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3.710 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx\)

Optimal. Leaf size=393 \[ \frac{3 c^5 d^5 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{128 g^{7/2} (c d f-a e g)^{5/2}}+\frac{3 c^4 d^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 g^3 \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g^3 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 g^3 \sqrt{d+e x} (f+g x)^3}-\frac{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5} \]

[Out]

-(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(16*g^3*Sqrt[d + e*x]*(f
+ g*x)^3) + (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)^2) + (3*c^4*d^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
 + c*d*e*x^2])/(128*g^3*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)) - (c*d*(a*d*e
 + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*g^2*(d + e*x)^(3/2)*(f + g*x)^4) - (
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*g*(d + e*x)^(5/2)*(f + g*x)^5) +
 (3*c^5*d^5*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])])/(128*g^(7/2)*(c*d*f - a*e*g)^(5/2))

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Rubi [A]  time = 1.83707, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{3 c^5 d^5 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{128 g^{7/2} (c d f-a e g)^{5/2}}+\frac{3 c^4 d^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 g^3 \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g^3 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 g^3 \sqrt{d+e x} (f+g x)^3}-\frac{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5} \]

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)^6),x]

[Out]

-(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(16*g^3*Sqrt[d + e*x]*(f
+ g*x)^3) + (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)^2) + (3*c^4*d^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
 + c*d*e*x^2])/(128*g^3*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)) - (c*d*(a*d*e
 + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*g^2*(d + e*x)^(3/2)*(f + g*x)^4) - (
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*g*(d + e*x)^(5/2)*(f + g*x)^5) +
 (3*c^5*d^5*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])])/(128*g^(7/2)*(c*d*f - a*e*g)^(5/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**6,x)

[Out]

Timed out

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Mathematica [A]  time = 1.42825, size = 231, normalized size = 0.59 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{\frac{15 c^4 d^4 (f+g x)^4}{(c d f-a e g)^2}+\frac{10 c^3 d^3 (f+g x)^3}{c d f-a e g}+336 c d (f+g x) (c d f-a e g)-128 (c d f-a e g)^2-248 c^2 d^2 (f+g x)^2}{5 g^3 (f+g x)^5 (a e+c d x)^2}-\frac{3 c^5 d^5 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{g^{7/2} (a e+c d x)^{5/2} (a e g-c d f)^{5/2}}\right )}{128 (d+e x)^{5/2}} \]

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)^6),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((-128*(c*d*f - a*e*g)^2 + 336*c*d*(c*d*f - a*e
*g)*(f + g*x) - 248*c^2*d^2*(f + g*x)^2 + (10*c^3*d^3*(f + g*x)^3)/(c*d*f - a*e*
g) + (15*c^4*d^4*(f + g*x)^4)/(c*d*f - a*e*g)^2)/(5*g^3*(a*e + c*d*x)^2*(f + g*x
)^5) - (3*c^5*d^5*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)^(5/2)*(a*e + c*d*x)^(5/2))))/(128*(d + e*x)^(5/2))

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Maple [B]  time = 0.049, size = 924, normalized size = 2.4 \[ \text{result too large to display} \]

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)^6,x)

[Out]

-1/640*(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^5*c^5*d^5*g^5+75*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c
*d*f)*g)^(1/2))*x^4*c^5*d^5*f*g^4+150*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)
*g)^(1/2))*x^3*c^5*d^5*f^2*g^3+150*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)
^(1/2))*x^2*c^5*d^5*f^3*g^2-15*x^4*c^4*d^4*g^4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*
g)^(1/2)+75*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*x*c^5*d^5*f^4*g
+10*x^3*a*c^3*d^3*e*g^4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-70*x^3*c^4*d^4
*f*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^5*d^5*f^5+248*x^2*a^2*c^2*d^2*e^2*g^4*(c*d*x+a*e)^(1/
2)*((a*e*g-c*d*f)*g)^(1/2)-466*x^2*a*c^3*d^3*e*f*g^3*(c*d*x+a*e)^(1/2)*((a*e*g-c
*d*f)*g)^(1/2)+128*x^2*c^4*d^4*f^2*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)
+336*x*a^3*c*d*e^3*g^4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-512*x*a^2*c^2*d
^2*e^2*f*g^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+46*x*a*c^3*d^3*e*f^2*g^2*
(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+70*x*c^4*d^4*f^3*g*(c*d*x+a*e)^(1/2)*(
(a*e*g-c*d*f)*g)^(1/2)+128*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^4*e^4*g^4
-176*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^3*c*d*e^3*f*g^3+8*((a*e*g-c*d*f
)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c^2*d^2*e^2*f^2*g^2+10*((a*e*g-c*d*f)*g)^(1/2)*
(c*d*x+a*e)^(1/2)*a*c^3*d^3*e*f^3*g+15*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)
*c^4*d^4*f^4)/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)^5/g^3/(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)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.31882, 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)^6),x, algorithm="fricas")

[Out]

