3.467 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx\)

Optimal. Leaf size=386 \[ \frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6}-\frac{\left (\frac{5 c}{a e}-\frac{7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 x^5} \]

[Out]

-((c*d^2 - a*e^2)^3*(5*c*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(512*a^3*d^4*e^3*x^2) + ((c*d^2 - a*e^2)*(5*c
*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(3/2))/(192*a^2*d^3*e^2*x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/
2)/(6*d*x^6) - (((5*c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(5/2))/(60*x^5) + ((c*d^2 - a*e^2)^5*(5*c*d^2 + 7*a*e^2)*ArcTanh[(2*a*d*e + (c
*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2])])/(1024*a^(7/2)*d^(9/2)*e^(7/2))

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Rubi [A]  time = 1.31555, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6}-\frac{\left (\frac{5 c}{a e}-\frac{7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)),x]

[Out]

-((c*d^2 - a*e^2)^3*(5*c*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(512*a^3*d^4*e^3*x^2) + ((c*d^2 - a*e^2)*(5*c
*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(3/2))/(192*a^2*d^3*e^2*x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/
2)/(6*d*x^6) - (((5*c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(5/2))/(60*x^5) + ((c*d^2 - a*e^2)^5*(5*c*d^2 + 7*a*e^2)*ArcTanh[(2*a*d*e + (c
*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2])])/(1024*a^(7/2)*d^(9/2)*e^(7/2))

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Rubi in Sympy [A]  time = 141.543, size = 362, normalized size = 0.94 \[ - \frac{\left (2 a d e + x \left (a e^{2} + c d^{2}\right )\right ) \left (\frac{7 e^{2}}{192 d^{3}} - \frac{c}{96 a d} - \frac{5 c^{2} d}{192 a^{2} e^{2}}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{x^{4}} + \frac{\left (\frac{7 e}{60 d^{2}} - \frac{c}{12 a e}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{x^{5}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{6 d x^{6}} + \frac{\left (a e^{2} - c d^{2}\right )^{3} \left (7 a e^{2} + 5 c d^{2}\right ) \left (2 a d e + x \left (a e^{2} + c d^{2}\right )\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{512 a^{3} d^{4} e^{3} x^{2}} - \frac{\left (a e^{2} - c d^{2}\right )^{5} \left (7 a e^{2} + 5 c d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{1024 a^{\frac{7}{2}} d^{\frac{9}{2}} e^{\frac{7}{2}}} \]

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)/x**7/(e*x+d),x)

[Out]

-(2*a*d*e + x*(a*e**2 + c*d**2))*(7*e**2/(192*d**3) - c/(96*a*d) - 5*c**2*d/(192
*a**2*e**2))*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/x**4 + (7*e/(60*d
**2) - c/(12*a*e))*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/x**5 - (a*d
*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(6*d*x**6) + (a*e**2 - c*d**2)**3*
(7*a*e**2 + 5*c*d**2)*(2*a*d*e + x*(a*e**2 + c*d**2))*sqrt(a*d*e + c*d*e*x**2 +
x*(a*e**2 + c*d**2))/(512*a**3*d**4*e**3*x**2) - (a*e**2 - c*d**2)**5*(7*a*e**2
+ 5*c*d**2)*atanh((2*a*d*e + x*(a*e**2 + c*d**2))/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqr
t(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(1024*a**(7/2)*d**(9/2)*e**(7/2))

