3.474 \(\int \frac{1}{x (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{\tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{a} d^{3/2} \sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.573941, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{\tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{a} d^{3/2} \sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 55.199, size = 133, normalized size = 0.93 \[ \frac{2 e \left (a e + c d x\right )}{d \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{\operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{\sqrt{a} d^{\frac{3}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.359109, size = 200, normalized size = 1.4 \[ \frac{\log (x) \sqrt{d+e x} \left (c d^2-a e^2\right ) \sqrt{a e+c d x}+\sqrt{d+e x} \left (a e^2-c d^2\right ) \sqrt{a e+c d x} \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )-2 \sqrt{a} \sqrt{d} e^{3/2} (a e+c d x)}{\sqrt{a} d^{3/2} \sqrt{e} \left (c d^2-a e^2\right ) \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 136, normalized size = 1. \[ -{\frac{1}{d}\ln \left ({\frac{1}{x} \left ( 2\,ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+2\,\sqrt{ade}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{ade}}}}+2\,{\frac{1}{d \left ( a{e}^{2}-c{d}^{2} \right ) }\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/d/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2))/x)+2/d/(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(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.372042, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{a d e} e -{\left (c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} d^{2} e^{2} +{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} -{\left (8 \, a^{2} d^{2} e^{2} +{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 8 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{a d e}}{x^{2}}\right )}{2 \,{\left (c d^{4} - a d^{2} e^{2} +{\left (c d^{3} e - a d e^{3}\right )} x\right )} \sqrt{a d e}}, -\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-a d e} e +{\left (c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-a d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} a d e}\right )}{{\left (c d^{4} - a d^{2} e^{2} +{\left (c d^{3} e - a d e^{3}\right )} x\right )} \sqrt{-a d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)*x),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

[Out]

Exception raised: TypeError