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

Optimal. Leaf size=323 \[ \frac{5 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{64 g^{7/2} (c d f-a e g)^{3/2}}+\frac{5 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g^3 \sqrt{d+e x} (f+g x) (c d f-a e g)}-\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 g^3 \sqrt{d+e x} (f+g x)^2}-\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4} \]

[Out]

(-5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*g^3*Sqrt[d + e*x]*(
f + g*x)^2) + (5*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*g^3*(c
*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)) - (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(3/2))/(24*g^2*(d + e*x)^(3/2)*(f + g*x)^3) - (a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(5/2)/(4*g*(d + e*x)^(5/2)*(f + g*x)^4) + (5*c^4*d^4*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])])/(64*g^(7/2)*(c*d*f - a*e*g)^(3/2))

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Rubi [A]  time = 1.62911, antiderivative size = 323, 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 \[ \frac{5 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{64 g^{7/2} (c d f-a e g)^{3/2}}+\frac{5 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g^3 \sqrt{d+e x} (f+g x) (c d f-a e g)}-\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 g^3 \sqrt{d+e x} (f+g x)^2}-\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4} \]

Antiderivative was successfully verified.

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

[Out]

(-5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*g^3*Sqrt[d + e*x]*(
f + g*x)^2) + (5*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*g^3*(c
*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)) - (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(3/2))/(24*g^2*(d + e*x)^(3/2)*(f + g*x)^3) - (a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(5/2)/(4*g*(d + e*x)^(5/2)*(f + g*x)^4) + (5*c^4*d^4*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])])/(64*g^(7/2)*(c*d*f - a*e*g)^(3/2))

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Rubi in Sympy [A]  time = 144.115, size = 311, normalized size = 0.96 \[ \frac{5 c^{4} d^{4} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{64 g^{\frac{7}{2}} \left (a e g - c d f\right )^{\frac{3}{2}}} - \frac{5 c^{3} d^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{64 g^{3} \sqrt{d + e x} \left (f + g x\right ) \left (a e g - c d f\right )} - \frac{5 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{32 g^{3} \sqrt{d + e x} \left (f + g x\right )^{2}} - \frac{5 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{24 g^{2} \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{3}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{4 g \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{4}} \]

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

[Out]

5*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**(7/2)*(a*e*g - c*d*f)**(3/2)) - 5*c**3*d**3
*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(64*g**3*sqrt(d + e*x)*(f + g*x)
*(a*e*g - c*d*f)) - 5*c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(
32*g**3*sqrt(d + e*x)*(f + g*x)**2) - 5*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*
d**2))**(3/2)/(24*g**2*(d + e*x)**(3/2)*(f + g*x)**3) - (a*d*e + c*d*e*x**2 + x*
(a*e**2 + c*d**2))**(5/2)/(4*g*(d + e*x)**(5/2)*(f + g*x)**4)

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Mathematica [A]  time = 1.04262, size = 204, normalized size = 0.63 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{5 c^4 d^4 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{g^{7/2} (a e+c d x)^{5/2} (a e g-c d f)^{3/2}}+\frac{\frac{15 c^3 d^3 (f+g x)^3}{c d f-a e g}+136 c d (f+g x) (c d f-a e g)-48 (c d f-a e g)^2-118 c^2 d^2 (f+g x)^2}{3 g^3 (f+g x)^4 (a e+c d x)^2}\right )}{64 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((-48*(c*d*f - a*e*g)^2 + 136*c*d*(c*d*f - a*e*
g)*(f + g*x) - 118*c^2*d^2*(f + g*x)^2 + (15*c^3*d^3*(f + g*x)^3)/(c*d*f - a*e*g
))/(3*g^3*(a*e + c*d*x)^2*(f + g*x)^4) + (5*c^4*d^4*ArcTanh[(Sqrt[g]*Sqrt[a*e +
c*d*x])/Sqrt[-(c*d*f) + a*e*g]])/(g^(7/2)*(-(c*d*f) + a*e*g)^(3/2)*(a*e + c*d*x)
^(5/2))))/(64*(d + e*x)^(5/2))

