3.1039 \(\int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \]

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(9/2)/(9*c^2*e)

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Rubi [A]  time = 0.0708369, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(9/2)/(9*c^2*e)

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Rubi in Sympy [A]  time = 18.4623, size = 31, normalized size = 0.91 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{9}{2}}}{9 c^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(9/2)/(9*c**2*e)

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Mathematica [A]  time = 0.0538787, size = 27, normalized size = 0.79 \[ \frac{(d+e x)^4 \left (c (d+e x)^2\right )^{5/2}}{9 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

((d + e*x)^4*(c*(d + e*x)^2)^(5/2))/(9*e)

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Maple [B]  time = 0.005, size = 117, normalized size = 3.4 \[{\frac{x \left ({e}^{8}{x}^{8}+9\,d{e}^{7}{x}^{7}+36\,{d}^{2}{e}^{6}{x}^{6}+84\,{d}^{3}{e}^{5}{x}^{5}+126\,{d}^{4}{e}^{4}{x}^{4}+126\,{d}^{5}{e}^{3}{x}^{3}+84\,{d}^{6}{e}^{2}{x}^{2}+36\,{d}^{7}ex+9\,{d}^{8} \right ) }{9\, \left ( ex+d \right ) ^{5}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x)

[Out]

1/9*x*(e^8*x^8+9*d*e^7*x^7+36*d^2*e^6*x^6+84*d^3*e^5*x^5+126*d^4*e^4*x^4+126*d^5
*e^3*x^3+84*d^6*e^2*x^2+36*d^7*e*x+9*d^8)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x
+d)^5

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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)^(5/2)*(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217874, size = 196, normalized size = 5.76 \[ \frac{{\left (c^{2} e^{8} x^{9} + 9 \, c^{2} d e^{7} x^{8} + 36 \, c^{2} d^{2} e^{6} x^{7} + 84 \, c^{2} d^{3} e^{5} x^{6} + 126 \, c^{2} d^{4} e^{4} x^{5} + 126 \, c^{2} d^{5} e^{3} x^{4} + 84 \, c^{2} d^{6} e^{2} x^{3} + 36 \, c^{2} d^{7} e x^{2} + 9 \, c^{2} d^{8} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{9 \,{\left (e x + d\right )}} \]

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)^(5/2)*(e*x + d)^3,x, algorithm="fricas")

[Out]

1/9*(c^2*e^8*x^9 + 9*c^2*d*e^7*x^8 + 36*c^2*d^2*e^6*x^7 + 84*c^2*d^3*e^5*x^6 + 1
26*c^2*d^4*e^4*x^5 + 126*c^2*d^5*e^3*x^4 + 84*c^2*d^6*e^2*x^3 + 36*c^2*d^7*e*x^2
+ 9*c^2*d^8*x)*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)

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Sympy [A]  time = 37.3179, size = 374, normalized size = 11. \[ \begin{cases} \frac{c^{2} d^{8} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9 e} + \frac{8 c^{2} d^{7} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{28 c^{2} d^{6} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{56 c^{2} d^{5} e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{70 c^{2} d^{4} e^{3} x^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{56 c^{2} d^{3} e^{4} x^{5} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{28 c^{2} d^{2} e^{5} x^{6} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{8 c^{2} d e^{6} x^{7} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{c^{2} e^{7} x^{8} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} & \text{for}\: e \neq 0 \\d^{3} x \left (c d^{2}\right )^{\frac{5}{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

Piecewise((c**2*d**8*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(9*e) + 8*c**2*d**7*
x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 28*c**2*d**6*e*x**2*sqrt(c*d**2 + 2
*c*d*e*x + c*e**2*x**2)/9 + 56*c**2*d**5*e**2*x**3*sqrt(c*d**2 + 2*c*d*e*x + c*e
**2*x**2)/9 + 70*c**2*d**4*e**3*x**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 +
56*c**2*d**3*e**4*x**5*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 28*c**2*d**2*e
**5*x**6*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 8*c**2*d*e**6*x**7*sqrt(c*d*
*2 + 2*c*d*e*x + c*e**2*x**2)/9 + c**2*e**7*x**8*sqrt(c*d**2 + 2*c*d*e*x + c*e**
2*x**2)/9, Ne(e, 0)), (d**3*x*(c*d**2)**(5/2), True))

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GIAC/XCAS [A]  time = 0.223368, size = 173, normalized size = 5.09 \[ \frac{1}{9} \,{\left (c^{2} d^{8} e^{\left (-1\right )} +{\left (8 \, c^{2} d^{7} +{\left (28 \, c^{2} d^{6} e +{\left (56 \, c^{2} d^{5} e^{2} +{\left (70 \, c^{2} d^{4} e^{3} +{\left (56 \, c^{2} d^{3} e^{4} +{\left (28 \, c^{2} d^{2} e^{5} +{\left (c^{2} x e^{7} + 8 \, c^{2} d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]

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)^(5/2)*(e*x + d)^3,x, algorithm="giac")

[Out]

1/9*(c^2*d^8*e^(-1) + (8*c^2*d^7 + (28*c^2*d^6*e + (56*c^2*d^5*e^2 + (70*c^2*d^4
*e^3 + (56*c^2*d^3*e^4 + (28*c^2*d^2*e^5 + (c^2*x*e^7 + 8*c^2*d*e^6)*x)*x)*x)*x)
*x)*x)*x)*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)