3.688 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^5} \, dx\)

Optimal. Leaf size=347 \[ \frac{5 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{64 g^{3/2} (c d f-a e g)^{7/2}}+\frac{5 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 g \sqrt{d+e x} (f+g x)^3 (c d f-a e g)}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt{d+e x} (f+g x)^4} \]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(4*g*Sqrt[d + e*x]*(f + g*x)^4) + (
c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*g*(c*d*f - a*e*g)*Sqrt[d +
e*x]*(f + g*x)^3) + (5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(96*
g*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^3*d^3*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2])/(64*g*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (5*c
^4*d^4*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f
- a*e*g]*Sqrt[d + e*x])])/(64*g^(3/2)*(c*d*f - a*e*g)^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.60925, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{5 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{64 g^{3/2} (c d f-a e g)^{7/2}}+\frac{5 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 g \sqrt{d+e x} (f+g x)^3 (c d f-a e g)}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt{d+e x} (f+g x)^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^5),x]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(4*g*Sqrt[d + e*x]*(f + g*x)^4) + (
c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*g*(c*d*f - a*e*g)*Sqrt[d +
e*x]*(f + g*x)^3) + (5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(96*
g*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^3*d^3*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2])/(64*g*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (5*c
^4*d^4*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f
- a*e*g]*Sqrt[d + e*x])])/(64*g^(3/2)*(c*d*f - a*e*g)^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 152.208, size = 328, normalized size = 0.95 \[ \frac{5 c^{4} d^{4} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{64 g^{\frac{3}{2}} \left (a e g - c d f\right )^{\frac{7}{2}}} - \frac{5 c^{3} d^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{64 g \sqrt{d + e x} \left (f + g x\right ) \left (a e g - c d f\right )^{3}} + \frac{5 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{96 g \sqrt{d + e x} \left (f + g x\right )^{2} \left (a e g - c d f\right )^{2}} - \frac{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{24 g \sqrt{d + e x} \left (f + g x\right )^{3} \left (a e g - c d f\right )} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 g \sqrt{d + e x} \left (f + g x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**5/(e*x+d)**(1/2),x)

[Out]

5*c**4*d**4*atanh(sqrt(g)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(sqrt(d
+ e*x)*sqrt(a*e*g - c*d*f)))/(64*g**(3/2)*(a*e*g - c*d*f)**(7/2)) - 5*c**3*d**3
*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(64*g*sqrt(d + e*x)*(f + g*x)*(a
*e*g - c*d*f)**3) + 5*c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(
96*g*sqrt(d + e*x)*(f + g*x)**2*(a*e*g - c*d*f)**2) - c*d*sqrt(a*d*e + c*d*e*x**
2 + x*(a*e**2 + c*d**2))/(24*g*sqrt(d + e*x)*(f + g*x)**3*(a*e*g - c*d*f)) - sqr
t(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*g*sqrt(d + e*x)*(f + g*x)**4)

_______________________________________________________________________________________

Mathematica [A]  time = 1.04662, size = 195, normalized size = 0.56 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{5 c^4 d^4 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{g^{3/2} \sqrt{a e+c d x} (a e g-c d f)^{7/2}}+\frac{\frac{15 c^3 d^3 (f+g x)^3}{(c d f-a e g)^3}+\frac{10 c^2 d^2 (f+g x)^2}{(c d f-a e g)^2}+\frac{8 c d (f+g x)}{c d f-a e g}-48}{3 g (f+g x)^4}\right )}{64 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^5),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((-48 + (8*c*d*(f + g*x))/(c*d*f - a*e*g) + (10*c
^2*d^2*(f + g*x)^2)/(c*d*f - a*e*g)^2 + (15*c^3*d^3*(f + g*x)^3)/(c*d*f - a*e*g)
^3)/(3*g*(f + g*x)^4) + (5*c^4*d^4*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[-(c*
d*f) + a*e*g]])/(g^(3/2)*(-(c*d*f) + a*e*g)^(7/2)*Sqrt[a*e + c*d*x])))/(64*Sqrt[
d + e*x])