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Maple [B]  time = 0.041, size = 665, normalized size = 2.1 \[{\frac{1}{192\,{g}^{3} \left ( aeg-cdf \right ) \left ( gx+f \right ) ^{4}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{4}{c}^{4}{d}^{4}{g}^{4}+60\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{3}{c}^{4}{d}^{4}f{g}^{3}+90\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}{c}^{4}{d}^{4}{f}^{2}{g}^{2}+60\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{4}{d}^{4}{f}^{3}g-15\,{x}^{3}{c}^{3}{d}^{3}{g}^{3}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{4}{d}^{4}{f}^{4}-118\,{x}^{2}a{c}^{2}{d}^{2}e{g}^{3}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+73\,{x}^{2}{c}^{3}{d}^{3}f{g}^{2}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-136\,x{a}^{2}cd{e}^{2}{g}^{3}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+36\,xa{c}^{2}{d}^{2}ef{g}^{2}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+55\,x{c}^{3}{d}^{3}{f}^{2}g\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-48\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{3}{e}^{3}{g}^{3}+8\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{2}cd{e}^{2}f{g}^{2}+10\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}a{c}^{2}{d}^{2}e{f}^{2}g+15\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{c}^{3}{d}^{3}{f}^{3} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \]

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

[Out]

1/192*(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^4*c^4*d^4*g^4+60*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+90*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+60*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-15*x^3*c^3*d^3*g^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2
)+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^4*d^4*f^4-118*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)+73*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)-136*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)+36*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)+55*x*c^3*d^3*f^2*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-48*((a*e*g-
c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^3*e^3*g^3+8*((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+10*((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+15*((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)/(c*d*x+a*e)^(1/2)/g^3/(a*e*g-c*d*f)/(g*x+f)^4/((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)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304298, 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)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^5),x, algorithm="fricas")

[Out]

[1/384*(2*(15*c^3*d^3*g^3*x^3 - 15*c^3*d^3*f^3 - 10*a*c^2*d^2*e*f^2*g - 8*a^2*c*
d*e^2*f*g^2 + 48*a^3*e^3*g^3 - (73*c^3*d^3*f*g^2 - 118*a*c^2*d^2*e*g^3)*x^2 - (5
5*c^3*d^3*f^2*g + 36*a*c^2*d^2*e*f*g^2 - 136*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) - 15*(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*d^2*f^5*g^3 - a*d*e*f^4*g^4 + (c*d*e*f*g^7 - a*e^2*g^8)*
x^5 + (4*c*d*e*f^2*g^6 - a*d*e*g^8 + (c*d^2 - 4*a*e^2)*f*g^7)*x^4 + 2*(3*c*d*e*f
^3*g^5 - 2*a*d*e*f*g^7 + (2*c*d^2 - 3*a*e^2)*f^2*g^6)*x^3 + 2*(2*c*d*e*f^4*g^4 -
3*a*d*e*f^2*g^6 + (3*c*d^2 - 2*a*e^2)*f^3*g^5)*x^2 + (c*d*e*f^5*g^3 - 4*a*d*e*f
^3*g^5 + (4*c*d^2 - a*e^2)*f^4*g^4)*x)*sqrt(-c*d*f*g + a*e*g^2)), 1/192*((15*c^3
*d^3*g^3*x^3 - 15*c^3*d^3*f^3 - 10*a*c^2*d^2*e*f^2*g - 8*a^2*c*d*e^2*f*g^2 + 48*
a^3*e^3*g^3 - (73*c^3*d^3*f*g^2 - 118*a*c^2*d^2*e*g^3)*x^2 - (55*c^3*d^3*f^2*g +
36*a*c^2*d^2*e*f*g^2 - 136*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) - 15*(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)*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*d^2*f^5*g^3 - a*d*e*f^4*g^4 + (c*d*e*f*g^7 - a*e^2*g^8)*x^5 + (4*c*d*e*f^2*g
^6 - a*d*e*g^8 + (c*d^2 - 4*a*e^2)*f*g^7)*x^4 + 2*(3*c*d*e*f^3*g^5 - 2*a*d*e*f*g
^7 + (2*c*d^2 - 3*a*e^2)*f^2*g^6)*x^3 + 2*(2*c*d*e*f^4*g^4 - 3*a*d*e*f^2*g^6 + (
3*c*d^2 - 2*a*e^2)*f^3*g^5)*x^2 + (c*d*e*f^5*g^3 - 4*a*d*e*f^3*g^5 + (4*c*d^2 -
a*e^2)*f^4*g^4)*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)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**5,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)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^5),x, algorithm="giac")

[Out]

Timed out