3.2074 \(\int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx\)
Optimal. Leaf size=90 \[ \frac{\left (c d^2-a e^2\right )^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac{2 c d \left (c d^2-a e^2\right ) (d+e x)^{m+4}}{e^3 (m+4)}+\frac{c^2 d^2 (d+e x)^{m+5}}{e^3 (m+5)} \]
[Out]
((c*d^2 - a*e^2)^2*(d + e*x)^(3 + m))/(e^3*(3 + m)) - (2*c*d*(c*d^2 - a*e^2)*(d
+ e*x)^(4 + m))/(e^3*(4 + m)) + (c^2*d^2*(d + e*x)^(5 + m))/(e^3*(5 + m))
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Rubi [A] time = 0.169438, antiderivative size = 90, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057
\[ \frac{\left (c d^2-a e^2\right )^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac{2 c d \left (c d^2-a e^2\right ) (d+e x)^{m+4}}{e^3 (m+4)}+\frac{c^2 d^2 (d+e x)^{m+5}}{e^3 (m+5)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
((c*d^2 - a*e^2)^2*(d + e*x)^(3 + m))/(e^3*(3 + m)) - (2*c*d*(c*d^2 - a*e^2)*(d
+ e*x)^(4 + m))/(e^3*(4 + m)) + (c^2*d^2*(d + e*x)^(5 + m))/(e^3*(5 + m))
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Rubi in Sympy [A] time = 44.0123, size = 78, normalized size = 0.87 \[ \frac{c^{2} d^{2} \left (d + e x\right )^{m + 5}}{e^{3} \left (m + 5\right )} + \frac{2 c d \left (d + e x\right )^{m + 4} \left (a e^{2} - c d^{2}\right )}{e^{3} \left (m + 4\right )} + \frac{\left (d + e x\right )^{m + 3} \left (a e^{2} - c d^{2}\right )^{2}}{e^{3} \left (m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
[Out]
c**2*d**2*(d + e*x)**(m + 5)/(e**3*(m + 5)) + 2*c*d*(d + e*x)**(m + 4)*(a*e**2 -
c*d**2)/(e**3*(m + 4)) + (d + e*x)**(m + 3)*(a*e**2 - c*d**2)**2/(e**3*(m + 3))
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Mathematica [A] time = 0.171974, size = 100, normalized size = 1.11 \[ \frac{(d+e x)^{m+3} \left (a^2 e^4 \left (m^2+9 m+20\right )-2 a c d e^2 (m+5) (d-e (m+3) x)+c^2 d^2 \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )\right )}{e^3 (m+3) (m+4) (m+5)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
((d + e*x)^(3 + m)*(a^2*e^4*(20 + 9*m + m^2) - 2*a*c*d*e^2*(5 + m)*(d - e*(3 + m
)*x) + c^2*d^2*(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 [B] time = 0.015, size = 183, normalized size = 2. \[{\frac{ \left ( ex+d \right ) ^{3+m} \left ({c}^{2}{d}^{2}{e}^{2}{m}^{2}{x}^{2}+2\,acd{e}^{3}{m}^{2}x+7\,{c}^{2}{d}^{2}{e}^{2}m{x}^{2}+{a}^{2}{e}^{4}{m}^{2}+16\,acd{e}^{3}mx-2\,{c}^{2}{d}^{3}emx+12\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+9\,{a}^{2}{e}^{4}m-2\,ac{d}^{2}{e}^{2}m+30\,xacd{e}^{3}-6\,x{c}^{2}{d}^{3}e+20\,{a}^{2}{e}^{4}-10\,ac{d}^{2}{e}^{2}+2\,{c}^{2}{d}^{4} \right ) }{{e}^{3} \left ({m}^{3}+12\,{m}^{2}+47\,m+60 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
[Out]
(e*x+d)^(3+m)*(c^2*d^2*e^2*m^2*x^2+2*a*c*d*e^3*m^2*x+7*c^2*d^2*e^2*m*x^2+a^2*e^4
*m^2+16*a*c*d*e^3*m*x-2*c^2*d^3*e*m*x+12*c^2*d^2*e^2*x^2+9*a^2*e^4*m-2*a*c*d^2*e
