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

Optimal. Leaf size=342 \[ \frac{35 c^2 d^2 g^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 (c d f-a e g)^{9/2}}+\frac{35 c d g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt{d+e x} (f+g x) (c d f-a e g)^4}+\frac{35 g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^3}+\frac{14 g \sqrt{d+e x}}{3 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]

[Out]

(-2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2)^(3/2)) + (14*g*Sqrt[d + e*x])/(3*(c*d*f - a*e*g)^2*(f + g*x)^2*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*g^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(6*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^2) + (35*c*d*g^2*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*(f
+ g*x)) + (35*c^2*d^2*g^(3/2)*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])])/(4*(c*d*f - a*e*g)^(9/2))

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Rubi [A]  time = 1.8413, antiderivative size = 342, 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 \[ \frac{35 c^2 d^2 g^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 (c d f-a e g)^{9/2}}+\frac{35 c d g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt{d+e x} (f+g x) (c d f-a e g)^4}+\frac{35 g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^3}+\frac{14 g \sqrt{d+e x}}{3 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(-2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2)^(3/2)) + (14*g*Sqrt[d + e*x])/(3*(c*d*f - a*e*g)^2*(f + g*x)^2*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*g^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(6*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^2) + (35*c*d*g^2*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*(f
+ g*x)) + (35*c^2*d^2*g^(3/2)*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])])/(4*(c*d*f - a*e*g)^(9/2))

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Rubi in Sympy [A]  time = 153.982, size = 332, normalized size = 0.97 \[ - \frac{35 c^{2} d^{2} g^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{4 \left (a e g - c d f\right )^{\frac{9}{2}}} + \frac{35 c d g^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 \sqrt{d + e x} \left (f + g x\right ) \left (a e g - c d f\right )^{4}} - \frac{35 g^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{6 \sqrt{d + e x} \left (f + g x\right )^{2} \left (a e g - c d f\right )^{3}} + \frac{14 g \sqrt{d + e x}}{3 \left (f + g x\right )^{2} \left (a e g - c d f\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 \left (f + g x\right )^{2} \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(g*x+f)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

-35*c**2*d**2*g**(3/2)*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)))/(4*(a*e*g - c*d*f)**(9/2)) + 35*c*d*g**
2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*sqrt(d + e*x)*(f + g*x)*(a*e
*g - c*d*f)**4) - 35*g**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(6*sqrt
(d + e*x)*(f + g*x)**2*(a*e*g - c*d*f)**3) + 14*g*sqrt(d + e*x)/(3*(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))) + 2*(d + e*x)
**(3/2)/(3*(f + g*x)**2*(a*e*g - c*d*f)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2
))**(3/2))

