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

Optimal. Leaf size=69 \[ \frac{c^2 d^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac{2 c^2 d (d+e x)^{m+4}}{e^3 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^3 (m+5)} \]

[Out]

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

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Rubi [A]  time = 0.106037, antiderivative size = 69, 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 \[ \frac{c^2 d^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac{2 c^2 d (d+e x)^{m+4}}{e^3 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^3 (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.1838, size = 61, normalized size = 0.88 \[ \frac{c^{2} d^{2} \left (d + e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} - \frac{2 c^{2} d \left (d + e x\right )^{m + 4}}{e^{3} \left (m + 4\right )} + \frac{c^{2} \left (d + e x\right )^{m + 5}}{e^{3} \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0462225, size = 60, normalized size = 0.87 \[ \frac{c^2 (d+e x)^{m+3} \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )}{e^3 (m+3) (m+4) (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^(3 + m)*(2*d^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2))/(e^
3*(3 + m)*(4 + m)*(5 + m))

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Maple [A]  time = 0.008, size = 76, normalized size = 1.1 \[{\frac{ \left ( ex+d \right ) ^{3+m} \left ({e}^{2}{m}^{2}{x}^{2}+7\,{e}^{2}m{x}^{2}-2\,demx+12\,{x}^{2}{e}^{2}-6\,dxe+2\,{d}^{2} \right ){c}^{2}}{{e}^{3} \left ({m}^{3}+12\,{m}^{2}+47\,m+60 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x+d)^(3+m)*(e^2*m^2*x^2+7*e^2*m*x^2-2*d*e*m*x+12*e^2*x^2-6*d*e*x+2*d^2)*c^2/e
^3/(m^3+12*m^2+47*m+60)

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Maxima [A]  time = 0.757079, size = 436, normalized size = 6.32 \[ \frac{{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} +{\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )}{\left (e x + d\right )}^{m} c^{2} d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac{2 \,{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} +{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \,{\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )}{\left (e x + d\right )}^{m} c^{2} d}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{3}} + \frac{{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} +{\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \,{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \,{\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )}{\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^
m*c^2*d^2/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 2*((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^2*d/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^3) + ((m^4 + 10*m^3 + 3
5*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^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^3)

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Fricas [A]  time = 0.231586, size = 271, normalized size = 3.93 \[ -\frac{{\left (2 \, c^{2} d^{4} e m x - 2 \, c^{2} d^{5} -{\left (c^{2} e^{5} m^{2} + 7 \, c^{2} e^{5} m + 12 \, c^{2} e^{5}\right )} x^{5} -{\left (3 \, c^{2} d e^{4} m^{2} + 19 \, c^{2} d e^{4} m + 30 \, c^{2} d e^{4}\right )} x^{4} -{\left (3 \, c^{2} d^{2} e^{3} m^{2} + 15 \, c^{2} d^{2} e^{3} m + 20 \, c^{2} d^{2} e^{3}\right )} x^{3} -{\left (c^{2} d^{3} e^{2} m^{2} + c^{2} d^{3} e^{2} m\right )} x^{2}\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*c^2*d^4*e*m*x - 2*c^2*d^5 - (c^2*e^5*m^2 + 7*c^2*e^5*m + 12*c^2*e^5)*x^5 - (
3*c^2*d*e^4*m^2 + 19*c^2*d*e^4*m + 30*c^2*d*e^4)*x^4 - (3*c^2*d^2*e^3*m^2 + 15*c
^2*d^2*e^3*m + 20*c^2*d^2*e^3)*x^3 - (c^2*d^3*e^2*m^2 + c^2*d^3*e^2*m)*x^2)*(e*x
+ d)^m/(e^3*m^3 + 12*e^3*m^2 + 47*e^3*m + 60*e^3)

