3.1030 \(\int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \]

[Out]

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

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Rubi [A]  time = 0.0210363, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.79506, size = 29, normalized size = 0.85 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}}{5 c e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(5/2)/(5*c*e)

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Mathematica [A]  time = 0.0253046, size = 23, normalized size = 0.68 \[ \frac{\left (c (d+e x)^2\right )^{5/2}}{5 c e} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)^2)^(5/2)/(5*c*e)

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Maple [B]  time = 0.005, size = 73, normalized size = 2.2 \[{\frac{x \left ({e}^{4}{x}^{4}+5\,d{e}^{3}{x}^{3}+10\,{d}^{2}{e}^{2}{x}^{2}+10\,{d}^{3}ex+5\,{d}^{4} \right ) }{5\, \left ( ex+d \right ) ^{3}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.680587, size = 41, normalized size = 1.21 \[ \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{5}{2}}}{5 \, c e} \]

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

[Out]

1/5*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2)/(c*e)

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Fricas [A]  time = 0.22605, size = 107, normalized size = 3.15 \[ \frac{{\left (c e^{4} x^{5} + 5 \, c d e^{3} x^{4} + 10 \, c d^{2} e^{2} x^{3} + 10 \, c d^{3} e x^{2} + 5 \, c d^{4} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{5 \,{\left (e x + d\right )}} \]

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

[Out]

1/5*(c*e^4*x^5 + 5*c*d*e^3*x^4 + 10*c*d^2*e^2*x^3 + 10*c*d^3*e*x^2 + 5*c*d^4*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 = 2.66249, size = 194, normalized size = 5.71 \[ \begin{cases} \frac{c d^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5 e} + \frac{4 c d^{3} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{6 c d^{2} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{4 c d e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{c e^{3} x^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} & \text{for}\: e \neq 0 \\d x \left (c d^{2}\right )^{\frac{3}{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((c*d**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(5*e) + 4*c*d**3*x*sqrt
(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5 + 6*c*d**2*e*x**2*sqrt(c*d**2 + 2*c*d*e*x +
c*e**2*x**2)/5 + 4*c*d*e**2*x**3*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5 + c*e
**3*x**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5, Ne(e, 0)), (d*x*(c*d**2)**(3/
2), True))

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GIAC/XCAS [A]  time = 0.21996, size = 89, normalized size = 2.62 \[ \frac{1}{5} \,{\left (c d^{4} e^{\left (-1\right )} +{\left (4 \, c d^{3} +{\left (6 \, c d^{2} e +{\left (c x e^{3} + 4 \, c d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]

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

[Out]

1/5*(c*d^4*e^(-1) + (4*c*d^3 + (6*c*d^2*e + (c*x*e^3 + 4*c*d*e^2)*x)*x)*x)*sqrt(
c*x^2*e^2 + 2*c*d*x*e + c*d^2)