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

Optimal. Leaf size=280 \[ \frac{5 c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{8 \sqrt{g} (c d f-a e g)^{7/2}}+\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^3 (c d f-a e g)} \]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f
+ g*x)^3) + (5*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*(c*d*f - a*e
*g)^2*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2])/(8*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (5*c^3*d^3*ArcTan[(Sq
rt[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*Sqrt[g]*(c*d*f - a*e*g)^(7/2))

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Rubi [A]  time = 1.37471, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{5 c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{8 \sqrt{g} (c d f-a e g)^{7/2}}+\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^3 (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f
+ g*x)^3) + (5*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*(c*d*f - a*e
*g)^2*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2])/(8*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (5*c^3*d^3*ArcTan[(Sq
rt[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*Sqrt[g]*(c*d*f - a*e*g)^(7/2))

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Rubi in Sympy [A]  time = 116.729, size = 269, normalized size = 0.96 \[ \frac{5 c^{3} d^{3} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{8 \sqrt{g} \left (a e g - c d f\right )^{\frac{7}{2}}} - \frac{5 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 \sqrt{d + e x} \left (f + g x\right ) \left (a e g - c d f\right )^{3}} + \frac{5 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{12 \sqrt{d + e x} \left (f + g x\right )^{2} \left (a e g - c d f\right )^{2}} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 \sqrt{d + e x} \left (f + g x\right )^{3} \left (a e g - c d f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)/(g*x+f)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

5*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*sqrt(g)*(a*e*g - c*d*f)**(7/2)) - 5*c**2*d**2*s
qrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(8*sqrt(d + e*x)*(f + g*x)*(a*e*g
- c*d*f)**3) + 5*c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(12*sqrt(d +
e*x)*(f + g*x)**2*(a*e*g - c*d*f)**2) - sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c
*d**2))/(3*sqrt(d + e*x)*(f + g*x)**3*(a*e*g - c*d*f))

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Mathematica [A]  time = 0.666635, size = 188, normalized size = 0.67 \[ \frac{\sqrt{d+e x} \left (\frac{(a e+c d x) \left (8 a^2 e^2 g^2-2 a c d e g (13 f+5 g x)+c^2 d^2 \left (33 f^2+40 f g x+15 g^2 x^2\right )\right )}{3 (f+g x)^3 (c d f-a e g)^3}+\frac{5 c^3 d^3 \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{\sqrt{g} (a e g-c d f)^{7/2}}\right )}{8 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(Sqrt[d + e*x]*(((a*e + c*d*x)*(8*a^2*e^2*g^2 - 2*a*c*d*e*g*(13*f + 5*g*x) + c^2
*d^2*(33*f^2 + 40*f*g*x + 15*g^2*x^2)))/(3*(c*d*f - a*e*g)^3*(f + g*x)^3) + (5*c
^3*d^3*Sqrt[a*e + c*d*x]*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d*f) + a*e
*g]])/(Sqrt[g]*(-(c*d*f) + a*e*g)^(7/2))))/(8*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.031, size = 450, normalized size = 1.6 \[{\frac{1}{24\, \left ( aeg-cdf \right ) ^{3} \left ( gx+f \right ) ^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{3}{c}^{3}{d}^{3}{g}^{3}+45\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}{c}^{3}{d}^{3}f{g}^{2}+45\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{3}{d}^{3}{f}^{2}g+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{3}{d}^{3}{f}^{3}-15\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{g}^{2}+10\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}xacde{g}^{2}-40\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}x{c}^{2}{d}^{2}fg-8\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{2}{e}^{2}{g}^{2}+26\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}acdefg-33\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{c}^{2}{d}^{2}{f}^{2} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

1/24*(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^3*c^3*d^3*g^3+45*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+45*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+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*
c^3*d^3*f^3-15*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x^2*c^2*d^2*g^2+10*((a*
e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x*a*c*d*e*g^2-40*((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+26*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*f*g-33*((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)/(c*d*x+a*e)^(1/2)/(a*e*g
-c*d*f)^3/(g*x+f)^3/((a*e*g-c*d*f)*g)^(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(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320529, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^4),x, algorithm="fricas")