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Mathematica [A]  time = 0.985426, size = 212, normalized size = 0.62 \[ \frac{\sqrt{d+e x} \left (\frac{(a e+c d x) \left (-\frac{8 c^2 d^2 (c d f-a e g)}{(a e+c d x)^2}+\frac{72 c^2 d^2 g}{a e+c d x}+\frac{6 g^2 (c d f-a e g)}{(f+g x)^2}+\frac{33 c d g^2}{f+g x}\right )}{3 (c d f-a e g)^4}-\frac{35 c^2 d^2 g^{3/2} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{(a e g-c d f)^{9/2}}\right )}{4 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)/((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(Sqrt[d + e*x]*(((a*e + c*d*x)*((-8*c^2*d^2*(c*d*f - a*e*g))/(a*e + c*d*x)^2 + (
72*c^2*d^2*g)/(a*e + c*d*x) + (6*g^2*(c*d*f - a*e*g))/(f + g*x)^2 + (33*c*d*g^2)
/(f + g*x)))/(3*(c*d*f - a*e*g)^4) - (35*c^2*d^2*g^(3/2)*Sqrt[a*e + c*d*x]*ArcTa
nh[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d*f) + a*e*g]])/(-(c*d*f) + a*e*g)^(9/2)
))/(4*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [B]  time = 0.047, size = 670, normalized size = 2. \[ -{\frac{1}{12\, \left ( cdx+ae \right ) ^{2} \left ( aeg-cdf \right ) ^{4} \left ( gx+f \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 105\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{3}{c}^{3}{d}^{3}{g}^{4}\sqrt{cdx+ae}+105\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}a{c}^{2}{d}^{2}e{g}^{4}\sqrt{cdx+ae}+210\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}{c}^{3}{d}^{3}f{g}^{3}\sqrt{cdx+ae}+210\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) xa{c}^{2}{d}^{2}ef{g}^{3}\sqrt{cdx+ae}+105\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{3}{d}^{3}{f}^{2}{g}^{2}\sqrt{cdx+ae}-105\,\sqrt{ \left ( aeg-cdf \right ) g}{x}^{3}{c}^{3}{d}^{3}{g}^{3}+105\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) a{c}^{2}{d}^{2}e{f}^{2}{g}^{2}\sqrt{cdx+ae}-140\,\sqrt{ \left ( aeg-cdf \right ) g}{x}^{2}a{c}^{2}{d}^{2}e{g}^{3}-175\,\sqrt{ \left ( aeg-cdf \right ) g}{x}^{2}{c}^{3}{d}^{3}f{g}^{2}-21\,\sqrt{ \left ( aeg-cdf \right ) g}x{a}^{2}cd{e}^{2}{g}^{3}-238\,\sqrt{ \left ( aeg-cdf \right ) g}xa{c}^{2}{d}^{2}ef{g}^{2}-56\,\sqrt{ \left ( aeg-cdf \right ) g}x{c}^{3}{d}^{3}{f}^{2}g+6\,\sqrt{ \left ( aeg-cdf \right ) g}{a}^{3}{e}^{3}{g}^{3}-39\,\sqrt{ \left ( aeg-cdf \right ) g}{a}^{2}cd{e}^{2}f{g}^{2}-80\,\sqrt{ \left ( aeg-cdf \right ) g}a{c}^{2}{d}^{2}e{f}^{2}g+8\,\sqrt{ \left ( aeg-cdf \right ) g}{c}^{3}{d}^{3}{f}^{3} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-1/12*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(105*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^4*(c*d*x+a*e)^(1/2)+105*arctanh(g*(c*d*x+a
*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*x^2*a*c^2*d^2*e*g^4*(c*d*x+a*e)^(1/2)+210*arc
tanh(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^3*(c*d*x+a*e)^
(1/2)+210*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*x*a*c^2*d^2*e*f*g
^3*(c*d*x+a*e)^(1/2)+105*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^2*(c*d*x+a*e)^(1/2)-105*((a*e*g-c*d*f)*g)^(1/2)*x^3*c^3*d^3*g^3+10
5*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e*f^2*g^2*(c*d*
x+a*e)^(1/2)-140*((a*e*g-c*d*f)*g)^(1/2)*x^2*a*c^2*d^2*e*g^3-175*((a*e*g-c*d*f)*
g)^(1/2)*x^2*c^3*d^3*f*g^2-21*((a*e*g-c*d*f)*g)^(1/2)*x*a^2*c*d*e^2*g^3-238*((a*
e*g-c*d*f)*g)^(1/2)*x*a*c^2*d^2*e*f*g^2-56*((a*e*g-c*d*f)*g)^(1/2)*x*c^3*d^3*f^2
*g+6*((a*e*g-c*d*f)*g)^(1/2)*a^3*e^3*g^3-39*((a*e*g-c*d*f)*g)^(1/2)*a^2*c*d*e^2*
f*g^2-80*((a*e*g-c*d*f)*g)^(1/2)*a*c^2*d^2*e*f^2*g+8*((a*e*g-c*d*f)*g)^(1/2)*c^3
*d^3*f^3)/(e*x+d)^(1/2)/(c*d*x+a*e)^2/(a*e*g-c*d*f)^4/(g*x+f)^2/((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((e*x + d)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312331, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^3),x, algorithm="fricas")

[Out]