_______________________________________________________________________________________

Maple [B]  time = 0.041, size = 696, normalized size = 2. \[{\frac{1}{192\, \left ( gx+f \right ) ^{4}g \left ( aeg-cdf \right ) \left ({a}^{2}{e}^{2}{g}^{2}-2\,acdefg+{c}^{2}{d}^{2}{f}^{2} \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{4}{c}^{4}{d}^{4}{g}^{4}+60\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{3}{c}^{4}{d}^{4}f{g}^{3}+90\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}{c}^{4}{d}^{4}{f}^{2}{g}^{2}+60\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{4}{d}^{4}{f}^{3}g-15\,{x}^{3}{c}^{3}{d}^{3}{g}^{3}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{4}{d}^{4}{f}^{4}+10\,{x}^{2}a{c}^{2}{d}^{2}e{g}^{3}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-55\,{x}^{2}{c}^{3}{d}^{3}f{g}^{2}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-8\,x{a}^{2}cd{e}^{2}{g}^{3}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+36\,xa{c}^{2}{d}^{2}ef{g}^{2}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-73\,x{c}^{3}{d}^{3}{f}^{2}g\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-48\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{3}{e}^{3}{g}^{3}+136\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{2}cd{e}^{2}f{g}^{2}-118\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}a{c}^{2}{d}^{2}e{f}^{2}g+15\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{c}^{3}{d}^{3}{f}^{3} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}{\frac{1}{\sqrt{cdx+ae}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^5/(e*x+d)^(1/2),x)

[Out]

1/192*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(15*arctanh(g*(c*d*x+a*e)^(1/2)/((
a*e*g-c*d*f)*g)^(1/2))*x^4*c^4*d^4*g^4+60*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*
d*f)*g)^(1/2))*x^3*c^4*d^4*f*g^3+90*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g
)^(1/2))*x^2*c^4*d^4*f^2*g^2+60*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1
/2))*x*c^4*d^4*f^3*g-15*x^3*c^3*d^3*g^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2
)+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^4*d^4*f^4+10*x^2*a*c
^2*d^2*e*g^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-55*x^2*c^3*d^3*f*g^2*(c*d
*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-8*x*a^2*c*d*e^2*g^3*(c*d*x+a*e)^(1/2)*((a*
e*g-c*d*f)*g)^(1/2)+36*x*a*c^2*d^2*e*f*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(
1/2)-73*x*c^3*d^3*f^2*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-48*((a*e*g-c*d
*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^3*e^3*g^3+136*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*
e)^(1/2)*a^2*c*d*e^2*f*g^2-118*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c^2*d
^2*e*f^2*g+15*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^3*d^3*f^3)/(e*x+d)^(1/
2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)^4/g/(a*e*g-c*d*f)/(a^2*e^2*g^2-2*a*c*d*e*f*g+
c^2*d^2*f^2)/(c*d*x+a*e)^(1/2)

_______________________________________________________________________________________

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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.313657, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^5),x, algorithm="fricas")

[Out]

