3.737 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{3/2}} \, dx\)

Optimal. Leaf size=158 \[ \frac{2 \sqrt{c} \sqrt{d} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{g^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt{d+e x} \sqrt{f+g x}} \]

[Out]

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

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Rubi [A]  time = 0.609772, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 \sqrt{c} \sqrt{d} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{g^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt{d+e x} \sqrt{f+g x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 60.8591, size = 151, normalized size = 0.96 \[ \frac{2 \sqrt{c} \sqrt{d} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{f + g x}}{\sqrt{g} \sqrt{a e + c d x}} \right )}}{g^{\frac{3}{2}} \sqrt{d + e x} \sqrt{a e + c d x}} - \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{g \sqrt{d + e x} \sqrt{f + g x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**(3/2)/(e*x+d)**(1/2),x)

[Out]

2*sqrt(c)*sqrt(d)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*atanh(sqrt(c)*s
qrt(d)*sqrt(f + g*x)/(sqrt(g)*sqrt(a*e + c*d*x)))/(g**(3/2)*sqrt(d + e*x)*sqrt(a
*e + c*d*x)) - 2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(g*sqrt(d + e*x)
*sqrt(f + g*x))

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Mathematica [A]  time = 0.189418, size = 147, normalized size = 0.93 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (\sqrt{c} \sqrt{d} \sqrt{f+g x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{g} \sqrt{f+g x} \sqrt{a e+c d x}+a e g+c d (f+2 g x)\right )-2 \sqrt{g} \sqrt{a e+c d x}\right )}{g^{3/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.037, size = 197, normalized size = 1.3 \[{\frac{1}{g}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( \ln \left ({\frac{1}{2} \left ( 2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc} \right ){\frac{1}{\sqrt{dgc}}}} \right ) xcdg+\ln \left ({\frac{1}{2} \left ( 2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc} \right ){\frac{1}{\sqrt{dgc}}}} \right ) cdf-2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc} \right ){\frac{1}{\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }}}{\frac{1}{\sqrt{dgc}}}{\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{gx+f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(3/2)/(e*x+d)^(1/2),x)

[Out]

(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*((g*x+f
)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*x*c*d*g+ln(1/2*(2*x*c*d*g+a*e
*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*c*d*f-2*((g
*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2)/((g*x+f)*(c*d*x+a*e))^(1/2
)/g/(e*x+d)^(1/2)/(g*x+f)^(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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.902083, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (e g x^{2} + d f +{\left (e f + d g\right )} x\right )} \sqrt{\frac{c d}{g}} \log \left (-\frac{8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 4 \,{\left (2 \, c d g^{2} x + c d f g + a e g^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} \sqrt{\frac{c d}{g}} + 8 \,{\left (c^{2} d^{2} e f g +{\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} +{\left (c^{2} d^{2} e f^{2} + 2 \,{\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g +{\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right ) - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{2 \,{\left (e g^{2} x^{2} + d f g +{\left (e f g + d g^{2}\right )} x\right )}}, \frac{{\left (e g x^{2} + d f +{\left (e f + d g\right )} x\right )} \sqrt{-\frac{c d}{g}} \arctan \left (\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} c d}{{\left (2 \, c d e g x^{2} + c d^{2} f + a d e g +{\left (c d e f +{\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x\right )} \sqrt{-\frac{c d}{g}}}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{e g^{2} x^{2} + d f g +{\left (e f g + d g^{2}\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**(3/2)/(e*x+d)**(1/2),x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(sqrt(d + e*x)*(f + g*x)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^(
3/2)), x)