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

Optimal. Leaf size=178 \[ -\frac{3 c d \sqrt{c d f-a e g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{5/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{g (d+e x)^{3/2} (f+g x)}+\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g^2 \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.878574, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{3 c d \sqrt{c d f-a e g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{5/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{g (d+e x)^{3/2} (f+g x)}+\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g^2 \sqrt{d+e x}} \]

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)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 82.0464, size = 170, normalized size = 0.96 \[ \frac{3 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{g^{2} \sqrt{d + e x}} - \frac{3 c d \sqrt{a e g - c d f} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{g^{\frac{5}{2}}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{g \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.742489, size = 127, normalized size = 0.71 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{\sqrt{g} (c d (3 f+2 g x)-a e g)}{f+g x}-\frac{3 c d \sqrt{a e g-c d f} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{\sqrt{a e+c d x}}\right )}{g^{5/2} \sqrt{d+e x}} \]

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)^2),x]

[Out]

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

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Maple [A]  time = 0.035, size = 306, normalized size = 1.7 \[{\frac{1}{{g}^{2} \left ( gx+f \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( -3\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) xacde{g}^{2}+3\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{2}{d}^{2}fg-3\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) acdefg+3\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{2}{d}^{2}{f}^{2}+2\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}xcdg-\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}aeg+3\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}cdf \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30806, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, c^{2} d^{2} e g x^{3} + 6 \, a c d^{2} e f - 2 \, a^{2} d e^{2} g + 3 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d g x + c d f\right )} \sqrt{e x + d} \sqrt{-\frac{c d f - a e g}{g}} \log \left (-\frac{c d e g x^{2} - c d^{2} f + 2 \, a d e g - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} g \sqrt{-\frac{c d f - a e g}{g}} -{\left (c d e f -{\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f +{\left (e f + d g\right )} x}\right ) + 2 \,{\left (3 \, c^{2} d^{2} e f +{\left (2 \, c^{2} d^{3} + a c d e^{2}\right )} g\right )} x^{2} + 2 \,{\left (3 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} f +{\left (a c d^{2} e - a^{2} e^{3}\right )} g\right )} x}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g^{3} x + f g^{2}\right )} \sqrt{e x + d}}, \frac{2 \, c^{2} d^{2} e g x^{3} + 3 \, a c d^{2} e f - a^{2} d e^{2} g - 3 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d g x + c d f\right )} \sqrt{e x + d} \sqrt{\frac{c d f - a e g}{g}} \arctan \left (-\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d f - a e g\right )} \sqrt{e x + d}}{{\left (c d e g x^{2} + a d e g +{\left (c d^{2} + a e^{2}\right )} g x\right )} \sqrt{\frac{c d f - a e g}{g}}}\right ) +{\left (3 \, c^{2} d^{2} e f +{\left (2 \, c^{2} d^{3} + a c d e^{2}\right )} g\right )} x^{2} +{\left (3 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} f +{\left (a c d^{2} e - a^{2} e^{3}\right )} g\right )} x}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g^{3} x + f g^{2}\right )} \sqrt{e x + d}}\right ] \]

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)^2),x, algorithm="fricas")

[Out]

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

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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)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**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((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)^2),x, algorithm="giac")

[Out]

Timed out