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

Optimal. Leaf size=351 \[ -\frac{2 g \sqrt{d+e x}}{\sqrt{f+g x} \left (a g^2+c f^2\right )}+\frac{\left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (a g^2+c f^2\right )}-\frac{\left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f} \left (a g^2+c f^2\right )} \]

[Out]

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

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Rubi [A]  time = 3.75604, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2 g \sqrt{d+e x}}{\sqrt{f+g x} \left (a g^2+c f^2\right )}+\frac{\left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (a g^2+c f^2\right )}-\frac{\left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f} \left (a g^2+c f^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 4.34368, size = 531, normalized size = 1.51 \[ \frac{\frac{\sqrt{\sqrt{c} d+i \sqrt{a} e} \left (\sqrt{a} g+i \sqrt{c} f\right ) \log \left (\frac{i \sqrt{a} \sqrt{\sqrt{c} f+i \sqrt{a} g} \left (2 \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}+i \sqrt{a} (d g+e f+2 e g x)+\sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x-i \sqrt{a}\right ) \left (\sqrt{c} d+i \sqrt{a} e\right )^{3/2}}\right )}{\sqrt{a} \sqrt{\sqrt{c} f+i \sqrt{a} g}}+\frac{\sqrt{\sqrt{c} d-i \sqrt{a} e} \left (\sqrt{a} g-i \sqrt{c} f\right ) \log \left (-\frac{i \sqrt{a} \sqrt{\sqrt{c} f-i \sqrt{a} g} \left (2 \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g}-i \sqrt{a} (d g+e (f+2 g x))+\sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x+i \sqrt{a}\right ) \left (\sqrt{c} d-i \sqrt{a} e\right )^{3/2}}\right )}{\sqrt{a} \sqrt{\sqrt{c} f-i \sqrt{a} g}}-\frac{4 g \sqrt{d+e x}}{\sqrt{f+g x}}}{2 \left (a g^2+c f^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((-4*g*Sqrt[d + e*x])/Sqrt[f + g*x] + (Sqrt[Sqrt[c]*d + I*Sqrt[a]*e]*(I*Sqrt[c]*
f + Sqrt[a]*g)*Log[(I*Sqrt[a]*Sqrt[Sqrt[c]*f + I*Sqrt[a]*g]*(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]*(e*f + d*g + 2*e*g*x)))/((Sqrt[c]*d + I*Sqr
t[a]*e)^(3/2)*((-I)*Sqrt[a] + Sqrt[c]*x))])/(Sqrt[a]*Sqrt[Sqrt[c]*f + I*Sqrt[a]*
g]) + (Sqrt[Sqrt[c]*d - I*Sqrt[a]*e]*((-I)*Sqrt[c]*f + Sqrt[a]*g)*Log[((-I)*Sqrt
[a]*Sqrt[Sqrt[c]*f - I*Sqrt[a]*g]*(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*Sqrt[a]*e)^(3/2)*(I*Sqrt[a] +
Sqrt[c]*x))])/(Sqrt[a]*Sqrt[Sqrt[c]*f - I*Sqrt[a]*g]))/(2*(c*f^2 + a*g^2))