[Out]

[1/48*(2*(15*c^2*d^2*g^2*x^2 + 33*c^2*d^2*f^2 - 26*a*c*d*e*f*g + 8*a^2*e^2*g^2 +
10*(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) - 15*(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^3*d^4*f^6 - 3*a*c^2*d^3*e*f^5*g + 3*a^2*c*d^2*e^2*f^4*g^2
- a^3*d*e^3*f^3*g^3 + (c^3*d^3*e*f^3*g^3 - 3*a*c^2*d^2*e^2*f^2*g^4 + 3*a^2*c*d*e
^3*f*g^5 - a^3*e^4*g^6)*x^4 + (3*c^3*d^3*e*f^4*g^2 - a^3*d*e^3*g^6 + (c^3*d^4 -
9*a*c^2*d^2*e^2)*f^3*g^3 - 3*(a*c^2*d^3*e - 3*a^2*c*d*e^3)*f^2*g^4 + 3*(a^2*c*d^
2*e^2 - a^3*e^4)*f*g^5)*x^3 + 3*(c^3*d^3*e*f^5*g - a^3*d*e^3*f*g^5 + (c^3*d^4 -
3*a*c^2*d^2*e^2)*f^4*g^2 - 3*(a*c^2*d^3*e - a^2*c*d*e^3)*f^3*g^3 + (3*a^2*c*d^2*
e^2 - a^3*e^4)*f^2*g^4)*x^2 + (c^3*d^3*e*f^6 - 3*a^3*d*e^3*f^2*g^4 + 3*(c^3*d^4
- a*c^2*d^2*e^2)*f^5*g - 3*(3*a*c^2*d^3*e - a^2*c*d*e^3)*f^4*g^2 + (9*a^2*c*d^2*
e^2 - a^3*e^4)*f^3*g^3)*x)*sqrt(-c*d*f*g + a*e*g^2)), 1/24*((15*c^2*d^2*g^2*x^2
+ 33*c^2*d^2*f^2 - 26*a*c*d*e*f*g + 8*a^2*e^2*g^2 + 10*(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) - 15*(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^3*d^4*f^6
- 3*a*c^2*d^3*e*f^5*g + 3*a^2*c*d^2*e^2*f^4*g^2 - a^3*d*e^3*f^3*g^3 + (c^3*d^3*e
*f^3*g^3 - 3*a*c^2*d^2*e^2*f^2*g^4 + 3*a^2*c*d*e^3*f*g^5 - a^3*e^4*g^6)*x^4 + (3
*c^3*d^3*e*f^4*g^2 - a^3*d*e^3*g^6 + (c^3*d^4 - 9*a*c^2*d^2*e^2)*f^3*g^3 - 3*(a*
c^2*d^3*e - 3*a^2*c*d*e^3)*f^2*g^4 + 3*(a^2*c*d^2*e^2 - a^3*e^4)*f*g^5)*x^3 + 3*
(c^3*d^3*e*f^5*g - a^3*d*e^3*f*g^5 + (c^3*d^4 - 3*a*c^2*d^2*e^2)*f^4*g^2 - 3*(a*
c^2*d^3*e - a^2*c*d*e^3)*f^3*g^3 + (3*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^4)*x^2 + (c
^3*d^3*e*f^6 - 3*a^3*d*e^3*f^2*g^4 + 3*(c^3*d^4 - a*c^2*d^2*e^2)*f^5*g - 3*(3*a*
c^2*d^3*e - a^2*c*d*e^3)*f^4*g^2 + (9*a^2*c*d^2*e^2 - a^3*e^4)*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((e*x+d)**(1/2)/(g*x+f)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(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(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^4),x, algorithm="giac")

[Out]

Timed out