^2*m+30*a*c*d*e^3*x-6*c^2*d^3*e*x+20*a^2*e^4-10*a*c*d^2*e^2+2*c^2*d^4)/e^3/(m^3+
12*m^2+47*m+60)
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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*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2*(e*x + d)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.227554, size = 647, normalized size = 7.19 \[ \frac{{\left (a^{2} d^{3} e^{4} m^{2} + 2 \, c^{2} d^{7} - 10 \, a c d^{5} e^{2} + 20 \, a^{2} d^{3} e^{4} +{\left (c^{2} d^{2} e^{5} m^{2} + 7 \, c^{2} d^{2} e^{5} m + 12 \, c^{2} d^{2} e^{5}\right )} x^{5} +{\left (30 \, c^{2} d^{3} e^{4} + 30 \, a c d e^{6} +{\left (3 \, c^{2} d^{3} e^{4} + 2 \, a c d e^{6}\right )} m^{2} +{\left (19 \, c^{2} d^{3} e^{4} + 16 \, a c d e^{6}\right )} m\right )} x^{4} +{\left (20 \, c^{2} d^{4} e^{3} + 80 \, a c d^{2} e^{5} + 20 \, a^{2} e^{7} +{\left (3 \, c^{2} d^{4} e^{3} + 6 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} m^{2} +{\left (15 \, c^{2} d^{4} e^{3} + 46 \, a c d^{2} e^{5} + 9 \, a^{2} e^{7}\right )} m\right )} x^{3} +{\left (60 \, a c d^{3} e^{4} + 60 \, a^{2} d e^{6} +{\left (c^{2} d^{5} e^{2} + 6 \, a c d^{3} e^{4} + 3 \, a^{2} d e^{6}\right )} m^{2} +{\left (c^{2} d^{5} e^{2} + 42 \, a c d^{3} e^{4} + 27 \, a^{2} d e^{6}\right )} m\right )} x^{2} -{\left (2 \, a c d^{5} e^{2} - 9 \, a^{2} d^{3} e^{4}\right )} m +{\left (60 \, a^{2} d^{2} e^{5} +{\left (2 \, a c d^{4} e^{3} + 3 \, a^{2} d^{2} e^{5}\right )} m^{2} -{\left (2 \, c^{2} d^{6} e - 10 \, a c d^{4} e^{3} - 27 \, a^{2} d^{2} e^{5}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2*(e*x + d)^m,x, algorithm="fricas")
[Out]
(a^2*d^3*e^4*m^2 + 2*c^2*d^7 - 10*a*c*d^5*e^2 + 20*a^2*d^3*e^4 + (c^2*d^2*e^5*m^
2 + 7*c^2*d^2*e^5*m + 12*c^2*d^2*e^5)*x^5 + (30*c^2*d^3*e^4 + 30*a*c*d*e^6 + (3*
c^2*d^3*e^4 + 2*a*c*d*e^6)*m^2 + (19*c^2*d^3*e^4 + 16*a*c*d*e^6)*m)*x^4 + (20*c^
2*d^4*e^3 + 80*a*c*d^2*e^5 + 20*a^2*e^7 + (3*c^2*d^4*e^3 + 6*a*c*d^2*e^5 + a^2*e
^7)*m^2 + (15*c^2*d^4*e^3 + 46*a*c*d^2*e^5 + 9*a^2*e^7)*m)*x^3 + (60*a*c*d^3*e^4
+ 60*a^2*d*e^6 + (c^2*d^5*e^2 + 6*a*c*d^3*e^4 + 3*a^2*d*e^6)*m^2 + (c^2*d^5*e^2
+ 42*a*c*d^3*e^4 + 27*a^2*d*e^6)*m)*x^2 - (2*a*c*d^5*e^2 - 9*a^2*d^3*e^4)*m + (
60*a^2*d^2*e^5 + (2*a*c*d^4*e^3 + 3*a^2*d^2*e^5)*m^2 - (2*c^2*d^6*e - 10*a*c*d^4
*e^3 - 27*a^2*d^2*e^5)*m)*x)*(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 = 13.3204, size = 2618, normalized size = 29.09 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**m*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
[Out]
Piecewise((c**2*d**4*d**m*x**3/3, Eq(e, 0)), (-a**2*e**4/(2*d**2*e**3 + 4*d*e**4
*x + 2*e**5*x**2) - 2*a*c*d**2*e**2/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) - 4
*a*c*d*e**3*x/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 2*c**2*d**4*log(d/e + x
