3.1078 \(\int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=21 \[ \frac{c^2 (d+e x)^{m+5}}{e (m+5)} \]

[Out]

(c^2*(d + e*x)^(5 + m))/(e*(5 + m))

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Rubi [A]  time = 0.0286727, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{c^2 (d+e x)^{m+5}}{e (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^(5 + m))/(e*(5 + m))

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Rubi in Sympy [A]  time = 20.1818, size = 15, normalized size = 0.71 \[ \frac{c^{2} \left (d + e x\right )^{m + 5}}{e \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*(d + e*x)**(m + 5)/(e*(m + 5))

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Mathematica [A]  time = 0.0289908, size = 22, normalized size = 1.05 \[ \frac{c^2 (d+e x)^{m+5}}{e m+5 e} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^(5 + m))/(5*e + e*m)

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Maple [A]  time = 0.003, size = 40, normalized size = 1.9 \[{\frac{ \left ( ex+d \right ) ^{1+m}{c}^{2} \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) ^{2}}{e \left ( 5+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x+d)^(1+m)*c^2*(e^2*x^2+2*d*e*x+d^2)^2/e/(5+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*e^2*x^2 + 2*c*d*e*x + c*d^2)^2*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241407, size = 115, normalized size = 5.48 \[ \frac{{\left (c^{2} e^{5} x^{5} + 5 \, c^{2} d e^{4} x^{4} + 10 \, c^{2} d^{2} e^{3} x^{3} + 10 \, c^{2} d^{3} e^{2} x^{2} + 5 \, c^{2} d^{4} e x + c^{2} d^{5}\right )}{\left (e x + d\right )}^{m}}{e m + 5 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^2*e^5*x^5 + 5*c^2*d*e^4*x^4 + 10*c^2*d^2*e^3*x^3 + 10*c^2*d^3*e^2*x^2 + 5*c^2
*d^4*e*x + c^2*d^5)*(e*x + d)^m/(e*m + 5*e)

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Sympy [A]  time = 3.92088, size = 185, normalized size = 8.81 \[ \begin{cases} \frac{c^{2} x}{d} & \text{for}\: e = 0 \wedge m = -5 \\c^{2} d^{4} d^{m} x & \text{for}\: e = 0 \\\frac{c^{2} \log{\left (\frac{d}{e} + x \right )}}{e} & \text{for}\: m = -5 \\\frac{c^{2} d^{5} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{5 c^{2} d^{4} e x \left (d + e x\right )^{m}}{e m + 5 e} + \frac{10 c^{2} d^{3} e^{2} x^{2} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{10 c^{2} d^{2} e^{3} x^{3} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{5 c^{2} d e^{4} x^{4} \left (d + e x\right )^{m}}{e m + 5 e} + \frac{c^{2} e^{5} x^{5} \left (d + e x\right )^{m}}{e m + 5 e} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((c**2*x/d, Eq(e, 0) & Eq(m, -5)), (c**2*d**4*d**m*x, Eq(e, 0)), (c**2*
log(d/e + x)/e, Eq(m, -5)), (c**2*d**5*(d + e*x)**m/(e*m + 5*e) + 5*c**2*d**4*e*
x*(d + e*x)**m/(e*m + 5*e) + 10*c**2*d**3*e**2*x**2*(d + e*x)**m/(e*m + 5*e) + 1
0*c**2*d**2*e**3*x**3*(d + e*x)**m/(e*m + 5*e) + 5*c**2*d*e**4*x**4*(d + e*x)**m
/(e*m + 5*e) + c**2*e**5*x**5*(d + e*x)**m/(e*m + 5*e), True))

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GIAC/XCAS [A]  time = 0.222281, size = 185, normalized size = 8.81 \[ \frac{c^{2} x^{5} e^{\left (m{\rm ln}\left (x e + d\right ) + 5\right )} + 5 \, c^{2} d x^{4} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} + 10 \, c^{2} d^{2} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 10 \, c^{2} d^{3} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 5 \, c^{2} d^{4} x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + c^{2} d^{5} e^{\left (m{\rm ln}\left (x e + d\right )\right )}}{m e + 5 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^2*x^5*e^(m*ln(x*e + d) + 5) + 5*c^2*d*x^4*e^(m*ln(x*e + d) + 4) + 10*c^2*d^2*
x^3*e^(m*ln(x*e + d) + 3) + 10*c^2*d^3*x^2*e^(m*ln(x*e + d) + 2) + 5*c^2*d^4*x*e
^(m*ln(x*e + d) + 1) + c^2*d^5*e^(m*ln(x*e + d)))/(m*e + 5*e)