[1/1280*(2*(15*c^4*d^4*g^4*x^4 - 15*c^4*d^4*f^4 - 10*a*c^3*d^3*e*f^3*g - 8*a^2*c
^2*d^2*e^2*f^2*g^2 + 176*a^3*c*d*e^3*f*g^3 - 128*a^4*e^4*g^4 + 10*(7*c^4*d^4*f*g
^3 - a*c^3*d^3*e*g^4)*x^3 - 2*(64*c^4*d^4*f^2*g^2 - 233*a*c^3*d^3*e*f*g^3 + 124*
a^2*c^2*d^2*e^2*g^4)*x^2 - 2*(35*c^4*d^4*f^3*g + 23*a*c^3*d^3*e*f^2*g^2 - 256*a^
2*c^2*d^2*e^2*f*g^3 + 168*a^3*c*d*e^3*g^4)*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^5*d^5*e*g^5*x^6 + c^5*d
^6*f^5 + (5*c^5*d^5*e*f*g^4 + c^5*d^6*g^5)*x^5 + 5*(2*c^5*d^5*e*f^2*g^3 + c^5*d^
6*f*g^4)*x^4 + 10*(c^5*d^5*e*f^3*g^2 + c^5*d^6*f^2*g^3)*x^3 + 5*(c^5*d^5*e*f^4*g
 + 2*c^5*d^6*f^3*g^2)*x^2 + (c^5*d^5*e*f^5 + 5*c^5*d^6*f^4*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^7*g^3 - 2*a*c*d^2*e*f^6
*g^4 + a^2*d*e^2*f^5*g^5 + (c^2*d^2*e*f^2*g^8 - 2*a*c*d*e^2*f*g^9 + a^2*e^3*g^10
)*x^6 + (5*c^2*d^2*e*f^3*g^7 + a^2*d*e^2*g^10 + (c^2*d^3 - 10*a*c*d*e^2)*f^2*g^8
 - (2*a*c*d^2*e - 5*a^2*e^3)*f*g^9)*x^5 + 5*(2*c^2*d^2*e*f^4*g^6 + a^2*d*e^2*f*g
^9 + (c^2*d^3 - 4*a*c*d*e^2)*f^3*g^7 - 2*(a*c*d^2*e - a^2*e^3)*f^2*g^8)*x^4 + 10
*(c^2*d^2*e*f^5*g^5 + a^2*d*e^2*f^2*g^8 + (c^2*d^3 - 2*a*c*d*e^2)*f^4*g^6 - (2*a
*c*d^2*e - a^2*e^3)*f^3*g^7)*x^3 + 5*(c^2*d^2*e*f^6*g^4 + 2*a^2*d*e^2*f^3*g^7 +
2*(c^2*d^3 - a*c*d*e^2)*f^5*g^5 - (4*a*c*d^2*e - a^2*e^3)*f^4*g^6)*x^2 + (c^2*d^
2*e*f^7*g^3 + 5*a^2*d*e^2*f^4*g^6 + (5*c^2*d^3 - 2*a*c*d*e^2)*f^6*g^4 - (10*a*c*
d^2*e - a^2*e^3)*f^5*g^5)*x)*sqrt(-c*d*f*g + a*e*g^2)), 1/640*((15*c^4*d^4*g^4*x
^4 - 15*c^4*d^4*f^4 - 10*a*c^3*d^3*e*f^3*g - 8*a^2*c^2*d^2*e^2*f^2*g^2 + 176*a^3
*c*d*e^3*f*g^3 - 128*a^4*e^4*g^4 + 10*(7*c^4*d^4*f*g^3 - a*c^3*d^3*e*g^4)*x^3 -
2*(64*c^4*d^4*f^2*g^2 - 233*a*c^3*d^3*e*f*g^3 + 124*a^2*c^2*d^2*e^2*g^4)*x^2 - 2
*(35*c^4*d^4*f^3*g + 23*a*c^3*d^3*e*f^2*g^2 - 256*a^2*c^2*d^2*e^2*f*g^3 + 168*a^
3*c*d*e^3*g^4)*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^5*d^5*e*g^5*x^6 + c^5*d^6*f^5 + (5*c^5*d^5*e*f*g^4 +
 c^5*d^6*g^5)*x^5 + 5*(2*c^5*d^5*e*f^2*g^3 + c^5*d^6*f*g^4)*x^4 + 10*(c^5*d^5*e*
f^3*g^2 + c^5*d^6*f^2*g^3)*x^3 + 5*(c^5*d^5*e*f^4*g + 2*c^5*d^6*f^3*g^2)*x^2 + (
c^5*d^5*e*f^5 + 5*c^5*d^6*f^4*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^2*d^3*f^7*g^3 - 2*a*c*d^2*e*f^6*g^4 + a^2*d*e^2*f^5*g^5 + (c^2*
d^2*e*f^2*g^8 - 2*a*c*d*e^2*f*g^9 + a^2*e^3*g^10)*x^6 + (5*c^2*d^2*e*f^3*g^7 + a
^2*d*e^2*g^10 + (c^2*d^3 - 10*a*c*d*e^2)*f^2*g^8 - (2*a*c*d^2*e - 5*a^2*e^3)*f*g
^9)*x^5 + 5*(2*c^2*d^2*e*f^4*g^6 + a^2*d*e^2*f*g^9 + (c^2*d^3 - 4*a*c*d*e^2)*f^3
*g^7 - 2*(a*c*d^2*e - a^2*e^3)*f^2*g^8)*x^4 + 10*(c^2*d^2*e*f^5*g^5 + a^2*d*e^2*
f^2*g^8 + (c^2*d^3 - 2*a*c*d*e^2)*f^4*g^6 - (2*a*c*d^2*e - a^2*e^3)*f^3*g^7)*x^3
 + 5*(c^2*d^2*e*f^6*g^4 + 2*a^2*d*e^2*f^3*g^7 + 2*(c^2*d^3 - a*c*d*e^2)*f^5*g^5
- (4*a*c*d^2*e - a^2*e^3)*f^4*g^6)*x^2 + (c^2*d^2*e*f^7*g^3 + 5*a^2*d*e^2*f^4*g^
6 + (5*c^2*d^3 - 2*a*c*d*e^2)*f^6*g^4 - (10*a*c*d^2*e - a^2*e^3)*f^5*g^5)*x)*sqr
t(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)**6,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)^6),x, algorithm="giac")

[Out]

Timed out

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