[1/384*(2*(15*c^3*d^3*g^3*x^3 - 15*c^3*d^3*f^3 + 118*a*c^2*d^2*e*f^2*g - 136*a^2
*c*d*e^2*f*g^2 + 48*a^3*e^3*g^3 + 5*(11*c^3*d^3*f*g^2 - 2*a*c^2*d^2*e*g^3)*x^2 +
(73*c^3*d^3*f^2*g - 36*a*c^2*d^2*e*f*g^2 + 8*a^2*c*d*e^2*g^3)*x)*sqrt(c*d*e*x^2
+ a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d) - 15*(c^4*d
^4*e*g^4*x^5 + c^4*d^5*f^4 + (4*c^4*d^4*e*f*g^3 + c^4*d^5*g^4)*x^4 + 2*(3*c^4*d^
4*e*f^2*g^2 + 2*c^4*d^5*f*g^3)*x^3 + 2*(2*c^4*d^4*e*f^3*g + 3*c^4*d^5*f^2*g^2)*x
^2 + (c^4*d^4*e*f^4 + 4*c^4*d^5*f^3*g)*x)*log((2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*(c*d*f*g - a*e*g^2)*sqrt(e*x + d) - (c*d*e*g*x^2 - c*d^2*f + 2*a*d*
e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x)*sqrt(-c*d*f*g + a*e*g^2))/(e*g*x^2 + d*
f + (e*f + d*g)*x)))/((c^3*d^4*f^7*g - 3*a*c^2*d^3*e*f^6*g^2 + 3*a^2*c*d^2*e^2*f
^5*g^3 - a^3*d*e^3*f^4*g^4 + (c^3*d^3*e*f^3*g^5 - 3*a*c^2*d^2*e^2*f^2*g^6 + 3*a^
2*c*d*e^3*f*g^7 - a^3*e^4*g^8)*x^5 + (4*c^3*d^3*e*f^4*g^4 - a^3*d*e^3*g^8 + (c^3
*d^4 - 12*a*c^2*d^2*e^2)*f^3*g^5 - 3*(a*c^2*d^3*e - 4*a^2*c*d*e^3)*f^2*g^6 + (3*
a^2*c*d^2*e^2 - 4*a^3*e^4)*f*g^7)*x^4 + 2*(3*c^3*d^3*e*f^5*g^3 - 2*a^3*d*e^3*f*g
^7 + (2*c^3*d^4 - 9*a*c^2*d^2*e^2)*f^4*g^4 - 3*(2*a*c^2*d^3*e - 3*a^2*c*d*e^3)*f
^3*g^5 + 3*(2*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^6)*x^3 + 2*(2*c^3*d^3*e*f^6*g^2 - 3
*a^3*d*e^3*f^2*g^6 + 3*(c^3*d^4 - 2*a*c^2*d^2*e^2)*f^5*g^3 - 3*(3*a*c^2*d^3*e -
2*a^2*c*d*e^3)*f^4*g^4 + (9*a^2*c*d^2*e^2 - 2*a^3*e^4)*f^3*g^5)*x^2 + (c^3*d^3*e
*f^7*g - 4*a^3*d*e^3*f^3*g^5 + (4*c^3*d^4 - 3*a*c^2*d^2*e^2)*f^6*g^2 - 3*(4*a*c^
2*d^3*e - a^2*c*d*e^3)*f^5*g^3 + (12*a^2*c*d^2*e^2 - a^3*e^4)*f^4*g^4)*x)*sqrt(-
c*d*f*g + a*e*g^2)), 1/192*((15*c^3*d^3*g^3*x^3 - 15*c^3*d^3*f^3 + 118*a*c^2*d^2
*e*f^2*g - 136*a^2*c*d*e^2*f*g^2 + 48*a^3*e^3*g^3 + 5*(11*c^3*d^3*f*g^2 - 2*a*c^
2*d^2*e*g^3)*x^2 + (73*c^3*d^3*f^2*g - 36*a*c^2*d^2*e*f*g^2 + 8*a^2*c*d*e^2*g^3)
*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*f*g - a*e*g^2)*sqrt(e*x
+ d) - 15*(c^4*d^4*e*g^4*x^5 + c^4*d^5*f^4 + (4*c^4*d^4*e*f*g^3 + c^4*d^5*g^4)*
x^4 + 2*(3*c^4*d^4*e*f^2*g^2 + 2*c^4*d^5*f*g^3)*x^3 + 2*(2*c^4*d^4*e*f^3*g + 3*c
^4*d^5*f^2*g^2)*x^2 + (c^4*d^4*e*f^4 + 4*c^4*d^5*f^3*g)*x)*arctan(sqrt(c*d*e*x^2
+ a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*f*g - a*e*g^2)*sqrt(e*x + d)/(c*d*e*g*x^2
+ a*d*e*g + (c*d^2 + a*e^2)*g*x)))/((c^3*d^4*f^7*g - 3*a*c^2*d^3*e*f^6*g^2 + 3*
a^2*c*d^2*e^2*f^5*g^3 - a^3*d*e^3*f^4*g^4 + (c^3*d^3*e*f^3*g^5 - 3*a*c^2*d^2*e^2
*f^2*g^6 + 3*a^2*c*d*e^3*f*g^7 - a^3*e^4*g^8)*x^5 + (4*c^3*d^3*e*f^4*g^4 - a^3*d
*e^3*g^8 + (c^3*d^4 - 12*a*c^2*d^2*e^2)*f^3*g^5 - 3*(a*c^2*d^3*e - 4*a^2*c*d*e^3
)*f^2*g^6 + (3*a^2*c*d^2*e^2 - 4*a^3*e^4)*f*g^7)*x^4 + 2*(3*c^3*d^3*e*f^5*g^3 -
2*a^3*d*e^3*f*g^7 + (2*c^3*d^4 - 9*a*c^2*d^2*e^2)*f^4*g^4 - 3*(2*a*c^2*d^3*e - 3
*a^2*c*d*e^3)*f^3*g^5 + 3*(2*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^6)*x^3 + 2*(2*c^3*d^
3*e*f^6*g^2 - 3*a^3*d*e^3*f^2*g^6 + 3*(c^3*d^4 - 2*a*c^2*d^2*e^2)*f^5*g^3 - 3*(3
*a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^4*g^4 + (9*a^2*c*d^2*e^2 - 2*a^3*e^4)*f^3*g^5)*x
^2 + (c^3*d^3*e*f^7*g - 4*a^3*d*e^3*f^3*g^5 + (4*c^3*d^4 - 3*a*c^2*d^2*e^2)*f^6*
g^2 - 3*(4*a*c^2*d^3*e - a^2*c*d*e^3)*f^5*g^3 + (12*a^2*c*d^2*e^2 - a^3*e^4)*f^4
*g^4)*x)*sqrt(c*d*f*g - a*e*g^2))]

_______________________________________________________________________________________

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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**5/(e*x+d)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^5),x, algorithm="giac")

[Out]

Timed out