3.472 \(\int \frac{x}{(d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=139 \[ \frac{\tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{c} \sqrt{d} e^{3/2}}-\frac{2 d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x) \left (c d^2-a e^2\right )} \]

[Out]

(-2*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e*(c*d^2 - a*e^2)*(d + e*x))
+ ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2])]/(Sqrt[c]*Sqrt[d]*e^(3/2))

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Rubi [A]  time = 0.304866, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079 \[ \frac{\tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{c} \sqrt{d} e^{3/2}}-\frac{2 d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x) \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e*(c*d^2 - a*e^2)*(d + e*x))
+ ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2])]/(Sqrt[c]*Sqrt[d]*e^(3/2))

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Rubi in Sympy [A]  time = 23.267, size = 129, normalized size = 0.93 \[ \frac{2 d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{e \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )} + \frac{\operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{\sqrt{c} \sqrt{d} e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(e*(d + e*x)*(a*e**2 - c*d**2
)) + atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e +
c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(sqrt(c)*sqrt(d)*e**(3/2))

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Mathematica [A]  time = 0.338025, size = 145, normalized size = 1.04 \[ \frac{\frac{2 d^{3/2} \sqrt{e} (a e+c d x)}{a e^2-c d^2}+\frac{\sqrt{d+e x} \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{\sqrt{c}}}{\sqrt{d} e^{3/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

((2*d^(3/2)*Sqrt[e]*(a*e + c*d*x))/(-(c*d^2) + a*e^2) + (Sqrt[a*e + c*d*x]*Sqrt[
d + e*x]*Log[a*e^2 + 2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] +
c*d*(d + 2*e*x)])/Sqrt[c])/(Sqrt[d]*e^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.014, size = 131, normalized size = 0.9 \[{\frac{1}{e}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{cde}}}}+2\,{\frac{d}{{e}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(x/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.384118, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d e} d -{\left (c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} \log \left (4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right )}{2 \,{\left (c d^{3} e - a d e^{3} +{\left (c d^{2} e^{2} - a e^{4}\right )} x\right )} \sqrt{c d e}}, -\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d e} d -{\left (c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right )}{{\left (c d^{3} e - a d e^{3} +{\left (c d^{2} e^{2} - a e^{4}\right )} x\right )} \sqrt{-c d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*e)*d - (c*d^3 - a*
d*e^2 + (c*d^2*e - a*e^3)*x)*log(4*(2*c^2*d^2*e^2*x + c^2*d^3*e + a*c*d*e^3)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) + (8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*
d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d*e)))/((c*d^3*e - a*d*e
^3 + (c*d^2*e^2 - a*e^4)*x)*sqrt(c*d*e)), -(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x)*sqrt(-c*d*e)*d - (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*arctan(1/2*(2
*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)
*x)*c*d*e)))/((c*d^3*e - a*d*e^3 + (c*d^2*e^2 - a*e^4)*x)*sqrt(-c*d*e))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError