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

Optimal. Leaf size=381 \[ \frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{9/2}}-\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^4}+\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^3}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}-\frac{1}{60} \left (\frac{5 a}{c d}+\frac{7 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \]

[Out]

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

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Rubi [A]  time = 0.852181, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132 \[ \frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{9/2}}-\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^4}+\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^3}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}-\frac{1}{60} \left (\frac{5 a}{c d}+\frac{7 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 86.2871, size = 360, normalized size = 0.94 \[ - \left (\frac{a}{12 c d} + \frac{7 d}{60 e^{2}}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}} + \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{6 c d e \left (d + e x\right )} - \frac{\left (a e^{2} - c d^{2}\right ) \left (5 a e^{2} + 7 c d^{2}\right ) \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{192 c^{2} d^{2} e^{3}} + \frac{\left (a e^{2} - c d^{2}\right )^{3} \left (5 a e^{2} + 7 c d^{2}\right ) \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{512 c^{3} d^{3} e^{4}} - \frac{\left (a e^{2} - c d^{2}\right )^{5} \left (5 a e^{2} + 7 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{1024 c^{\frac{7}{2}} d^{\frac{7}{2}} e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.921525, size = 386, normalized size = 1.01 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{2 \left (75 a^5 e^{10}-5 a^4 c d e^8 (49 d+10 e x)+10 a^3 c^2 d^2 e^6 \left (15 d^2+16 d e x+4 e^2 x^2\right )-6 a^2 c^3 d^3 e^4 \left (91 d^3-58 d^2 e x-564 d e^2 x^2-360 e^3 x^3\right )+a c^4 d^4 e^2 \left (415 d^4-272 d^3 e x+216 d^2 e^2 x^2+4448 d e^3 x^3+3200 e^4 x^4\right )+c^5 d^5 \left (-105 d^5+70 d^4 e x-56 d^3 e^2 x^2+48 d^2 e^3 x^3+1664 d e^4 x^4+1280 e^5 x^5\right )\right )}{15 c^3 d^3 e^4 (d+e x) (a e+c d x)}+\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^5 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{7/2} d^{7/2} e^{9/2} (d+e x)^{3/2} (a e+c d x)^{3/2}}\right )}{1024} \]

Antiderivative was successfully verified.

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((2*(75*a^5*e^10 - 5*a^4*c*d*e^8*(49*d + 10*e*x
) + 10*a^3*c^2*d^2*e^6*(15*d^2 + 16*d*e*x + 4*e^2*x^2) - 6*a^2*c^3*d^3*e^4*(91*d
^3 - 58*d^2*e*x - 564*d*e^2*x^2 - 360*e^3*x^3) + a*c^4*d^4*e^2*(415*d^4 - 272*d^
3*e*x + 216*d^2*e^2*x^2 + 4448*d*e^3*x^3 + 3200*e^4*x^4) + c^5*d^5*(-105*d^5 + 7
0*d^4*e*x - 56*d^3*e^2*x^2 + 48*d^2*e^3*x^3 + 1664*d*e^4*x^4 + 1280*e^5*x^5)))/(
15*c^3*d^3*e^4*(a*e + c*d*x)*(d + e*x)) + ((c*d^2 - a*e^2)^5*(7*c*d^2 + 5*a*e^2)
*Log[a*e^2 + 2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + c*d*(d
+ 2*e*x)])/(c^(7/2)*d^(7/2)*e^(9/2)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/1024

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Maple [B]  time = 0.017, size = 2411, normalized size = 6.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/128*d^7/e^4*c^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/16*e*a^2/c*(c
*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-5/192/e^3*c*d^4*(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(3/2)+5/512/e^4*c^2*d^7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+5/
192*e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+5/192/e*d^2*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(3/2)*a+5/48*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a+1/1
2/c/d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a-1/8*d*a*(c*d*e*(x+d/e)^2+(a*e^2-
c*d^2)*(x+d/e))^(3/2)*x+1/16*d^4/e^3*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(
3/2)-3/256/d*e^6*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)-15/256*d^7/e^2*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*e^4*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)+9/64*d^4/e*a*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+5/256*e^5/
c^2/d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4-5/64/e*c*d^4*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+15/512/e^2*c^2*d^7*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)*a-5/96*e
^2/c/d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2-5/1024*e^8/c^3/d^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)*a^6+15/512*e^6/c^2/d*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)*a^5-75/1024*e^4/c*d*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)*a^4+3/64*d^5/e^2*a*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^
(1/2)-15/128*d^3*e^2*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^6/e^3*c^2*(c*d
*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+3/256*d^9/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)-9/64*d^2*e*a^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-3
/64*d*e^2*a^3/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+1/8*d^3/e^2*c*(c*d
*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x+3/64*e^3*a^3/c*(c*d*e*(x+d/e)^2+(a*e
^2-c*d^2)*(x+d/e))^(1/2)*x-5/1024/e^4*c^3*d^9*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)-5/192*e^3/c^
2/d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+5/256*e^2/c*d*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*a^3-5/96/e^2*c*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/
2)*x+5/256/e^3*c^2*d^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+5/512*e^6/c^3/d
^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5-15/512*e^4/c^2/d*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*a^4-15/512/e^2*c*d^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)*a-5/64*e^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+15/128*e*d^2*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+25/256*e^2*d^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)*
a^3-75/1024*c*d^5*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)*a^2+3/128/d*e^4*a^4/c^2*(c*d*e*(x+d/e)^2
+(a*e^2-c*d^2)*(x+d/e))^(1/2)+15/128*d^5*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)
-1/5*d/e^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)+1/6/e*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(5/2)*x+5/256*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+1
/12/e^2*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)

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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)*x/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.355523, 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)*x/(e*x + d),x, algorithm="fricas")

[Out]

[1/30720*(4*(1280*c^5*d^5*e^5*x^5 - 105*c^5*d^10 + 415*a*c^4*d^8*e^2 - 546*a^2*c
^3*d^6*e^4 + 150*a^3*c^2*d^4*e^6 - 245*a^4*c*d^2*e^8 + 75*a^5*e^10 + 128*(13*c^5
*d^6*e^4 + 25*a*c^4*d^4*e^6)*x^4 + 16*(3*c^5*d^7*e^3 + 278*a*c^4*d^5*e^5 + 135*a
^2*c^3*d^3*e^7)*x^3 - 8*(7*c^5*d^8*e^2 - 27*a*c^4*d^6*e^4 - 423*a^2*c^3*d^4*e^6
- 5*a^3*c^2*d^2*e^8)*x^2 + 2*(35*c^5*d^9*e - 136*a*c^4*d^7*e^3 + 174*a^2*c^3*d^5
*e^5 + 80*a^3*c^2*d^3*e^7 - 25*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x)*sqrt(c*d*e) - 15*(7*c^6*d^12 - 30*a*c^5*d^10*e^2 + 45*a^2*c^4*d^8*e^4
- 20*a^3*c^3*d^6*e^6 - 15*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 - 5*a^6*e^12)*log
(-4*(2*c^2*d^2*e^2*x + c^2*d^3*e + a*c*d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x) + (8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*
e + a*c*d*e^3)*x)*sqrt(c*d*e)))/(sqrt(c*d*e)*c^3*d^3*e^4), 1/15360*(2*(1280*c^5*
d^5*e^5*x^5 - 105*c^5*d^10 + 415*a*c^4*d^8*e^2 - 546*a^2*c^3*d^6*e^4 + 150*a^3*c
^2*d^4*e^6 - 245*a^4*c*d^2*e^8 + 75*a^5*e^10 + 128*(13*c^5*d^6*e^4 + 25*a*c^4*d^
4*e^6)*x^4 + 16*(3*c^5*d^7*e^3 + 278*a*c^4*d^5*e^5 + 135*a^2*c^3*d^3*e^7)*x^3 -
8*(7*c^5*d^8*e^2 - 27*a*c^4*d^6*e^4 - 423*a^2*c^3*d^4*e^6 - 5*a^3*c^2*d^2*e^8)*x
^2 + 2*(35*c^5*d^9*e - 136*a*c^4*d^7*e^3 + 174*a^2*c^3*d^5*e^5 + 80*a^3*c^2*d^3*
e^7 - 25*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e
) + 15*(7*c^6*d^12 - 30*a*c^5*d^10*e^2 + 45*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6
- 15*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 - 5*a^6*e^12)*arctan(1/2*(2*c*d*e*x +
c*d^2 + a*e^2)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*e))
)/(sqrt(-c*d*e)*c^3*d^3*e^4)]

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

[Out]

Timed out

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GIAC/XCAS [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)*x/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: TypeError