3.687 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^4} \, dx\)

Optimal. Leaf size=277 \[ \frac{c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{8 g^{3/2} (c d f-a e g)^{5/2}}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 g \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt{d+e x} (f+g x)^3} \]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*g*Sqrt[d + e*x]*(f + g*x)^3) + (
c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*g*(c*d*f - a*e*g)*Sqrt[d +
e*x]*(f + g*x)^2) + (c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*g*(
c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)) + (c^3*d^3*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])])/(8*g^(3/
2)*(c*d*f - a*e*g)^(5/2))

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Rubi [A]  time = 1.15375, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{8 g^{3/2} (c d f-a e g)^{5/2}}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 g \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt{d+e x} (f+g x)^3} \]

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)^4),x]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*g*Sqrt[d + e*x]*(f + g*x)^3) + (
c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*g*(c*d*f - a*e*g)*Sqrt[d +
e*x]*(f + g*x)^2) + (c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*g*(
c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)) + (c^3*d^3*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])])/(8*g^(3/
2)*(c*d*f - a*e*g)^(5/2))

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Rubi in Sympy [A]  time = 116.688, size = 257, normalized size = 0.93 \[ - \frac{c^{3} d^{3} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{8 g^{\frac{3}{2}} \left (a e g - c d f\right )^{\frac{5}{2}}} + \frac{c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 g \sqrt{d + e x} \left (f + g x\right ) \left (a e g - c d f\right )^{2}} - \frac{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{12 g \sqrt{d + e x} \left (f + g x\right )^{2} \left (a e g - c d f\right )} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 g \sqrt{d + e x} \left (f + g x\right )^{3}} \]

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)**4/(e*x+d)**(1/2),x)

[Out]

-c**3*d**3*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)))/(8*g**(3/2)*(a*e*g - c*d*f)**(5/2)) + c**2*d**2*sqr
t(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(8*g*sqrt(d + e*x)*(f + g*x)*(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))/(12*g*sqrt(d +
e*x)*(f + g*x)**2*(a*e*g - c*d*f)) - sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d*
*2))/(3*g*sqrt(d + e*x)*(f + g*x)**3)

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Mathematica [A]  time = 0.703858, size = 168, normalized size = 0.61 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{\frac{3 c^2 d^2 (f+g x)^2}{(c d f-a e g)^2}+\frac{2 c d (f+g x)}{c d f-a e g}-8}{3 g (f+g x)^3}-\frac{c^3 d^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{g^{3/2} \sqrt{a e+c d x} (a e g-c d f)^{5/2}}\right )}{8 \sqrt{d+e x}} \]

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)^4),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((-8 + (2*c*d*(f + g*x))/(c*d*f - a*e*g) + (3*c^2
*d^2*(f + g*x)^2)/(c*d*f - a*e*g)^2)/(3*g*(f + g*x)^3) - (c^3*d^3*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)^(5/2)
*Sqrt[a*e + c*d*x])))/(8*Sqrt[d + e*x])

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Maple [A]  time = 0.042, size = 453, normalized size = 1.6 \[ -{\frac{1}{24\, \left ( gx+f \right ) ^{3}g \left ( aeg-cdf \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 3\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{3}{c}^{3}{d}^{3}{g}^{3}+9\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}{c}^{3}{d}^{3}f{g}^{2}+9\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{3}{d}^{3}{f}^{2}g+3\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{3}{d}^{3}{f}^{3}-3\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{g}^{2}+2\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}xacde{g}^{2}-8\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}x{c}^{2}{d}^{2}fg+8\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{2}{e}^{2}{g}^{2}-14\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}acdefg+3\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{c}^{2}{d}^{2}{f}^{2} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}{\frac{1}{\sqrt{cdx+ae}}}} \]

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)^4/(e*x+d)^(1/2),x)

[Out]

