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

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

[Out]

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

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Rubi [A]  time = 4.0677, 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 \[ -\frac{2 e \sqrt{f+g x}}{\sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{\left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \left (a e^2+c d^2\right ) \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \left (a e^2+c d^2\right ) \sqrt{\sqrt{-a} g+\sqrt{c} f}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

[Out]

Timed out

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Mathematica [C]  time = 4.4792, size = 531, normalized size = 1.51 \[ \frac{\frac{\left (\sqrt{a} e+i \sqrt{c} d\right ) \sqrt{\sqrt{c} f+i \sqrt{a} g} \log \left (\frac{i \sqrt{a} \sqrt{\sqrt{c} d+i \sqrt{a} e} \left (2 \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}+i \sqrt{a} (d g+e f+2 e g x)+\sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x-i \sqrt{a}\right ) \left (\sqrt{c} f+i \sqrt{a} g\right )^{3/2}}\right )}{\sqrt{a} \sqrt{\sqrt{c} d+i \sqrt{a} e}}+\frac{\left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\sqrt{c} f-i \sqrt{a} g} \log \left (-\frac{i \sqrt{a} \sqrt{\sqrt{c} d-i \sqrt{a} e} \left (2 \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g}-i \sqrt{a} (d g+e (f+2 g x))+\sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x+i \sqrt{a}\right ) \left (\sqrt{c} f-i \sqrt{a} g\right )^{3/2}}\right )}{\sqrt{a} \sqrt{\sqrt{c} d-i \sqrt{a} e}}-\frac{4 e \sqrt{f+g x}}{\sqrt{d+e x}}}{2 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.1, size = 5383, normalized size = 15.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^(1/2)/(e*x+d)^(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. \[ \int \frac{\sqrt{g x + f}}{{\left (c x^{2} + a\right )}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 54.5613, size = 7852, normalized size = 22.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*((c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-((c^2*d^3 - 3*a*c*d*e^2)*f +
(3*a*c*d^2*e - a^2*e^3)*g + (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 +
a^4*e^6)*sqrt(-((9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*(3*c^3*d^5
*e - 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*
c*d^2*e^4)*g^2)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c
^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*
a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6))*log(((3*c*d^2*e^2 - a*e^4)*f^2 + 2
*(c*d^3*e + a*d*e^3)*f*g - (c*d^4 - 3*a*d^2*e^2)*g^2 + 2*((3*c^2*d^4*e - 4*a*c*d
^2*e^3 + a^2*e^5)*f - (c^2*d^5 - 4*a*c*d^3*e^2 + 3*a^2*d*e^4)*g - 2*(a*c^3*d^7*e
+ 3*a^2*c^2*d^5*e^3 + 3*a^3*c*d^3*e^5 + a^4*d*e^7)*sqrt(-((9*c^3*d^4*e^2 - 6*a*
c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5
)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*d^12 + 6*a^2*c
^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a
^6*c*d^2*e^10 + a^7*e^12)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-((c^2*d^3 - 3*a*c*
d*e^2)*f + (3*a*c*d^2*e - a^2*e^3)*g + (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*
d^2*e^4 + a^4*e^6)*sqrt(-((9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*
(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^
2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4
+ 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^
3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)) + 2*((3*c*d^2*e^2 - a*e^
4)*f*g - (c*d^3*e - 3*a*d*e^3)*g^2)*x + (2*(c^3*d^7 + 3*a*c^2*d^5*e^2 + 3*a^2*c*
d^3*e^4 + a^3*d*e^6)*f + ((c^3*d^6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e
^7)*f + (c^3*d^7 + 3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 + a^3*d*e^6)*g)*x)*sqrt(-((
9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a*c^2*d^3
*e^3 + 3*a^2*c*d*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(
a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a
^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/x) - (c*d^3 + a*d*e^2 + (c*d^2*e
+ a*e^3)*x)*sqrt(-((c^2*d^3 - 3*a*c*d*e^2)*f + (3*a*c*d^2*e - a^2*e^3)*g + (a*c
^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-((9*c^3*d^4*e^2 -
6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 + 3*a^2*c*d
*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*d^12 + 6*a
^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 +
6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4
+ a^4*e^6))*log(((3*c*d^2*e^2 - a*e^4)*f^2 + 2*(c*d^3*e + a*d*e^3)*f*g - (c*d^4
- 3*a*d^2*e^2)*g^2 - 2*((3*c^2*d^4*e - 4*a*c*d^2*e^3 + a^2*e^5)*f - (c^2*d^5 -
4*a*c*d^3*e^2 + 3*a^2*d*e^4)*g - 2*(a*c^3*d^7*e + 3*a^2*c^2*d^5*e^3 + 3*a^3*c*d^
3*e^5 + a^4*d*e^7)*sqrt(-((9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*
(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^
2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4
+ 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(
e*x + d)*sqrt(g*x + f)*sqrt(-((c^2*d^3 - 3*a*c*d*e^2)*f + (3*a*c*d^2*e - a^2*e^3
)*g + (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-((9*c^3*
