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

Optimal. Leaf size=222 \[ -\frac{3 \sqrt{c} \sqrt{d} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{g^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{3 c d \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g^2 \sqrt{d+e x}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{g (d+e x)^{3/2} \sqrt{f+g x}} \]

[Out]

(3*c*d*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^2*Sqrt[d +
e*x]) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(g*(d + e*x)^(3/2)*Sqr
t[f + g*x]) - (3*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]
*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(g^(5/2)*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.934953, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.104 \[ -\frac{3 \sqrt{c} \sqrt{d} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{g^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{3 c d \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g^2 \sqrt{d+e x}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{g (d+e x)^{3/2} \sqrt{f+g x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*c*d*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^2*Sqrt[d +
e*x]) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(g*(d + e*x)^(3/2)*Sqr
t[f + g*x]) - (3*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]
*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(g^(5/2)*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 87.7019, size = 216, normalized size = 0.97 \[ \frac{3 \sqrt{c} \sqrt{d} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{f + g x}}{\sqrt{g} \sqrt{a e + c d x}} \right )}}{g^{\frac{5}{2}} \sqrt{d + e x} \sqrt{a e + c d x}} + \frac{3 c d \sqrt{f + g x} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{g^{2} \sqrt{d + e x}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{g \left (d + e x\right )^{\frac{3}{2}} \sqrt{f + g x}} \]

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

[Out]

3*sqrt(c)*sqrt(d)*(a*e*g - c*d*f)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))
*atanh(sqrt(c)*sqrt(d)*sqrt(f + g*x)/(sqrt(g)*sqrt(a*e + c*d*x)))/(g**(5/2)*sqrt
(d + e*x)*sqrt(a*e + c*d*x)) + 3*c*d*sqrt(f + g*x)*sqrt(a*d*e + c*d*e*x**2 + x*(
a*e**2 + c*d**2))/(g**2*sqrt(d + e*x)) - 2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d
**2))**(3/2)/(g*(d + e*x)**(3/2)*sqrt(f + g*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.57936, size = 164, normalized size = 0.74 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{2 (c d (3 f+g x)-2 a e g)}{g^2 \sqrt{f+g x} (a e+c d x)}-\frac{3 \sqrt{c} \sqrt{d} (c d f-a e g) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{g} \sqrt{f+g x} \sqrt{a e+c d x}+a e g+c d (f+2 g x)\right )}{g^{5/2} (a e+c d x)^{3/2}}\right )}{2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((2*(-2*a*e*g + c*d*(3*f + g*x)))/(g^2*(a*e + c
*d*x)*Sqrt[f + g*x]) - (3*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)*Log[a*e*g + 2*Sqrt[c]*
Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x]*Sqrt[f + g*x] + c*d*(f + 2*g*x)])/(g^(5/2)*(a*
e + c*d*x)^(3/2))))/(2*(d + e*x)^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.037, size = 383, normalized size = 1.7 \[{\frac{1}{2\,{g}^{2}} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ) xacde{g}^{2}-3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ) x{c}^{2}{d}^{2}fg+3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ) aegfcd-3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){f}^{2}{c}^{2}{d}^{2}+2\,g\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }xcd\sqrt{dgc}-4\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }aeg\sqrt{dgc}+6\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }fcd\sqrt{dgc} \right ) \sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }}}{\frac{1}{\sqrt{dgc}}}{\frac{1}{\sqrt{gx+f}}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(3/2),x)

[Out]