[1/24*(105*(c^4*d^4*e*g^3*x^5 + a^2*c^2*d^3*e^2*f^2*g + (2*c^4*d^4*e*f*g^2 + (c^
4*d^5 + 2*a*c^3*d^3*e^2)*g^3)*x^4 + (c^4*d^4*e*f^2*g + 2*(c^4*d^5 + 2*a*c^3*d^3*
e^2)*f*g^2 + (2*a*c^3*d^4*e + a^2*c^2*d^2*e^3)*g^3)*x^3 + (a^2*c^2*d^3*e^2*g^3 +
(c^4*d^5 + 2*a*c^3*d^3*e^2)*f^2*g + 2*(2*a*c^3*d^4*e + a^2*c^2*d^2*e^3)*f*g^2)*
x^2 + (2*a^2*c^2*d^3*e^2*f*g^2 + (2*a*c^3*d^4*e + a^2*c^2*d^2*e^3)*f^2*g)*x)*sqr
t(-g/(c*d*f - a*e*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)*(c*d*f - a*e*g)*sqrt(e*x + d)*sqrt(-g/(c*d*f - a*e
*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*(
105*c^3*d^3*g^3*x^3 - 8*c^3*d^3*f^3 + 80*a*c^2*d^2*e*f^2*g + 39*a^2*c*d*e^2*f*g^
2 - 6*a^3*e^3*g^3 + 35*(5*c^3*d^3*f*g^2 + 4*a*c^2*d^2*e*g^3)*x^2 + 7*(8*c^3*d^3*
f^2*g + 34*a*c^2*d^2*e*f*g^2 + 3*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(e*x + d))/(a^2*c^4*d^5*e^2*f^6 - 4*a^3*c^3*d^4*e^3*f^5*g +
6*a^4*c^2*d^3*e^4*f^4*g^2 - 4*a^5*c*d^2*e^5*f^3*g^3 + a^6*d*e^6*f^2*g^4 + (c^6*
d^6*e*f^4*g^2 - 4*a*c^5*d^5*e^2*f^3*g^3 + 6*a^2*c^4*d^4*e^3*f^2*g^4 - 4*a^3*c^3*
d^3*e^4*f*g^5 + a^4*c^2*d^2*e^5*g^6)*x^5 + (2*c^6*d^6*e*f^5*g + (c^6*d^7 - 6*a*c
^5*d^5*e^2)*f^4*g^2 - 4*(a*c^5*d^6*e - a^2*c^4*d^4*e^3)*f^3*g^3 + 2*(3*a^2*c^4*d
^5*e^2 + 2*a^3*c^3*d^3*e^4)*f^2*g^4 - 2*(2*a^3*c^3*d^4*e^3 + 3*a^4*c^2*d^2*e^5)*
f*g^5 + (a^4*c^2*d^3*e^4 + 2*a^5*c*d*e^6)*g^6)*x^4 + (c^6*d^6*e*f^6 + 2*c^6*d^7*
f^5*g - 6*a^4*c^2*d^3*e^4*f*g^5 - 3*(2*a*c^5*d^6*e + 3*a^2*c^4*d^4*e^3)*f^4*g^2
+ 4*(a^2*c^4*d^5*e^2 + 4*a^3*c^3*d^3*e^4)*f^3*g^3 + (4*a^3*c^3*d^4*e^3 - 9*a^4*c
^2*d^2*e^5)*f^2*g^4 + (2*a^5*c*d^2*e^5 + a^6*e^7)*g^6)*x^3 - (6*a^2*c^4*d^4*e^3*
f^5*g - 2*a^6*e^7*f*g^5 - a^6*d*e^6*g^6 - (c^6*d^7 + 2*a*c^5*d^5*e^2)*f^6 + (9*a
^2*c^4*d^5*e^2 - 4*a^3*c^3*d^3*e^4)*f^4*g^2 - 4*(4*a^3*c^3*d^4*e^3 + a^4*c^2*d^2
*e^5)*f^3*g^3 + 3*(3*a^4*c^2*d^3*e^4 + 2*a^5*c*d*e^6)*f^2*g^4)*x^2 + (2*a^6*d*e^
6*f*g^5 + (2*a*c^5*d^6*e + a^2*c^4*d^4*e^3)*f^6 - 2*(3*a^2*c^4*d^5*e^2 + 2*a^3*c
^3*d^3*e^4)*f^5*g + 2*(2*a^3*c^3*d^4*e^3 + 3*a^4*c^2*d^2*e^5)*f^4*g^2 + 4*(a^4*c
^2*d^3*e^4 - a^5*c*d*e^6)*f^3*g^3 - (6*a^5*c*d^2*e^5 - a^6*e^7)*f^2*g^4)*x), -1/
12*(105*(c^4*d^4*e*g^3*x^5 + a^2*c^2*d^3*e^2*f^2*g + (2*c^4*d^4*e*f*g^2 + (c^4*d