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Maple [B]  time = 0.07, size = 5383, normalized size = 15.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 74.9854, size = 7889, normalized size = 22.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*((c*f^3 + a*f*g^2 + (c*f^2*g + a*g^3)*x)*sqrt(-(c^2*d*f^3 + 3*a*c*e*f^2*g -
3*a*c*d*f*g^2 - a^2*e*g^3 + (a*c^3*f^6 + 3*a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*g^4 +
a^4*g^6)*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d
*e*f*g^5 + a^2*c*d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2 - 3*(2*a*c^2*d^2
- 3*a^2*c*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a^3*c^4*f^8*g^4 +
20*a^4*c^3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^7*g^12)))/(a*c^3*
f^6 + 3*a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^6))*log((c*e^2*f^4 - 2*c*d*e*f
^3*g - 2*a*d*e*f*g^3 + a*d^2*g^4 - 3*(c*d^2 + a*e^2)*f^2*g^2 + 2*(c^2*e*f^5 - 3*
c^2*d*f^4*g - 4*a*c*e*f^3*g^2 + 4*a*c*d*f^2*g^3 + 3*a^2*e*f*g^4 - a^2*d*g^5 + 2*
(a*c^3*f^7*g + 3*a^2*c^2*f^5*g^3 + 3*a^3*c*f^3*g^5 + a^4*f*g^7)*sqrt(-(c^3*e^2*f
^6 - 6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6
+ 3*(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(
a*c^6*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a
^5*c^2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^7*g^12)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt
(-(c^2*d*f^3 + 3*a*c*e*f^2*g - 3*a*c*d*f*g^2 - a^2*e*g^3 + (a*c^3*f^6 + 3*a^2*c^
2*f^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^6)*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20
*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*
e^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^
10*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*
f^2*g^10 + a^7*g^12)))/(a*c^3*f^6 + 3*a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^
6)) + 2*(c*e^2*f^3*g - 3*c*d*e*f^2*g^2 - 3*a*e^2*f*g^3 + a*d*e*g^4)*x - (2*c^3*d
*f^7 + 6*a*c^2*d*f^5*g^2 + 6*a^2*c*d*f^3*g^4 + 2*a^3*d*f*g^6 + (c^3*e*f^7 + c^3*
d*f^6*g + 3*a*c^2*e*f^5*g^2 + 3*a*c^2*d*f^4*g^3 + 3*a^2*c*e*f^3*g^4 + 3*a^2*c*d*
f^2*g^5 + a^3*e*f*g^6 + a^3*d*g^7)*x)*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20*
a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*e
^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^1
0*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*f
^2*g^10 + a^7*g^12)))/x) - (c*f^3 + a*f*g^2 + (c*f^2*g + a*g^3)*x)*sqrt(-(c^2*d*
f^3 + 3*a*c*e*f^2*g - 3*a*c*d*f*g^2 - a^2*e*g^3 + (a*c^3*f^6 + 3*a^2*c^2*f^4*g^2
+ 3*a^3*c*f^2*g^4 + a^4*g^6)*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20*a*c^2*d*
e*f^3*g^3 - 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*
g^2 - 3*(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^10*g^2 +
15*a^3*c^4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*f^2*g^10
+ a^7*g^12)))/(a*c^3*f^6 + 3*a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^6))*log((
c*e^2*f^4 - 2*c*d*e*f^3*g - 2*a*d*e*f*g^3 + a*d^2*g^4 - 3*(c*d^2 + a*e^2)*f^2*g^
2 - 2*(c^2*e*f^5 - 3*c^2*d*f^4*g - 4*a*c*e*f^3*g^2 + 4*a*c*d*f^2*g^3 + 3*a^2*e*f
*g^4 - a^2*d*g^5 + 2*(a*c^3*f^7*g + 3*a^2*c^2*f^5*g^3 + 3*a^3*c*f^3*g^5 + a^4*f*
g^7)*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f
*g^5 + a^2*c*d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*
a^2*c*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a
^4*c^3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^7*g^12)))*sqrt(e*x +
d)*sqrt(g*x + f)*sqrt(-(c^2*d*f^3 + 3*a*c*e*f^2*g - 3*a*c*d*f*g^2 - a^2*e*g^3 +
(a*c^3*f^6 + 3*a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^6)*sqrt(-(c^3*e^2*f^6 -
6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6 + 3*
(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(a*c^
6*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a^5*c
^2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^7*g^12)))/(a*c^3*f^6 + 3*a^2*c^2*f^4*g^2 + 3*a
^3*c*f^2*g^4 + a^4*g^6)) + 2*(c*e^2*f^3*g - 3*c*d*e*f^2*g^2 - 3*a*e^2*f*g^3 + a*
d*e*g^4)*x - (2*c^3*d*f^7 + 6*a*c^2*d*f^5*g^2 + 6*a^2*c*d*f^3*g^4 + 2*a^3*d*f*g^
6 + (c^3*e*f^7 + c^3*d*f^6*g + 3*a*c^2*e*f^5*g^2 + 3*a*c^2*d*f^4*g^3 + 3*a^2*c*e
*f^3*g^4 + 3*a^2*c*d*f^2*g^5 + a^3*e*f*g^6 + a^3*d*g^7)*x)*sqrt(-(c^3*e^2*f^6 -
6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6 + 3*(
3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(a*c^6
*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a^5*c^
2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^7*g^12)))/x) + (c*f^3 + a*f*g^2 + (c*f^2*g + a*
