*d**3*d**m*x**4/4, Eq(e, 0)), (6*c**3*d**3*log(d/e + x)/(6*d**3*e
**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*c**3*d**3/(6*d**3*e**4
+ 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c**3*d**2*e*x*log(d/e + x
)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 27*c**3*d**2*e
*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c**3*d*e**
2*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3
) + 18*c**3*d*e**2*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*
x**3) + 6*c**3*e**3*x**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*
x**2 + 6*e**7*x**3), Eq(m, -7)), (-6*c**3*d**3*log(d/e + x)/(2*d**2*e**4 + 4*d*e
**5*x + 2*e**6*x**2) - 15*c**3*d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 1
2*c**3*d**2*e*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 24*c**3*
d**2*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c**3*d*e**2*x**2*log(d/e +
x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c**3*d*e**2*x**2/(2*d**2*e**4 +
4*d*e**5*x + 2*e**6*x**2) + 2*c**3*e**3*x**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6
*x**2), Eq(m, -6)), (6*c**3*d**3*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 12*c**3*d*
*3/(2*d*e**4 + 2*e**5*x) + 6*c**3*d**2*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) +
6*c**3*d**2*e*x/(2*d*e**4 + 2*e**5*x) - 3*c**3*d*e**2*x**2/(2*d*e**4 + 2*e**5*x)
+ c**3*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -5)), (-c**3*d**3*log(d/e + x)/e*
*4 + c**3*d**2*x/e**3 - c**3*d*x**2/(2*e**2) + c**3*x**3/(3*e), Eq(m, -4)), (-6*
c**3*d**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m +
840*e**4) + 6*c**3*d**6*e*m*x*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*
m**2 + 638*e**4*m + 840*e**4) - 3*c**3*d**5*e**2*m**2*x**2*(d + e*x)**m/(e**4*m*
*4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 3*c**3*d**5*e**2*m*
x**2*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e
**4) + c**3*d**4*e**3*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**
4*m**2 + 638*e**4*m + 840*e**4) + 3*c**3*d**4*e**3*m**2*x**3*(d + e*x)**m/(e**4*
m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 2*c**3*d**4*e**3*
m*x**3*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840
*e**4) + 4*c**3*d**3*e**4*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179
*e**4*m**2 + 638*e**4*m + 840*e**4) + 42*c**3*d**3*e**4*m**2*x**4*(d + e*x)**m/(
e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 158*c**3*d**
3*e**4*m*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*
m + 840*e**4) + 210*c**3*d**3*e**4*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 +
179*e**4*m**2 + 638*e**4*m + 840*e**4) + 6*c**3*d**2*e**5*m**3*x**5*(d + e*x)**
m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 78*c**3*d
**2*e**5*m**2*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*
e**4*m + 840*e**4) + 342*c**3*d**2*e**5*m*x**5*(d + e*x)**m/(e**4*m**4 + 22*e**4
*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 504*c**3*d**2*e**5*x**5*(d + e*
x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 4*c**
3*d*e**6*m**3*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*
e**4*m + 840*e**4) + 57*c**3*d*e**6*m**2*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*
m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 269*c**3*d*e**6*m*x**6*(d + e*x)
**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 420*c**
3*d*e**6*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*
m + 840*e**4) + c**3*e**7*m**3*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179
*e**4*m**2 + 638*e**4*m + 840*e**4) + 15*c**3*e**7*m**2*x**7*(d + e*x)**m/(e**4*
m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 74*c**3*e**7*m*x*
*7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**
4) + 120*c**3*e**7*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 +
638*e**4*m + 840*e**4), True))