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Sympy [A]  time = 7.5607, size = 1012, normalized size = 14.67 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((c**2*d**2*d**m*x**3/3, Eq(e, 0)), (2*c**2*d**2*log(d/e + x)/(2*d**2*e
**3 + 4*d*e**4*x + 2*e**5*x**2) + 3*c**2*d**2/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5
*x**2) + 4*c**2*d*e*x*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 4*
c**2*d*e*x/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 2*c**2*e**2*x**2*log(d/e +
x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2), Eq(m, -5)), (-6*c**2*d**2*log(d/e
+ x)/(3*d*e**3 + 3*e**4*x) - 2*c**2*d**2/(3*d*e**3 + 3*e**4*x) - 6*c**2*d*e*x*lo
g(d/e + x)/(3*d*e**3 + 3*e**4*x) + 4*c**2*d*e*x/(3*d*e**3 + 3*e**4*x) + 3*c**2*e
**2*x**2/(3*d*e**3 + 3*e**4*x), Eq(m, -4)), (c**2*d**2*log(d/e + x)/e**3 - c**2*
d*x/e**2 + c**2*x**2/(2*e), Eq(m, -3)), (2*c**2*d**5*(d + e*x)**m/(e**3*m**3 + 1
2*e**3*m**2 + 47*e**3*m + 60*e**3) - 2*c**2*d**4*e*m*x*(d + e*x)**m/(e**3*m**3 +
12*e**3*m**2 + 47*e**3*m + 60*e**3) + c**2*d**3*e**2*m**2*x**2*(d + e*x)**m/(e*
*3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + c**2*d**3*e**2*m*x**2*(d + e*x)*
*m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 3*c**2*d**2*e**3*m**2*x**3
*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 15*c**2*d**2*e*
*3*m*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 20*c**
2*d**2*e**3*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) +
3*c**2*d*e**4*m**2*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60
*e**3) + 19*c**2*d*e**4*m*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*
m + 60*e**3) + 30*c**2*d*e**4*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e
**3*m + 60*e**3) + c**2*e**5*m**2*x**5*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 +
47*e**3*m + 60*e**3) + 7*c**2*e**5*m*x**5*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2
+ 47*e**3*m + 60*e**3) + 12*c**2*e**5*x**5*(d + e*x)**m/(e**3*m**3 + 12*e**3*m*
*2 + 47*e**3*m + 60*e**3), True))

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GIAC/XCAS [A]  time = 0.215189, size = 429, normalized size = 6.22 \[ \frac{c^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (x e + d\right ) + 5\right )} + 3 \, c^{2} d m^{2} x^{4} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} + 3 \, c^{2} d^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + c^{2} d^{3} m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 7 \, c^{2} m x^{5} e^{\left (m{\rm ln}\left (x e + d\right ) + 5\right )} + 19 \, c^{2} d m x^{4} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} + 15 \, c^{2} d^{2} m x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + c^{2} d^{3} m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - 2 \, c^{2} d^{4} m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 12 \, c^{2} x^{5} e^{\left (m{\rm ln}\left (x e + d\right ) + 5\right )} + 30 \, c^{2} d x^{4} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} + 20 \, c^{2} d^{2} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 2 \, c^{2} d^{5} e^{\left (m{\rm ln}\left (x e + d\right )\right )}}{m^{3} e^{3} + 12 \, m^{2} e^{3} + 47 \, m e^{3} + 60 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^2*m^2*x^5*e^(m*ln(x*e + d) + 5) + 3*c^2*d*m^2*x^4*e^(m*ln(x*e + d) + 4) + 3*c
^2*d^2*m^2*x^3*e^(m*ln(x*e + d) + 3) + c^2*d^3*m^2*x^2*e^(m*ln(x*e + d) + 2) + 7
*c^2*m*x^5*e^(m*ln(x*e + d) + 5) + 19*c^2*d*m*x^4*e^(m*ln(x*e + d) + 4) + 15*c^2
*d^2*m*x^3*e^(m*ln(x*e + d) + 3) + c^2*d^3*m*x^2*e^(m*ln(x*e + d) + 2) - 2*c^2*d
^4*m*x*e^(m*ln(x*e + d) + 1) + 12*c^2*x^5*e^(m*ln(x*e + d) + 5) + 30*c^2*d*x^4*e
^(m*ln(x*e + d) + 4) + 20*c^2*d^2*x^3*e^(m*ln(x*e + d) + 3) + 2*c^2*d^5*e^(m*ln(
x*e + d)))/(m^3*e^3 + 12*m^2*e^3 + 47*m*e^3 + 60*e^3)