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

Optimal. Leaf size=240 \[ \frac{\sqrt{\sqrt{c} f-\sqrt{-a} g} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{c} d-\sqrt{-a} e}}-\frac{\sqrt{\sqrt{-a} g+\sqrt{c} f} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{-a} e+\sqrt{c} d}} \]

[Out]

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

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Rubi [A]  time = 0.918239, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{\sqrt{\sqrt{c} f-\sqrt{-a} g} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{c} d-\sqrt{-a} e}}-\frac{\sqrt{\sqrt{-a} g+\sqrt{c} f} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{-a} e+\sqrt{c} d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 77.3828, size = 209, normalized size = 0.87 \[ - \frac{\sqrt{\sqrt{c} f + g \sqrt{- a}} \operatorname{atanh}{\left (\frac{\sqrt{d + e x} \sqrt{\sqrt{c} f + g \sqrt{- a}}}{\sqrt{f + g x} \sqrt{\sqrt{c} d + e \sqrt{- a}}} \right )}}{\sqrt{c} \sqrt{- a} \sqrt{\sqrt{c} d + e \sqrt{- a}}} + \frac{\sqrt{\sqrt{c} f - g \sqrt{- a}} \operatorname{atanh}{\left (\frac{\sqrt{d + e x} \sqrt{\sqrt{c} f - g \sqrt{- a}}}{\sqrt{f + g x} \sqrt{\sqrt{c} d - e \sqrt{- a}}} \right )}}{\sqrt{c} \sqrt{- a} \sqrt{\sqrt{c} d - e \sqrt{- a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 2.74332, size = 496, normalized size = 2.07 \[ \frac{i \left (\frac{\left (c f+i \sqrt{a} \sqrt{c} g\right ) \log \left (\frac{i \sqrt{a} \sqrt{c} \left (2 \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}+i \sqrt{a} (d g+e (f+2 g x))+\sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x-i \sqrt{a}\right ) \sqrt{\sqrt{c} d+i \sqrt{a} e} \left (\sqrt{c} f+i \sqrt{a} g\right )^{3/2}}\right )}{\sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}}-\frac{\sqrt{c} \sqrt{\sqrt{c} f-i \sqrt{a} g} \log \left (-\frac{\sqrt{a} \sqrt{c} \left (2 i \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g}+\sqrt{a} (d g+e (f+2 g x))+i \sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x+i \sqrt{a}\right ) \sqrt{\sqrt{c} d-i \sqrt{a} e} \left (\sqrt{c} f-i \sqrt{a} g\right )^{3/2}}\right )}{\sqrt{\sqrt{c} d-i \sqrt{a} e}}\right )}{2 \sqrt{a} c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.045, size = 1383, normalized size = 5.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 9.80736, size = 2593, normalized size = 10.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(-(c*d*f + a*e*g + (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g +
d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2))
*log(-(e^2*f^2 - d^2*g^2 + 2*(c*d*e*f - c*d^2*g - (a*c^2*d^2*e + a^2*c*e^3)*sqrt
(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))*
sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-(c*d*f + a*e*g + (a*c^2*d^2 + a^2*c*e^2)*sqrt(
-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(
a*c^2*d^2 + a^2*c*e^2)) + 2*(e^2*f*g - d*e*g^2)*x + (2*(c^2*d^3 + a*c*d*e^2)*f +
((c^2*d^2*e + a*c*e^3)*f + (c^2*d^3 + a*c*d*e^2)*g)*x)*sqrt(-(e^2*f^2 - 2*d*e*f
*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/x) + 1/4*sqrt(-(c*d*
f + a*e*g + (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3
*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2))*log(-(e^2*f^2 -
d^2*g^2 - 2*(c*d*e*f - c*d^2*g - (a*c^2*d^2*e + a^2*c*e^3)*sqrt(-(e^2*f^2 - 2*d
*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))*sqrt(e*x + d)*sq
rt(g*x + f)*sqrt(-(c*d*f + a*e*g + (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(e^2*f^2 - 2*d*
e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*
c*e^2)) + 2*(e^2*f*g - d*e*g^2)*x + (2*(c^2*d^3 + a*c*d*e^2)*f + ((c^2*d^2*e + a
*c*e^3)*f + (c^2*d^3 + a*c*d*e^2)*g)*x)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a
*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/x) - 1/4*sqrt(-(c*d*f + a*e*g - (a*c
^2*d^2 + a^2*c*e^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2
*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2))*log(-(e^2*f^2 - d^2*g^2 + 2*(c*
d*e*f - c*d^2*g + (a*c^2*d^2*e + a^2*c*e^3)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2
)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt
(-(c*d*f + a*e*g - (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)
/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2)) + 2*(e^2
*f*g - d*e*g^2)*x - (2*(c^2*d^3 + a*c*d*e^2)*f + ((c^2*d^2*e + a*c*e^3)*f + (c^2
*d^3 + a*c*d*e^2)*g)*x)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2
*c^2*d^2*e^2 + a^3*c*e^4)))/x) + 1/4*sqrt(-(c*d*f + a*e*g - (a*c^2*d^2 + a^2*c*e
^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c
*e^4)))/(a*c^2*d^2 + a^2*c*e^2))*log(-(e^2*f^2 - d^2*g^2 - 2*(c*d*e*f - c*d^2*g
+ (a*c^2*d^2*e + a^2*c*e^3)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2
*a^2*c^2*d^2*e^2 + a^3*c*e^4)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-(c*d*f + a*e*g
- (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*
a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2)) + 2*(e^2*f*g - d*e*g^2)*
x - (2*(c^2*d^3 + a*c*d*e^2)*f + ((c^2*d^2*e + a*c*e^3)*f + (c^2*d^3 + a*c*d*e^2
)*g)*x)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a
^3*c*e^4)))/x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError