3.698 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx\)

Optimal. Leaf size=335 \[ \frac{3 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{64 g^{5/2} (c d f-a e g)^{5/2}}+\frac{3 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g^2 \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 g^2 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 g^2 \sqrt{d+e x} (f+g x)^3}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4} \]

[Out]

-(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*g^2*Sqrt[d + e*x]*(f + g*x
)^3) + (c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*g^2*(c*d*f - a*
e*g)*Sqrt[d + e*x]*(f + g*x)^2) + (3*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2])/(64*g^2*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)) - (a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(3/2)/(4*g*(d + e*x)^(3/2)*(f + g*x)^4) + (3*c^4*d^4*ArcT
an[(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]*Sq
rt[d + e*x])])/(64*g^(5/2)*(c*d*f - a*e*g)^(5/2))

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Rubi [A]  time = 1.50351, antiderivative size = 335, 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 \[ \frac{3 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{64 g^{5/2} (c d f-a e g)^{5/2}}+\frac{3 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g^2 \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 g^2 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 g^2 \sqrt{d+e x} (f+g x)^3}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4} \]

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)^5),x]

[Out]

-(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*g^2*Sqrt[d + e*x]*(f + g*x
)^3) + (c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*g^2*(c*d*f - a*
e*g)*Sqrt[d + e*x]*(f + g*x)^2) + (3*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2])/(64*g^2*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)) - (a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(3/2)/(4*g*(d + e*x)^(3/2)*(f + g*x)^4) + (3*c^4*d^4*ArcT
an[(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]*Sq
rt[d + e*x])])/(64*g^(5/2)*(c*d*f - a*e*g)^(5/2))

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Rubi in Sympy [A]  time = 150.577, size = 320, normalized size = 0.96 \[ - \frac{3 c^{4} d^{4} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{64 g^{\frac{5}{2}} \left (a e g - c d f\right )^{\frac{5}{2}}} + \frac{3 c^{3} d^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{64 g^{2} \sqrt{d + e x} \left (f + g x\right ) \left (a e g - c d f\right )^{2}} - \frac{c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{32 g^{2} \sqrt{d + e x} \left (f + g x\right )^{2} \left (a e g - c d f\right )} - \frac{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 g^{2} \sqrt{d + e x} \left (f + g x\right )^{3}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{4 g \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{4}} \]

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)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**5,x)

[Out]

-3*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**(5/2)*(a*e*g - c*d*f)**(5/2)) + 3*c**3*d**
3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(64*g**2*sqrt(d + e*x)*(f + g*x
)*(a*e*g - c*d*f)**2) - c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))
/(32*g**2*sqrt(d + e*x)*(f + g*x)**2*(a*e*g - c*d*f)) - c*d*sqrt(a*d*e + c*d*e*x
**2 + x*(a*e**2 + c*d**2))/(8*g**2*sqrt(d + e*x)*(f + g*x)**3) - (a*d*e + c*d*e*
x**2 + x*(a*e**2 + c*d**2))**(3/2)/(4*g*(d + e*x)**(3/2)*(f + g*x)**4)

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Mathematica [A]  time = 1.0119, size = 201, normalized size = 0.6 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{\frac{3 c^3 d^3 (f+g x)^3}{(c d f-a e g)^2}+\frac{2 c^2 d^2 (f+g x)^2}{c d f-a e g}+16 (c d f-a e g)-24 c d (f+g x)}{g^2 (f+g x)^4 (a e+c d x)}-\frac{3 c^4 d^4 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{g^{5/2} (a e+c d x)^{3/2} (a e g-c d f)^{5/2}}\right )}{64 (d+e x)^{3/2}} \]

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)^5),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((16*(c*d*f - a*e*g) - 24*c*d*(f + g*x) + (2*c^
2*d^2*(f + g*x)^2)/(c*d*f - a*e*g) + (3*c^3*d^3*(f + g*x)^3)/(c*d*f - a*e*g)^2)/
(g^2*(a*e + c*d*x)*(f + g*x)^4) - (3*c^4*d^4*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])
/Sqrt[-(c*d*f) + a*e*g]])/(g^(5/2)*(-(c*d*f) + a*e*g)^(5/2)*(a*e + c*d*x)^(3/2))
))/(64*(d + e*x)^(3/2))

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Maple [B]  time = 0.045, size = 665, normalized size = 2. \[ -{\frac{1}{64\, \left ( gx+f \right ) ^{4}{g}^{2} \left ( aeg-cdf \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 3\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{4}{c}^{4}{d}^{4}{g}^{4}+12\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{3}{c}^{4}{d}^{4}f{g}^{3}+18\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}{c}^{4}{d}^{4}{f}^{2}{g}^{2}+12\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{4}{d}^{4}{f}^{3}g-3\,{x}^{3}{c}^{3}{d}^{3}{g}^{3}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+3\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{4}{d}^{4}{f}^{4}+2\,{x}^{2}a{c}^{2}{d}^{2}e{g}^{3}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-11\,{x}^{2}{c}^{3}{d}^{3}f{g}^{2}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+24\,x{a}^{2}cd{e}^{2}{g}^{3}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-44\,xa{c}^{2}{d}^{2}ef{g}^{2}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+11\,x{c}^{3}{d}^{3}{f}^{2}g\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+16\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{3}{e}^{3}{g}^{3}-24\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{2}cd{e}^{2}f{g}^{2}+2\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}a{c}^{2}{d}^{2}e{f}^{2}g+3\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{c}^{3}{d}^{3}{f}^{3} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}{\frac{1}{\sqrt{cdx+ae}}}} \]