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Mathematica [A]  time = 0.897661, size = 450, normalized size = 1.17 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (-2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^5 e^5 \left (1280 d^5+1664 d^4 e x+48 d^3 e^2 x^2-56 d^2 e^3 x^3+70 d e^4 x^4-105 e^5 x^5\right )+a^4 c d^2 e^4 x \left (3200 d^4+4448 d^3 e x+216 d^2 e^2 x^2-272 d e^3 x^3+415 e^4 x^4\right )+6 a^3 c^2 d^4 e^3 x^2 \left (360 d^3+564 d^2 e x+58 d e^2 x^2-91 e^3 x^3\right )+10 a^2 c^3 d^6 e^2 x^3 \left (4 d^2+16 d e x+15 e^2 x^2\right )-5 a c^4 d^8 e x^4 (10 d+49 e x)+75 c^5 d^{10} x^5\right )-15 x^6 \log (x) \left (c d^2-a e^2\right )^5 \left (7 a e^2+5 c d^2\right )+15 x^6 \left (c d^2-a e^2\right )^5 \left (7 a e^2+5 c d^2\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )\right )}{15360 a^{7/2} d^{9/2} e^{7/2} x^6 (d+e x)^{3/2} (a e+c d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*(-2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*S
qrt[d + e*x]*(75*c^5*d^10*x^5 - 5*a*c^4*d^8*e*x^4*(10*d + 49*e*x) + 10*a^2*c^3*d
^6*e^2*x^3*(4*d^2 + 16*d*e*x + 15*e^2*x^2) + 6*a^3*c^2*d^4*e^3*x^2*(360*d^3 + 56
4*d^2*e*x + 58*d*e^2*x^2 - 91*e^3*x^3) + a^4*c*d^2*e^4*x*(3200*d^4 + 4448*d^3*e*
x + 216*d^2*e^2*x^2 - 272*d*e^3*x^3 + 415*e^4*x^4) + a^5*e^5*(1280*d^5 + 1664*d^
4*e*x + 48*d^3*e^2*x^2 - 56*d^2*e^3*x^3 + 70*d*e^4*x^4 - 105*e^5*x^5)) - 15*(c*d
^2 - a*e^2)^5*(5*c*d^2 + 7*a*e^2)*x^6*Log[x] + 15*(c*d^2 - a*e^2)^5*(5*c*d^2 + 7
*a*e^2)*x^6*Log[c*d^2*x + 2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e
*x] + a*e*(2*d + e*x)]))/(15360*a^(7/2)*d^(9/2)*e^(7/2)*x^6*(a*e + c*d*x)^(3/2)*
(d + e*x)^(3/2))

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Maple [B]  time = 0.081, size = 4735, normalized size = 12.3 \[ \text{output too large to display} \]

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)/x^7/(e*x+d),x)

[Out]

1/16/d^4*e^5*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-3/128/d*e^4*c^2*(c*
d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+7/1536/d^6*a*e^7*(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(3/2)+7/512/d^5*a^2*e^8*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-35
/384/d^4*e^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c+17/60/d^3/a/x^5*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-59/320/d/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5
/2)*c^3+1/512*d^3/a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^4+25/512/d*e^4*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2-1/5/d^7*e^6*(c*d*e*(x+d/e)^2+(a*e^2-c
*d^2)*(x+d/e))^(5/2)-101/512/d^7*e^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+3/6
4/d^3*e^6*a*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-15/128/d^5*e^10*a^3*
ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d
^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-3/64/d^2*e^5*c^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2
)*(x+d/e))^(1/2)*x+3/256*d*e^4*c^3*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e
)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+3/64/d^7*e^
10*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8/d^7*e^8*a*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)*x+15/128/d^5*e^10*a^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c
*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-3/256*d*e^4*c
^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(1/2))/(c*d*e)^(1/2)+1/16/d^8*e^9*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3
/2)+9/64/d^6*e^9*a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-3/128/d^9*e^12*a^
4/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-5/1536*d^6/a^5/e^5*(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(3/2)*c^6-1/512*d^5/a^6/e^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(5/2)*c^6+1017/2560/d^7/a*e^4/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-7/1024/
d^4*a^3*e^9/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)-185/1536/d^5*e^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(3/2)*x*c+15/512/d^2*e^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^2+5/25
6*d^2*e^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^3+89/320/d^3/a^2/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(7/2)*c+381/1280/d^3/a^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^2+
35/768*d/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*c^4-397/960/d^5/a*e^4*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-35/1536/d^2/a*e^3*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(3/2)*c^2-1/64*d/a*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^3-5/
64/d^3*a*e^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c-2681/7680/d^3/a^2*e^2*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2-221/7680*d/a^4/e^2*(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(5/2)*c^4+1/64*d^5/a^3/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
*c^5-57/160/d^4/a*e/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+377/960/d^5/a*e^
2/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-5/256/a*e^3*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*x*c^3-81/1280/a^4/e*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2
)*x-45/1024*a*e^5/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^2-11/480/a^4/e^3/x^2*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(7/2)*c^3-1/32/a^3/e^3/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7
/2)*c^2-1/6/d^2/a/e/x^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+1/120*d^3/a^5/e^
4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^5-5/512*d^7/a^4/e^4*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*c^6-1543/3840/d^6/a*e^3/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(7/2)+49/1536*d^2/a^3/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^4+7/384*
d^4/a^4/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^5+1/8/d^5*e^6*c*(c*d*e*(x+
d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x-1/16/d^8*e^9*a^2/c*(c*d*e*(x+d/e)^2+(a*e^2
-c*d^2)*(x+d/e))^(3/2)-9/64/d^6*e^9*a^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^
(1/2)*x+3/128/d^9*e^12*a^4/c^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-5/1
536*d^5/a^5/e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*c^6+3211/7680/d^5/a^2*
e^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c+7/256*d^2/a^2*e*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*x*c^4-5/512*d^6/a^4/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(1/2)*x*c^6-25/768/d/a^2*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*c^3+3/6
4/d^8*e^11*a^3/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+9/64/d^4*e^7*a*
c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-3/256/d^9*e^14*a^5/c^2*ln((1/2
*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+
d/e))^(1/2))/(c*d*e)^(1/2)+15/128/d^3*e^8*a^2*c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*
c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2
)-15/256/d*e^6*a*c^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e
*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+15/256/d^7*e^12*a^4/c*ln(
(1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)
*(x+d/e))^(1/2))/(c*d*e)^(1/2)+3/256/d^9*e^14*a^5/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*
d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-15/1
28/d^3*e^8*a^2*c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-3/64/d^8*e^11*a^3/c*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*x+15/256/d*e^6*a*c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)
^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-15/256/d^7*e^12*a^
4/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2))/(c*d*e)^(1/2)+81/1280/d/a^4/e^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(7/2)*c^3-89/7680*d/a^5/e^4/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^4+1/512*
d^3/a^6/e^6/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^5-11/30/d^4/a^2*e/x^2*(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c-65/512/d^4*a*e^7*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)*x*c+29/320/d/a^3/e^2/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2
)*c^2+1/192*d/a^4/e^4/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^3+89/7680*d^
2/a^5/e^3*c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x-43/240/d^2/a^2/e/x^4*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c+3/512*d^4/a^3/e*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*x*c^5-113/640/d^2/a^3/e/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/
2)*c^2-381/1280/d^2/a^3*e*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x-65/1536/
d^3/a*e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*c^2+15/512/d^2*a^2*e^7/(a*d*
e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2))/x)*c+15/1024*d^4/a*e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(
a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^4-1/512*d^4/a^6/e^5*c
^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+1/768*d^2/a^5/e^5/x^2*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(7/2)*c^4-9/512*d^6/a^2/e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+
c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^5+5/1024*
d^8/a^3/e^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^6+1/12/d/a^2/e^2/x^5*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(7/2)*c-1017/2560/d^6/a*e^5*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)
*x-3211/7680/d^4/a^2*e^3*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+43/1536*d
^3/a^4/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*c^5-1/8/d^7*e^8*a*(c*d*e*(x
+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x-3/64/d^7*e^10*a^3/c*(c*d*e*(x+d/e)^2+(a*e
^2-c*d^2)*(x+d/e))^(1/2)

_______________________________________________________________________________________

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)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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)*x^7),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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)/x**7/(e*x+d),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 77.5188, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)*x^7),x, algorithm="giac")

[Out]

Done