)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 3*c**2*d**4/(2*d**2*e**3 + 4*d*e**4
*x + 2*e**5*x**2) + 4*c**2*d**3*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**3*e*x/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 2*c**2*d*
*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)),
(-a**2*d*e**4/(3*d**2*e**3 + 3*d*e**4*x) + 2*a**2*e**5*x/(3*d**2*e**3 + 3*d*e**4
*x) + 6*a*c*d**3*e**2*log(d/e + x)/(3*d**2*e**3 + 3*d*e**4*x) + 2*a*c*d**3*e**2/
(3*d**2*e**3 + 3*d*e**4*x) + 6*a*c*d**2*e**3*x*log(d/e + x)/(3*d**2*e**3 + 3*d*e
**4*x) - 4*a*c*d**2*e**3*x/(3*d**2*e**3 + 3*d*e**4*x) - 6*c**2*d**5*log(d/e + x)
/(3*d**2*e**3 + 3*d*e**4*x) - 2*c**2*d**5/(3*d**2*e**3 + 3*d*e**4*x) - 6*c**2*d*
*4*e*x*log(d/e + x)/(3*d**2*e**3 + 3*d*e**4*x) + 4*c**2*d**4*e*x/(3*d**2*e**3 +
3*d*e**4*x) + 3*c**2*d**3*e**2*x**2/(3*d**2*e**3 + 3*d*e**4*x), Eq(m, -4)), (a**
2*e*log(d/e + x) - 2*a*c*d**2*log(d/e + x)/e + 2*a*c*d*x + c**2*d**4*log(d/e + x
)/e**3 - c**2*d**3*x/e**2 + c**2*d**2*x**2/(2*e), Eq(m, -3)), (a**2*d**3*e**4*m*
*2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 9*a**2*d**3*e
**4*m*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 20*a**2*d*
*3*e**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 3*a**2*d
**2*e**5*m**2*x*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) +
27*a**2*d**2*e**5*m*x*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e*
*3) + 60*a**2*d**2*e**5*x*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 6
0*e**3) + 3*a**2*d*e**6*m**2*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e*
*3*m + 60*e**3) + 27*a**2*d*e**6*m*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 +
47*e**3*m + 60*e**3) + 60*a**2*d*e**6*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m*
*2 + 47*e**3*m + 60*e**3) + a**2*e**7*m**2*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**
3*m**2 + 47*e**3*m + 60*e**3) + 9*a**2*e**7*m*x**3*(d + e*x)**m/(e**3*m**3 + 12*
e**3*m**2 + 47*e**3*m + 60*e**3) + 20*a**2*e**7*x**3*(d + e*x)**m/(e**3*m**3 + 1
2*e**3*m**2 + 47*e**3*m + 60*e**3) - 2*a*c*d**5*e**2*m*(d + e*x)**m/(e**3*m**3 +
12*e**3*m**2 + 47*e**3*m + 60*e**3) - 10*a*c*d**5*e**2*(d + e*x)**m/(e**3*m**3
+ 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 2*a*c*d**4*e**3*m**2*x*(d + e*x)**m/(e**
3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 10*a*c*d**4*e**3*m*x*(d + e*x)**m
/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 6*a*c*d**3*e**4*m**2*x**2*(d
+ e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 42*a*c*d**3*e**4*m
*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 60*a*c*d**
3*e**4*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 6*a*
c*d**2*e**5*m**2*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e*
*3) + 46*a*c*d**2*e**5*m*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m
