3.1062 \(\int \frac{(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=42 \[ \frac{(d+e x) \log (d+e x)}{c e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

((d + e*x)*Log[d + e*x])/(c*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

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Rubi [A]  time = 0.0795238, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ \frac{(d+e x) \log (d+e x)}{c e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)*Log[d + e*x])/(c*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

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Rubi in Sympy [A]  time = 18.4379, size = 39, normalized size = 0.93 \[ \frac{\left (d + e x\right )^{3} \log{\left (d + e x \right )}}{e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**3*log(d + e*x)/(e*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(3/2))

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Mathematica [A]  time = 0.00559426, size = 31, normalized size = 0.74 \[ \frac{(d+e x) \log (d+e x)}{c e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)*Log[d + e*x])/(c*e*Sqrt[c*(d + e*x)^2])

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Maple [A]  time = 0.004, size = 40, normalized size = 1. \[{\frac{ \left ( ex+d \right ) ^{3}\ln \left ( ex+d \right ) }{e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.689228, size = 166, normalized size = 3.95 \[ \frac{3 \, c^{2} d^{2} e^{4}}{2 \, \left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{2}} + \frac{2 \, c d e^{3} x}{\left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{2}} + \frac{e^{2} \log \left (x + \frac{d}{e}\right )}{\left (c e^{2}\right )^{\frac{3}{2}}} - \frac{2 \, d}{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} + \frac{d^{2}}{2 \, \left (c e^{2}\right )^{\frac{3}{2}}{\left (x + \frac{d}{e}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="maxima")

[Out]

3/2*c^2*d^2*e^4/((c*e^2)^(7/2)*(x + d/e)^2) + 2*c*d*e^3*x/((c*e^2)^(5/2)*(x + d/
e)^2) + e^2*log(x + d/e)/(c*e^2)^(3/2) - 2*d/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2
)*c*e) + 1/2*d^2/((c*e^2)^(3/2)*(x + d/e)^2)

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Fricas [A]  time = 0.23333, size = 62, normalized size = 1.48 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{c^{2} e^{2} x + c^{2} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="fricas")

[Out]

sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*log(e*x + d)/(c^2*e^2*x + c^2*d*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\left (c \left (d + e x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**2/(c*(d + e*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.308968, size = 119, normalized size = 2.83 \[ \frac{2 \,{\left (C_{0} d e^{\left (-1\right )} + C_{0} x\right )}}{\sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} - \frac{e^{\left (-1\right )}{\rm ln}\left ({\left | -\sqrt{c} d e^{2} -{\left (\sqrt{c} x e - \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}\right )} e^{2} \right |}\right )}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="giac")

[Out]

2*(C_0*d*e^(-1) + C_0*x)/sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2) - e^(-1)*ln(abs(-sq
rt(c)*d*e^2 - (sqrt(c)*x*e - sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2))*e^2))/c^(3/2)