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GIAC/XCAS [A]  time = 0.220341, size = 775, normalized size = 8.16 \[ \frac{c^{3} m^{3} x^{7} e^{\left (m{\rm ln}\left (x e + d\right ) + 7\right )} + 4 \, c^{3} d m^{3} x^{6} e^{\left (m{\rm ln}\left (x e + d\right ) + 6\right )} + 6 \, c^{3} d^{2} m^{3} x^{5} e^{\left (m{\rm ln}\left (x e + d\right ) + 5\right )} + 4 \, c^{3} d^{3} m^{3} x^{4} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} + c^{3} d^{4} m^{3} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 15 \, c^{3} m^{2} x^{7} e^{\left (m{\rm ln}\left (x e + d\right ) + 7\right )} + 57 \, c^{3} d m^{2} x^{6} e^{\left (m{\rm ln}\left (x e + d\right ) + 6\right )} + 78 \, c^{3} d^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (x e + d\right ) + 5\right )} + 42 \, c^{3} d^{3} m^{2} x^{4} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} + 3 \, c^{3} d^{4} m^{2} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} - 3 \, c^{3} d^{5} m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 74 \, c^{3} m x^{7} e^{\left (m{\rm ln}\left (x e + d\right ) + 7\right )} + 269 \, c^{3} d m x^{6} e^{\left (m{\rm ln}\left (x e + d\right ) + 6\right )} + 342 \, c^{3} d^{2} m x^{5} e^{\left (m{\rm ln}\left (x e + d\right ) + 5\right )} + 158 \, c^{3} d^{3} m x^{4} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} + 2 \, c^{3} d^{4} m x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} - 3 \, c^{3} d^{5} m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 6 \, c^{3} d^{6} m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 120 \, c^{3} x^{7} e^{\left (m{\rm ln}\left (x e + d\right ) + 7\right )} + 420 \, c^{3} d x^{6} e^{\left (m{\rm ln}\left (x e + d\right ) + 6\right )} + 504 \, c^{3} d^{2} x^{5} e^{\left (m{\rm ln}\left (x e + d\right ) + 5\right )} + 210 \, c^{3} d^{3} x^{4} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} - 6 \, c^{3} d^{7} e^{\left (m{\rm ln}\left (x e + d\right )\right )}}{m^{4} e^{4} + 22 \, m^{3} e^{4} + 179 \, m^{2} e^{4} + 638 \, m e^{4} + 840 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e*x^2 + c*d*x)^3*(e*x + d)^m,x, algorithm="giac")

[Out]

(c^3*m^3*x^7*e^(m*ln(x*e + d) + 7) + 4*c^3*d*m^3*x^6*e^(m*ln(x*e + d) + 6) + 6*c
^3*d^2*m^3*x^5*e^(m*ln(x*e + d) + 5) + 4*c^3*d^3*m^3*x^4*e^(m*ln(x*e + d) + 4) +
c^3*d^4*m^3*x^3*e^(m*ln(x*e + d) + 3) + 15*c^3*m^2*x^7*e^(m*ln(x*e + d) + 7) +
57*c^3*d*m^2*x^6*e^(m*ln(x*e + d) + 6) + 78*c^3*d^2*m^2*x^5*e^(m*ln(x*e + d) + 5
) + 42*c^3*d^3*m^2*x^4*e^(m*ln(x*e + d) + 4) + 3*c^3*d^4*m^2*x^3*e^(m*ln(x*e + d
) + 3) - 3*c^3*d^5*m^2*x^2*e^(m*ln(x*e + d) + 2) + 74*c^3*m*x^7*e^(m*ln(x*e + d)
+ 7) + 269*c^3*d*m*x^6*e^(m*ln(x*e + d) + 6) + 342*c^3*d^2*m*x^5*e^(m*ln(x*e +
d) + 5) + 158*c^3*d^3*m*x^4*e^(m*ln(x*e + d) + 4) + 2*c^3*d^4*m*x^3*e^(m*ln(x*e
+ d) + 3) - 3*c^3*d^5*m*x^2*e^(m*ln(x*e + d) + 2) + 6*c^3*d^6*m*x*e^(m*ln(x*e +
d) + 1) + 120*c^3*x^7*e^(m*ln(x*e + d) + 7) + 420*c^3*d*x^6*e^(m*ln(x*e + d) + 6
) + 504*c^3*d^2*x^5*e^(m*ln(x*e + d) + 5) + 210*c^3*d^3*x^4*e^(m*ln(x*e + d) + 4
) - 6*c^3*d^7*e^(m*ln(x*e + d)))/(m^4*e^4 + 22*m^3*e^4 + 179*m^2*e^4 + 638*m*e^4
+ 840*e^4)