3.1051 \(\int \frac{(d+e x)^4}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c^2 e} \]

[Out]

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

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Rubi [A]  time = 0.0751221, antiderivative size = 39, 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 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c^2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 18.7292, size = 34, normalized size = 0.87 \[ \frac{\left (d + e x\right )^{5}}{4 e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0165342, size = 27, normalized size = 0.69 \[ \frac{(d+e x)^5}{4 e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

(d + e*x)^5/(4*e*Sqrt[c*(d + e*x)^2])

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Maple [A]  time = 0.006, size = 60, normalized size = 1.5 \[{\frac{x \left ({e}^{3}{x}^{3}+4\,d{e}^{2}{x}^{2}+6\,{d}^{2}ex+4\,{d}^{3} \right ) \left ( ex+d \right ) }{4}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.692744, size = 246, normalized size = 6.31 \[ \frac{3 \, c^{2} d^{4} e^{4} \log \left (x + \frac{d}{e}\right )}{2 \, \left (c e^{2}\right )^{\frac{5}{2}}} - \frac{3 \, c d^{3} e^{3} x}{2 \, \left (c e^{2}\right )^{\frac{3}{2}}} + \frac{3 \, d^{2} e^{2} x^{2}}{4 \, \sqrt{c e^{2}}} - \frac{3}{2} \, d^{4} \sqrt{\frac{1}{c e^{2}}} \log \left (x + \frac{d}{e}\right ) + \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} e^{2} x^{3}}{4 \, c} + \frac{3 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d e x^{2}}{4 \, c} + \frac{5 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d^{3}}{2 \, c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.221624, size = 89, normalized size = 2.28 \[ \frac{{\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \,{\left (c e x + c d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{4}}{\sqrt{c \left (d + e x\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

Integral((d + e*x)**4/sqrt(c*(d + e*x)**2), x)

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GIAC/XCAS [A]  time = 0.29677, size = 85, normalized size = 2.18 \[ \frac{1}{4} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (\frac{d^{3} e^{\left (-1\right )}}{c} +{\left (x{\left (\frac{x e^{2}}{c} + \frac{3 \, d e}{c}\right )} + \frac{3 \, d^{2}}{c}\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")

[Out]

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