3.1937 \(\int \frac{d+e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=134 \[ \frac{\left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} \sqrt{e}}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d} \]

[Out]

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

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Rubi [A]  time = 0.169036, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{\left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} \sqrt{e}}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.8428, size = 124, normalized size = 0.93 \[ \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{c d} - \frac{\left (a e^{2} - c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2 c^{\frac{3}{2}} d^{\frac{3}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.23346, size = 156, normalized size = 1.16 \[ \frac{\sqrt{d+e x} \left (c d^2-a e^2\right ) \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )+2 \sqrt{c} \sqrt{d} \sqrt{e} (d+e x) (a e+c d x)}{2 c^{3/2} d^{3/2} \sqrt{e} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 171, normalized size = 1.3 \[{\frac{d}{2}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}+{\frac{1}{cd}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}-{\frac{a{e}^{2}}{2\,cd}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251253, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (c d^{2} - a e^{2}\right )} \log \left (-4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right ) - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d e}}{4 \, \sqrt{c d e} c d}, \frac{{\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d e}}{2 \, \sqrt{-c d e} c d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.265989, size = 181, normalized size = 1.35 \[ -\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{c d} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{2 \, c^{2} d^{2}} + \frac{\sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(c*d^2 - a*e^2)*sqrt(c*d)*e^(-1/2)*ln(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqr
t(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x))*c*d*e - sqrt(c*d
)*a*e^(5/2)))/(c^2*d^2) + sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)/(c*d)