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

Optimal. Leaf size=161 \[ \frac{2 \sqrt{g} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x} \sqrt{f+g x}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(-2*Sqrt[d + e*x]*Sqrt[f + g*x])/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
]) + (2*Sqrt[g]*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])])/(c^(3/2)*d^(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 = 0.617123, antiderivative size = 161, 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 \[ \frac{2 \sqrt{g} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x} \sqrt{f+g x}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*Sqrt[f + g*x])/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
]) + (2*Sqrt[g]*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])])/(c^(3/2)*d^(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 = 62.4654, size = 153, normalized size = 0.95 \[ - \frac{2 \sqrt{d + e x} \sqrt{f + g x}}{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{2 \sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{f + g x}}{\sqrt{g} \sqrt{a e + c d x}} \right )}}{c^{\frac{3}{2}} d^{\frac{3}{2}} \sqrt{d + e x} \sqrt{a e + c d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.189023, size = 131, normalized size = 0.81 \[ \frac{\sqrt{d+e x} \left (\sqrt{g} \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{g} \sqrt{f+g x} \sqrt{a e+c d x}+a e g+c d (f+2 g x)\right )-2 \sqrt{c} \sqrt{d} \sqrt{f+g x}\right )}{c^{3/2} d^{3/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.034, size = 210, normalized size = 1.3 \[{\frac{1}{ \left ( cdx+ae \right ) dc}\sqrt{gx+f}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( \ln \left ({\frac{1}{2} \left ( 2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc} \right ){\frac{1}{\sqrt{dgc}}}} \right ) xcdg+\ln \left ({\frac{1}{2} \left ( 2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc} \right ){\frac{1}{\sqrt{dgc}}}} \right ) aeg-2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc} \right ){\frac{1}{\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }}}{\frac{1}{\sqrt{dgc}}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.898196, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{\frac{g}{c d}} \log \left (-\frac{8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 8 \,{\left (c^{2} d^{2} e f g +{\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + 4 \,{\left (2 \, c^{2} d^{2} g x + c^{2} d^{2} f + a c d e g\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} \sqrt{\frac{g}{c d}} +{\left (c^{2} d^{2} e f^{2} + 2 \,{\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g +{\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right ) - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{2 \,{\left (c^{2} d^{2} e x^{2} + a c d^{2} e +{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )}}, \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-\frac{g}{c d}} \arctan \left (\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} g}{{\left (2 \, c d e g x^{2} + c d^{2} f + a d e g +{\left (c d e f +{\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x\right )} \sqrt{-\frac{g}{c d}}}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{c^{2} d^{2} e x^{2} + a c d^{2} e +{\left (c^{2} d^{3} + a c d e^{2}\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(g/(c*d))*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 + 8*(c^2*d^2*e*f*g + (c^2*
d^3 + a*c*d*e^2)*g^2)*x^2 + 4*(2*c^2*d^2*g*x + c^2*d^2*f + a*c*d*e*g)*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(g/(c*d)) + (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*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^2*d^2*e*x^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^2)*x), ((c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)*sqrt(-g/(c*d))*arctan(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)*g/((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(-g/(c*d)))) - 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^2*d^2*e*x^2 + a*c*d^2*e + (
c^2*d^3 + a*c*d*e^2)*x)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.632856, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x