+ 60*e**3) + 80*a*c*d**2*e**5*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*
e**3*m + 60*e**3) + 2*a*c*d*e**6*m**2*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**
2 + 47*e**3*m + 60*e**3) + 16*a*c*d*e**6*m*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**
3*m**2 + 47*e**3*m + 60*e**3) + 30*a*c*d*e**6*x**4*(d + e*x)**m/(e**3*m**3 + 12*
e**3*m**2 + 47*e**3*m + 60*e**3) + 2*c**2*d**7*(d + e*x)**m/(e**3*m**3 + 12*e**3
*m**2 + 47*e**3*m + 60*e**3) - 2*c**2*d**6*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**5*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**5*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**4*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**4*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**4
*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**3*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**3*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**3*e**4*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 4
7*e**3*m + 60*e**3) + c**2*d**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*d**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*d**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 = 0.220364, size = 1188, normalized size = 13.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2*(e*x + d)^m,x, algorithm="giac")
[Out]
(c^2*d^2*m^2*x^5*e^(m*ln(x*e + d) + 5) + 3*c^2*d^3*m^2*x^4*e^(m*ln(x*e + d) + 4)
+ 3*c^2*d^4*m^2*x^3*e^(m*ln(x*e + d) + 3) + c^2*d^5*m^2*x^2*e^(m*ln(x*e + d) +
2) + 7*c^2*d^2*m*x^5*e^(m*ln(x*e + d) + 5) + 19*c^2*d^3*m*x^4*e^(m*ln(x*e + d) +
4) + 15*c^2*d^4*m*x^3*e^(m*ln(x*e + d) + 3) + c^2*d^5*m*x^2*e^(m*ln(x*e + d) +
2) - 2*c^2*d^6*m*x*e^(m*ln(x*e + d) + 1) + 2*a*c*d*m^2*x^4*e^(m*ln(x*e + d) + 6)
+ 6*a*c*d^2*m^2*x^3*e^(m*ln(x*e + d) + 5) + 12*c^2*d^2*x^5*e^(m*ln(x*e + d) + 5
) + 6*a*c*d^3*m^2*x^2*e^(m*ln(x*e + d) + 4) + 30*c^2*d^3*x^4*e^(m*ln(x*e + d) +
4) + 2*a*c*d^4*m^2*x*e^(m*ln(x*e + d) + 3) + 20*c^2*d^4*x^3*e^(m*ln(x*e + d) + 3
) + 2*c^2*d^7*e^(m*ln(x*e + d)) + 16*a*c*d*m*x^4*e^(m*ln(x*e + d) + 6) + 46*a*c*
d^2*m*x^3*e^(m*ln(x*e + d) + 5) + 42*a*c*d^3*m*x^2*e^(m*ln(x*e + d) + 4) + 10*a*
c*d^4*m*x*e^(m*ln(x*e + d) + 3) - 2*a*c*d^5*m*e^(m*ln(x*e + d) + 2) + a^2*m^2*x^
3*e^(m*ln(x*e + d) + 7) + 3*a^2*d*m^2*x^2*e^(m*ln(x*e + d) + 6) + 30*a*c*d*x^4*e
^(m*ln(x*e + d) + 6) + 3*a^2*d^2*m^2*x*e^(m*ln(x*e + d) + 5) + 80*a*c*d^2*x^3*e^
(m*ln(x*e + d) + 5) + a^2*d^3*m^2*e^(m*ln(x*e + d) + 4) + 60*a*c*d^3*x^2*e^(m*ln
(x*e + d) + 4) - 10*a*c*d^5*e^(m*ln(x*e + d) + 2) + 9*a^2*m*x^3*e^(m*ln(x*e + d)
+ 7) + 27*a^2*d*m*x^2*e^(m*ln(x*e + d) + 6) + 27*a^2*d^2*m*x*e^(m*ln(x*e + d) +
5) + 9*a^2*d^3*m*e^(m*ln(x*e + d) + 4) + 20*a^2*x^3*e^(m*ln(x*e + d) + 7) + 60*
a^2*d*x^2*e^(m*ln(x*e + d) + 6) + 60*a^2*d^2*x*e^(m*ln(x*e + d) + 5) + 20*a^2*d^
3*e^(m*ln(x*e + d) + 4))/(m^3*e^3 + 12*m^2*e^3 + 47*m*e^3 + 60*e^3)