-1/24*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(3*arctanh(g*(c*d*x+a*e)^(1/2)/((a
*e*g-c*d*f)*g)^(1/2))*x^3*c^3*d^3*g^3+9*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*
f)*g)^(1/2))*x^2*c^3*d^3*f*g^2+9*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(
1/2))*x*c^3*d^3*f^2*g+3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3
*d^3*f^3-3*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x^2*c^2*d^2*g^2+2*((a*e*g-c
*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x*a*c*d*e*g^2-8*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+
a*e)^(1/2)*x*c^2*d^2*f*g+8*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^2*g^2
-14*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*f*g+3*((a*e*g-c*d*f)*g)^(1
/2)*(c*d*x+a*e)^(1/2)*c^2*d^2*f^2)/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)
^3/g/(a*e*g-c*d*f)^2/(c*d*x+a*e)^(1/2)

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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)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302511, 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)^4),x, algorithm="fricas")

[Out]

[1/48*(2*(3*c^2*d^2*g^2*x^2 - 3*c^2*d^2*f^2 + 14*a*c*d*e*f*g - 8*a^2*e^2*g^2 + 2
*(4*c^2*d^2*f*g - a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq
rt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d) + 3*(c^3*d^3*e*g^3*x^4 + c^3*d^4*f^3 + (3*c
^3*d^3*e*f*g^2 + c^3*d^4*g^3)*x^3 + 3*(c^3*d^3*e*f^2*g + c^3*d^4*f*g^2)*x^2 + (c
^3*d^3*e*f^3 + 3*c^3*d^4*f^2*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^2*d^3*f^5*g - 2*a*c*d^2*e*f^4*g^2 + a^2*d*e^2*f^3*g^3 + (c^2*
d^2*e*f^2*g^4 - 2*a*c*d*e^2*f*g^5 + a^2*e^3*g^6)*x^4 + (3*c^2*d^2*e*f^3*g^3 + a^
2*d*e^2*g^6 + (c^2*d^3 - 6*a*c*d*e^2)*f^2*g^4 - (2*a*c*d^2*e - 3*a^2*e^3)*f*g^5)
*x^3 + 3*(c^2*d^2*e*f^4*g^2 + a^2*d*e^2*f*g^5 + (c^2*d^3 - 2*a*c*d*e^2)*f^3*g^3
- (2*a*c*d^2*e - a^2*e^3)*f^2*g^4)*x^2 + (c^2*d^2*e*f^5*g + 3*a^2*d*e^2*f^2*g^4
+ (3*c^2*d^3 - 2*a*c*d*e^2)*f^4*g^2 - (6*a*c*d^2*e - a^2*e^3)*f^3*g^3)*x)*sqrt(-
c*d*f*g + a*e*g^2)), 1/24*((3*c^2*d^2*g^2*x^2 - 3*c^2*d^2*f^2 + 14*a*c*d*e*f*g -
8*a^2*e^2*g^2 + 2*(4*c^2*d^2*f*g - a*c*d*e*g^2)*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) - 3*(c^3*d^3*e*g^3*x^4 + c
^3*d^4*f^3 + (3*c^3*d^3*e*f*g^2 + c^3*d^4*g^3)*x^3 + 3*(c^3*d^3*e*f^2*g + c^3*d^
4*f*g^2)*x^2 + (c^3*d^3*e*f^3 + 3*c^3*d^4*f^2*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^2*d^3*f^5*g - 2*a*c*d^2*e*f^4*g^2 + a^2*d*e^2*f
^3*g^3 + (c^2*d^2*e*f^2*g^4 - 2*a*c*d*e^2*f*g^5 + a^2*e^3*g^6)*x^4 + (3*c^2*d^2*
e*f^3*g^3 + a^2*d*e^2*g^6 + (c^2*d^3 - 6*a*c*d*e^2)*f^2*g^4 - (2*a*c*d^2*e - 3*a
^2*e^3)*f*g^5)*x^3 + 3*(c^2*d^2*e*f^4*g^2 + a^2*d*e^2*f*g^5 + (c^2*d^3 - 2*a*c*d
*e^2)*f^3*g^3 - (2*a*c*d^2*e - a^2*e^3)*f^2*g^4)*x^2 + (c^2*d^2*e*f^5*g + 3*a^2*
d*e^2*f^2*g^4 + (3*c^2*d^3 - 2*a*c*d*e^2)*f^4*g^2 - (6*a*c*d^2*e - a^2*e^3)*f^3*
g^3)*x)*sqrt(c*d*f*g - a*e*g^2))]

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**4/(e*x+d)**(1/2),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(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^4),x, algorithm="giac")

[Out]

Timed out