1/2*(3*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)
)/(d*g*c)^(1/2))*x*a*c*d*e*g^2-3*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*((g*x+f)*(c*d*x
+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*x*c^2*d^2*f*g+3*ln(1/2*(2*x*c*d*g+a*e
*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*a*e*g*f*c*d
-3*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d
*g*c)^(1/2))*f^2*c^2*d^2+2*g*((g*x+f)*(c*d*x+a*e))^(1/2)*x*c*d*(d*g*c)^(1/2)-4*(
(g*x+f)*(c*d*x+a*e))^(1/2)*a*e*g*(d*g*c)^(1/2)+6*((g*x+f)*(c*d*x+a*e))^(1/2)*f*c
*d*(d*g*c)^(1/2))*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/((g*x+f)*(c*d*x+a*e))^
(1/2)/(d*g*c)^(1/2)/g^2/(g*x+f)^(1/2)/(e*x+d)^(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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.941302, size = 1, normalized size = 0. \[ \left [\frac{4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d g x + 3 \, c d f - 2 \, a e g\right )} \sqrt{e x + d} \sqrt{g x + f} - 3 \,{\left (c d^{2} f^{2} - a d e f g +{\left (c d e f g - a e^{2} g^{2}\right )} x^{2} +{\left (c d e f^{2} - a d e g^{2} +{\left (c d^{2} - a e^{2}\right )} f g\right )} x\right )} \sqrt{\frac{c d}{g}} \log \left (-\frac{8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 4 \,{\left (2 \, c d g^{2} x + c d f g + a e g^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} \sqrt{\frac{c d}{g}} + 8 \,{\left (c^{2} d^{2} e f g +{\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} +{\left (c^{2} d^{2} e f^{2} + 2 \,{\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g +{\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{4 \,{\left (e g^{3} x^{2} + d f g^{2} +{\left (e f g^{2} + d g^{3}\right )} x\right )}}, \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d g x + 3 \, c d f - 2 \, a e g\right )} \sqrt{e x + d} \sqrt{g x + f} - 3 \,{\left (c d^{2} f^{2} - a d e f g +{\left (c d e f g - a e^{2} g^{2}\right )} x^{2} +{\left (c d e f^{2} - a d e g^{2} +{\left (c d^{2} - a e^{2}\right )} f g\right )} x\right )} \sqrt{-\frac{c d}{g}} \arctan \left (\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} c d}{{\left (2 \, c d e g x^{2} + c d^{2} f + a d e g +{\left (c d e f +{\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x\right )} \sqrt{-\frac{c d}{g}}}\right )}{2 \,{\left (e g^{3} x^{2} + d f g^{2} +{\left (e f g^{2} + d g^{3}\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*g*x + 3*c*d*f - 2*a*e*g
)*sqrt(e*x + d)*sqrt(g*x + f) - 3*(c*d^2*f^2 - a*d*e*f*g + (c*d*e*f*g - a*e^2*g^
2)*x^2 + (c*d*e*f^2 - a*d*e*g^2 + (c*d^2 - a*e^2)*f*g)*x)*sqrt(c*d/g)*log(-(8*c^
2*d^2*e*g^2*x^3 + c^2*d^3*f^2 + 6*a*c*d^2*e*f*g + a^2*d*e^2*g^2 + 4*(2*c*d*g^2*x
+ c*d*f*g + a*e*g^2)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*
sqrt(g*x + f)*sqrt(c*d/g) + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^2 +
(c^2*d^2*e*f^2 + 2*(4*c^2*d^3 + 3*a*c*d*e^2)*f*g + (8*a*c*d^2*e + a^2*e^3)*g^2)*
x)/(e*x + d)))/(e*g^3*x^2 + d*f*g^2 + (e*f*g^2 + d*g^3)*x), 1/2*(2*sqrt(c*d*e*x^
2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*g*x + 3*c*d*f - 2*a*e*g)*sqrt(e*x + d)*sqrt(
g*x + f) - 3*(c*d^2*f^2 - a*d*e*f*g + (c*d*e*f*g - a*e^2*g^2)*x^2 + (c*d*e*f^2 -
a*d*e*g^2 + (c*d^2 - a*e^2)*f*g)*x)*sqrt(-c*d/g)*arctan(2*sqrt(c*d*e*x^2 + a*d*
e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*c*d/((2*c*d*e*g*x^2 + c*d^2*f
+ a*d*e*g + (c*d*e*f + (2*c*d^2 + a*e^2)*g)*x)*sqrt(-c*d/g))))/(e*g^3*x^2 + d*f
*g^2 + (e*f*g^2 + d*g^3)*x)]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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