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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^5,x)

[Out]

-1/64*(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^4*c^4*d^4*g^4+12*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+18*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+12*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-3*x^3*c^3*d^3*g^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+
3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^4*d^4*f^4+2*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)-11*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)+24*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)-44*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
)+11*x*c^3*d^3*f^2*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+16*((a*e*g-c*d*f)
*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^3*e^3*g^3-24*((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+2*((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+3*((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^2/(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((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)^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.307154, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^5),x, algorithm="fricas")

[Out]

[1/128*(2*(3*c^3*d^3*g^3*x^3 - 3*c^3*d^3*f^3 - 2*a*c^2*d^2*e*f^2*g + 24*a^2*c*d*
e^2*f*g^2 - 16*a^3*e^3*g^3 + (11*c^3*d^3*f*g^2 - 2*a*c^2*d^2*e*g^3)*x^2 - (11*c^
3*d^3*f^2*g - 44*a*c^2*d^2*e*f*g^2 + 24*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) + 3*(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^2*d^3*f^6*g^2 - 2*a*c*d^2*e*f^5*g^3 + a^2*d*e^2*f^4*g^4 + (c^
2*d^2*e*f^2*g^6 - 2*a*c*d*e^2*f*g^7 + a^2*e^3*g^8)*x^5 + (4*c^2*d^2*e*f^3*g^5 +
a^2*d*e^2*g^8 + (c^2*d^3 - 8*a*c*d*e^2)*f^2*g^6 - 2*(a*c*d^2*e - 2*a^2*e^3)*f*g^
7)*x^4 + 2*(3*c^2*d^2*e*f^4*g^4 + 2*a^2*d*e^2*f*g^7 + 2*(c^2*d^3 - 3*a*c*d*e^2)*
f^3*g^5 - (4*a*c*d^2*e - 3*a^2*e^3)*f^2*g^6)*x^3 + 2*(2*c^2*d^2*e*f^5*g^3 + 3*a^
2*d*e^2*f^2*g^6 + (3*c^2*d^3 - 4*a*c*d*e^2)*f^4*g^4 - 2*(3*a*c*d^2*e - a^2*e^3)*
f^3*g^5)*x^2 + (c^2*d^2*e*f^6*g^2 + 4*a^2*d*e^2*f^3*g^5 + 2*(2*c^2*d^3 - a*c*d*e
^2)*f^5*g^3 - (8*a*c*d^2*e - a^2*e^3)*f^4*g^4)*x)*sqrt(-c*d*f*g + a*e*g^2)), 1/6
4*((3*c^3*d^3*g^3*x^3 - 3*c^3*d^3*f^3 - 2*a*c^2*d^2*e*f^2*g + 24*a^2*c*d*e^2*f*g
^2 - 16*a^3*e^3*g^3 + (11*c^3*d^3*f*g^2 - 2*a*c^2*d^2*e*g^3)*x^2 - (11*c^3*d^3*f
^2*g - 44*a*c^2*d^2*e*f*g^2 + 24*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) - 3*(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)*sq
rt(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^6*g^2 - 2*a*c*d^2*e*f^5*g^3 + a^2*d*e^2*f^4*g^4 + (c^2*d^2*e*f^2
*g^6 - 2*a*c*d*e^2*f*g^7 + a^2*e^3*g^8)*x^5 + (4*c^2*d^2*e*f^3*g^5 + a^2*d*e^2*g
^8 + (c^2*d^3 - 8*a*c*d*e^2)*f^2*g^6 - 2*(a*c*d^2*e - 2*a^2*e^3)*f*g^7)*x^4 + 2*
(3*c^2*d^2*e*f^4*g^4 + 2*a^2*d*e^2*f*g^7 + 2*(c^2*d^3 - 3*a*c*d*e^2)*f^3*g^5 - (
4*a*c*d^2*e - 3*a^2*e^3)*f^2*g^6)*x^3 + 2*(2*c^2*d^2*e*f^5*g^3 + 3*a^2*d*e^2*f^2
*g^6 + (3*c^2*d^3 - 4*a*c*d*e^2)*f^4*g^4 - 2*(3*a*c*d^2*e - a^2*e^3)*f^3*g^5)*x^
2 + (c^2*d^2*e*f^6*g^2 + 4*a^2*d*e^2*f^3*g^5 + 2*(2*c^2*d^3 - a*c*d*e^2)*f^5*g^3
- (8*a*c*d^2*e - a^2*e^3)*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)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**5,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((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)^5),x, algorithm="giac")

[Out]

Timed out