d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 +
3*a^2*c*d*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*
d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2
*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3
*c*d^2*e^4 + a^4*e^6)) + 2*((3*c*d^2*e^2 - a*e^4)*f*g - (c*d^3*e - 3*a*d*e^3)*g^
2)*x + (2*(c^3*d^7 + 3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 + a^3*d*e^6)*f + ((c^3*d^
6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e^7)*f + (c^3*d^7 + 3*a*c^2*d^5*e^
2 + 3*a^2*c*d^3*e^4 + a^3*d*e^6)*g)*x)*sqrt(-((9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 +
a^2*c*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*f*g + (c^3*
d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 +
15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10
+ a^7*e^12)))/x) + (c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-((c^2*d^3 - 3*
a*c*d*e^2)*f + (3*a*c*d^2*e - a^2*e^3)*g - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^
3*c*d^2*e^4 + a^4*e^6)*sqrt(-((9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2
- 2*(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^
4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*
e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(
a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6))*log(((3*c*d^2*e^2 -
a*e^4)*f^2 + 2*(c*d^3*e + a*d*e^3)*f*g - (c*d^4 - 3*a*d^2*e^2)*g^2 + 2*((3*c^2*d
^4*e - 4*a*c*d^2*e^3 + a^2*e^5)*f - (c^2*d^5 - 4*a*c*d^3*e^2 + 3*a^2*d*e^4)*g +
2*(a*c^3*d^7*e + 3*a^2*c^2*d^5*e^3 + 3*a^3*c*d^3*e^5 + a^4*d*e^7)*sqrt(-((9*c^3*
d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 +
3*a^2*c*d*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*
d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2
*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-((c^
2*d^3 - 3*a*c*d*e^2)*f + (3*a*c*d^2*e - a^2*e^3)*g - (a*c^3*d^6 + 3*a^2*c^2*d^4*
e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-((9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c
*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*f*g + (c^3*d^6 -
6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^
3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7
*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)) + 2*((3*c*
d^2*e^2 - a*e^4)*f*g - (c*d^3*e - 3*a*d*e^3)*g^2)*x - (2*(c^3*d^7 + 3*a*c^2*d^5*
e^2 + 3*a^2*c*d^3*e^4 + a^3*d*e^6)*f + ((c^3*d^6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d
^2*e^5 + a^3*e^7)*f + (c^3*d^7 + 3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 + a^3*d*e^6)*
g)*x)*sqrt(-((9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*(3*c^3*d^5*e
- 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*c*d
^2*e^4)*g^2)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*
d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/x) - (c*d^3 + a*d*
e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-((c^2*d^3 - 3*a*c*d*e^2)*f + (3*a*c*d^2*e - a^2
*e^3)*g - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-((9*
c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a*c^2*d^3*e
^3 + 3*a^2*c*d*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(a*
c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5
*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3
*a^3*c*d^2*e^4 + a^4*e^6))*log(((3*c*d^2*e^2 - a*e^4)*f^2 + 2*(c*d^3*e + a*d*e^3
)*f*g - (c*d^4 - 3*a*d^2*e^2)*g^2 - 2*((3*c^2*d^4*e - 4*a*c*d^2*e^3 + a^2*e^5)*f
- (c^2*d^5 - 4*a*c*d^3*e^2 + 3*a^2*d*e^4)*g + 2*(a*c^3*d^7*e + 3*a^2*c^2*d^5*e^
3 + 3*a^3*c*d^3*e^5 + a^4*d*e^7)*sqrt(-((9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c
*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*f*g + (c^3*d^6 -
6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^
3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7
*e^12)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-((c^2*d^3 - 3*a*c*d*e^2)*f + (3*a*c*d
^2*e - a^2*e^3)*g - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*
sqrt(-((9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a
*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4
)*g^2)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^
6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d
^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)) + 2*((3*c*d^2*e^2 - a*e^4)*f*g - (c*d^3*e -
3*a*d*e^3)*g^2)*x - (2*(c^3*d^7 + 3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 + a^3*d*e^6
)*f + ((c^3*d^6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e^7)*f + (c^3*d^7 +
3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 + a^3*d*e^6)*g)*x)*sqrt(-((9*c^3*d^4*e^2 - 6*a
*c^2*d^2*e^4 + a^2*c*e^6)*f^2 - 2*(3*c^3*d^5*e - 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^
5)*f*g + (c^3*d^6 - 6*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)*g^2)/(a*c^6*d^12 + 6*a^2*
c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*
a^6*c*d^2*e^10 + a^7*e^12)))/x) + 8*sqrt(e*x + d)*sqrt(g*x + f)*e)/(c*d^3 + a*d*
e^2 + (c*d^2*e + a*e^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((g*x+f)**(1/2)/(e*x+d)**(3/2)/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 6.67416, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x