^5 + 2*a*c^3*d^3*e^2)*g^3)*x^4 + (c^4*d^4*e*f^2*g + 2*(c^4*d^5 + 2*a*c^3*d^3*e^2
)*f*g^2 + (2*a*c^3*d^4*e + a^2*c^2*d^2*e^3)*g^3)*x^3 + (a^2*c^2*d^3*e^2*g^3 + (c
^4*d^5 + 2*a*c^3*d^3*e^2)*f^2*g + 2*(2*a*c^3*d^4*e + a^2*c^2*d^2*e^3)*f*g^2)*x^2
+ (2*a^2*c^2*d^3*e^2*f*g^2 + (2*a*c^3*d^4*e + a^2*c^2*d^2*e^3)*f^2*g)*x)*sqrt(g
/(c*d*f - a*e*g))*arctan(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)
*x)*sqrt(g/(c*d*f - a*e*g)))) - (105*c^3*d^3*g^3*x^3 - 8*c^3*d^3*f^3 + 80*a*c^2*
d^2*e*f^2*g + 39*a^2*c*d*e^2*f*g^2 - 6*a^3*e^3*g^3 + 35*(5*c^3*d^3*f*g^2 + 4*a*c
^2*d^2*e*g^3)*x^2 + 7*(8*c^3*d^3*f^2*g + 34*a*c^2*d^2*e*f*g^2 + 3*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(e*x + d))/(a^2*c^4*d^5*e^
2*f^6 - 4*a^3*c^3*d^4*e^3*f^5*g + 6*a^4*c^2*d^3*e^4*f^4*g^2 - 4*a^5*c*d^2*e^5*f^
3*g^3 + a^6*d*e^6*f^2*g^4 + (c^6*d^6*e*f^4*g^2 - 4*a*c^5*d^5*e^2*f^3*g^3 + 6*a^2
*c^4*d^4*e^3*f^2*g^4 - 4*a^3*c^3*d^3*e^4*f*g^5 + a^4*c^2*d^2*e^5*g^6)*x^5 + (2*c
^6*d^6*e*f^5*g + (c^6*d^7 - 6*a*c^5*d^5*e^2)*f^4*g^2 - 4*(a*c^5*d^6*e - a^2*c^4*
d^4*e^3)*f^3*g^3 + 2*(3*a^2*c^4*d^5*e^2 + 2*a^3*c^3*d^3*e^4)*f^2*g^4 - 2*(2*a^3*
c^3*d^4*e^3 + 3*a^4*c^2*d^2*e^5)*f*g^5 + (a^4*c^2*d^3*e^4 + 2*a^5*c*d*e^6)*g^6)*
x^4 + (c^6*d^6*e*f^6 + 2*c^6*d^7*f^5*g - 6*a^4*c^2*d^3*e^4*f*g^5 - 3*(2*a*c^5*d^
6*e + 3*a^2*c^4*d^4*e^3)*f^4*g^2 + 4*(a^2*c^4*d^5*e^2 + 4*a^3*c^3*d^3*e^4)*f^3*g
^3 + (4*a^3*c^3*d^4*e^3 - 9*a^4*c^2*d^2*e^5)*f^2*g^4 + (2*a^5*c*d^2*e^5 + a^6*e^
7)*g^6)*x^3 - (6*a^2*c^4*d^4*e^3*f^5*g - 2*a^6*e^7*f*g^5 - a^6*d*e^6*g^6 - (c^6*
d^7 + 2*a*c^5*d^5*e^2)*f^6 + (9*a^2*c^4*d^5*e^2 - 4*a^3*c^3*d^3*e^4)*f^4*g^2 - 4
*(4*a^3*c^3*d^4*e^3 + a^4*c^2*d^2*e^5)*f^3*g^3 + 3*(3*a^4*c^2*d^3*e^4 + 2*a^5*c*
d*e^6)*f^2*g^4)*x^2 + (2*a^6*d*e^6*f*g^5 + (2*a*c^5*d^6*e + a^2*c^4*d^4*e^3)*f^6
- 2*(3*a^2*c^4*d^5*e^2 + 2*a^3*c^3*d^3*e^4)*f^5*g + 2*(2*a^3*c^3*d^4*e^3 + 3*a^
4*c^2*d^2*e^5)*f^4*g^2 + 4*(a^4*c^2*d^3*e^4 - a^5*c*d*e^6)*f^3*g^3 - (6*a^5*c*d^
2*e^5 - a^6*e^7)*f^2*g^4)*x)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 13.2964, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^3),x, algorithm="giac")

[Out]

sage0*x