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

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

[Out]

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

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Rubi [A]  time = 0.845938, 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 \[ \frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{-a} g+\sqrt{c} f}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 80.0216, size = 209, normalized size = 0.87 \[ \frac{\sqrt{\sqrt{c} d - e \sqrt{- a}} \operatorname{atanh}{\left (\frac{\sqrt{d + e x} \sqrt{\sqrt{c} f - g \sqrt{- a}}}{\sqrt{f + g x} \sqrt{\sqrt{c} d - e \sqrt{- a}}} \right )}}{\sqrt{c} \sqrt{- a} \sqrt{\sqrt{c} f - g \sqrt{- a}}} - \frac{\sqrt{\sqrt{c} d + e \sqrt{- a}} \operatorname{atanh}{\left (\frac{\sqrt{d + e x} \sqrt{\sqrt{c} f + g \sqrt{- a}}}{\sqrt{f + g x} \sqrt{\sqrt{c} d + e \sqrt{- a}}} \right )}}{\sqrt{c} \sqrt{- a} \sqrt{\sqrt{c} f + g \sqrt{- a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 2.52612, size = 496, normalized size = 2.07 \[ \frac{i \left (\frac{\left (c d+i \sqrt{a} \sqrt{c} e\right ) \log \left (\frac{i \sqrt{a} \sqrt{c} \left (2 \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}+i \sqrt{a} (d g+e (f+2 g x))+\sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x-i \sqrt{a}\right ) \left (\sqrt{c} d+i \sqrt{a} e\right )^{3/2} \sqrt{\sqrt{c} f+i \sqrt{a} g}}\right )}{\sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}}-\frac{\sqrt{c} \sqrt{\sqrt{c} d-i \sqrt{a} e} \log \left (-\frac{\sqrt{a} \sqrt{c} \left (2 i \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g}+\sqrt{a} (d g+e (f+2 g x))+i \sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x+i \sqrt{a}\right ) \left (\sqrt{c} d-i \sqrt{a} e\right )^{3/2} \sqrt{\sqrt{c} f-i \sqrt{a} g}}\right )}{\sqrt{\sqrt{c} f-i \sqrt{a} g}}\right )}{2 \sqrt{a} c} \]

Antiderivative was successfully verified.

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

[Out]

((I/2)*(((c*d + I*Sqrt[a]*Sqrt[c]*e)*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[c]*d + I*Sqr
t[a]*e)^(3/2)*Sqrt[Sqrt[c]*f + I*Sqrt[a]*g]*((-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]*
d - I*Sqrt[a]*e]*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[c]*d - I*Sqrt[a]*e)^(3/2)*Sq
rt[Sqrt[c]*f - I*Sqrt[a]*g]*(I*Sqrt[a] + Sqrt[c]*x)))])/Sqrt[Sqrt[c]*f - I*Sqrt[
a]*g]))/(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((e*x+d)^(1/2)/(c*x^2+a)/(g*x+f)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 10.4801, size = 2583, normalized size = 10.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: RuntimeError