g^3)*x)*sqrt(-(c^2*d*f^3 + 3*a*c*e*f^2*g - 3*a*c*d*f*g^2 - a^2*e*g^3 - (a*c^3*f^
6 + 3*a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^6)*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*
e*f^5*g + 20*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6 + 3*(3*c^3*d^
2 - 2*a*c^2*e^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(a*c^6*f^12 +
6*a^2*c^5*f^10*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a^5*c^2*f^4*g^
8 + 6*a^6*c*f^2*g^10 + a^7*g^12)))/(a*c^3*f^6 + 3*a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*
g^4 + a^4*g^6))*log((c*e^2*f^4 - 2*c*d*e*f^3*g - 2*a*d*e*f*g^3 + a*d^2*g^4 - 3*(
c*d^2 + a*e^2)*f^2*g^2 + 2*(c^2*e*f^5 - 3*c^2*d*f^4*g - 4*a*c*e*f^3*g^2 + 4*a*c*
d*f^2*g^3 + 3*a^2*e*f*g^4 - a^2*d*g^5 - 2*(a*c^3*f^7*g + 3*a^2*c^2*f^5*g^3 + 3*a
^3*c*f^3*g^5 + a^4*f*g^7)*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^
3*g^3 - 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2
- 3*(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a
^3*c^4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^
7*g^12)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-(c^2*d*f^3 + 3*a*c*e*f^2*g - 3*a*c*d
*f*g^2 - a^2*e*g^3 - (a*c^3*f^6 + 3*a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^6)
*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f*g^5
+ a^2*c*d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*a^2*
c*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a^4*c
^3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^7*g^12)))/(a*c^3*f^6 + 3*
a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^6)) + 2*(c*e^2*f^3*g - 3*c*d*e*f^2*g^2
- 3*a*e^2*f*g^3 + a*d*e*g^4)*x + (2*c^3*d*f^7 + 6*a*c^2*d*f^5*g^2 + 6*a^2*c*d*f
^3*g^4 + 2*a^3*d*f*g^6 + (c^3*e*f^7 + c^3*d*f^6*g + 3*a*c^2*e*f^5*g^2 + 3*a*c^2*
d*f^4*g^3 + 3*a^2*c*e*f^3*g^4 + 3*a^2*c*d*f^2*g^5 + a^3*e*f*g^6 + a^3*d*g^7)*x)*
sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f*g^5
+ a^2*c*d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*a^2*c
*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a^4*c^
3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^7*g^12)))/x) - (c*f^3 + a*
f*g^2 + (c*f^2*g + a*g^3)*x)*sqrt(-(c^2*d*f^3 + 3*a*c*e*f^2*g - 3*a*c*d*f*g^2 -
a^2*e*g^3 - (a*c^3*f^6 + 3*a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^6)*sqrt(-(c
^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f*g^5 + a^2*c*
d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^
2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a^4*c^3*f^6*g^
6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^7*g^12)))/(a*c^3*f^6 + 3*a^2*c^2*f
^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^6))*log((c*e^2*f^4 - 2*c*d*e*f^3*g - 2*a*d*e*f*
g^3 + a*d^2*g^4 - 3*(c*d^2 + a*e^2)*f^2*g^2 - 2*(c^2*e*f^5 - 3*c^2*d*f^4*g - 4*a
*c*e*f^3*g^2 + 4*a*c*d*f^2*g^3 + 3*a^2*e*f*g^4 - a^2*d*g^5 - 2*(a*c^3*f^7*g + 3*
a^2*c^2*f^5*g^3 + 3*a^3*c*f^3*g^5 + a^4*f*g^7)*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^
5*g + 20*a*c^2*d*e*f^3*g^3 - 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6 + 3*(3*c^3*d^2 -
2*a*c^2*e^2)*f^4*g^2 - 3*(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^
2*c^5*f^10*g^2 + 15*a^3*c^4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 +
6*a^6*c*f^2*g^10 + a^7*g^12)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-(c^2*d*f^3 + 3*
a*c*e*f^2*g - 3*a*c*d*f*g^2 - a^2*e*g^3 - (a*c^3*f^6 + 3*a^2*c^2*f^4*g^2 + 3*a^3
*c*f^2*g^4 + a^4*g^6)*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^3*g^
3 - 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2 - 3*
(2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a^3*c
^4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^7*g^
12)))/(a*c^3*f^6 + 3*a^2*c^2*f^4*g^2 + 3*a^3*c*f^2*g^4 + a^4*g^6)) + 2*(c*e^2*f^
3*g - 3*c*d*e*f^2*g^2 - 3*a*e^2*f*g^3 + a*d*e*g^4)*x + (2*c^3*d*f^7 + 6*a*c^2*d*
f^5*g^2 + 6*a^2*c*d*f^3*g^4 + 2*a^3*d*f*g^6 + (c^3*e*f^7 + c^3*d*f^6*g + 3*a*c^2
*e*f^5*g^2 + 3*a*c^2*d*f^4*g^3 + 3*a^2*c*e*f^3*g^4 + 3*a^2*c*d*f^2*g^5 + a^3*e*f
*g^6 + a^3*d*g^7)*x)*sqrt(-(c^3*e^2*f^6 - 6*c^3*d*e*f^5*g + 20*a*c^2*d*e*f^3*g^3
- 6*a^2*c*d*e*f*g^5 + a^2*c*d^2*g^6 + 3*(3*c^3*d^2 - 2*a*c^2*e^2)*f^4*g^2 - 3*(
2*a*c^2*d^2 - 3*a^2*c*e^2)*f^2*g^4)/(a*c^6*f^12 + 6*a^2*c^5*f^10*g^2 + 15*a^3*c^
4*f^8*g^4 + 20*a^4*c^3*f^6*g^6 + 15*a^5*c^2*f^4*g^8 + 6*a^6*c*f^2*g^10 + a^7*g^1
2)))/x) + 8*sqrt(e*x + d)*sqrt(g*x + f)*g)/(c*f^3 + a*f*g^2 + (c*f^2*g + a*g^3)*
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)**(1/2)/(g*x+f)**(3/2)/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out