3.1953 \(\int \frac{(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=294 \[ \frac{35 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}}+\frac{35 e^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^4 d^4}+\frac{35 e^2 (d+e x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c^3 d^3}-\frac{14 e (d+e x)^3}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^5}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*(d + e*x)^5)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (14*e*(
d + e*x)^3)/(3*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e^2*(c
*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c^4*d^4) + (35*e^2
*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(6*c^3*d^3) + (35*e^(3/2
)*(c*d^2 - a*e^2)^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])])/(8*c^(9/2)*d^(9/2))

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Rubi [A]  time = 0.631074, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{35 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}}+\frac{35 e^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^4 d^4}+\frac{35 e^2 (d+e x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c^3 d^3}-\frac{14 e (d+e x)^3}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^5}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^5)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (14*e*(
d + e*x)^3)/(3*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e^2*(c
*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c^4*d^4) + (35*e^2
*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(6*c^3*d^3) + (35*e^(3/2
)*(c*d^2 - a*e^2)^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])])/(8*c^(9/2)*d^(9/2))

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Rubi in Sympy [A]  time = 101.59, size = 286, normalized size = 0.97 \[ - \frac{2 \left (d + e x\right )^{5}}{3 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{14 e \left (d + e x\right )^{3}}{3 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{35 e^{2} \left (d + e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{6 c^{3} d^{3}} - \frac{35 e^{2} \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 c^{4} d^{4}} + \frac{35 e^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{8 c^{\frac{9}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

-2*(d + e*x)**5/(3*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)) - 14*e
*(d + e*x)**3/(3*c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 35*
e**2*(d + e*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(6*c**3*d**3) - 35
*e**2*(a*e**2 - c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*c**4*d
**4) + 35*e**(3/2)*(a*e**2 - c*d**2)**2*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*s
qrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(8*c**(9
/2)*d**(9/2))

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Mathematica [A]  time = 0.639048, size = 233, normalized size = 0.79 \[ \frac{1}{8} \sqrt{(d+e x) (a e+c d x)} \left (\frac{35 e^{3/2} \left (c d^2-a e^2\right )^2 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{9/2} d^{9/2} \sqrt{d+e x} \sqrt{a e+c d x}}-\frac{160 e \left (c d^2-a e^2\right )^2}{3 c^4 d^4 (a e+c d x)}-\frac{16 \left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^2}+\frac{26 c d^2 e^2-22 a e^4}{c^4 d^4}+\frac{4 e^3 x}{c^3 d^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((26*c*d^2*e^2 - 22*a*e^4)/(c^4*d^4) + (4*e^3*x)/
(c^3*d^3) - (16*(c*d^2 - a*e^2)^3)/(3*c^4*d^4*(a*e + c*d*x)^2) - (160*e*(c*d^2 -
a*e^2)^2)/(3*c^4*d^4*(a*e + c*d*x)) + (35*e^(3/2)*(c*d^2 - a*e^2)^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^(9/2)*d^(9/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/8

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Maple [B]  time = 0.056, size = 4008, normalized size = 13.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^6/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

35/16*e^9/d^3/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2)*a^4-35/24*e^7/d^3/c^3*x^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2
-35/4*e^4/d^2/c^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a-7/4*e^6/d^2/c^2*x^4/(a*e*d+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(3/2)*a-35/8*e^2/c^2*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3
5/16*e*d^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)+35/8*e^2/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)+35/16*e*d/c^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2)+17/4*e^4/c*x^4/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+625/64*e^4/c
^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+253/384*d^8*c/(-a^2*e^4+2*a*c*d^2
*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/128*d^4/c/(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(3/2)-35/16*e^3/d/c^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)*a+1/2*e^5*x^5/d/c/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-115/32*e^3*d^5/(-a^2
*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a-35/4*e^6
/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*
a^2+35/16*e^11/d^5/c^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)*a^5+253/24*e^2*d^8*c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+253/192*e*d^7*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*
d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-415/8*e^10/c^2/(-a^2*e^4+2*a*c*d^
2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4+65/8*e^6*d^4/(-a^
2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2-35/
384*e^14/d^6/c^6/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(3/2)*a^7+427/384*e^12/d^4/c^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^6-165/128*e^4*d^4/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^
4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+45/32*e^5/d/c^3*x/(a*e*d+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(3/2)*a^2-35/8*e^5*d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*
e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-261/128*e^10/d^2/c^4/(-a^2*e^4+2*a*c*d^
2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^5+1255/384*e^6*d^2/c^2/
(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3-35/
48*e^15/d^5/c^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2)*a^7-261/16*e^11/d/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+35/16*e^8/d^4/c^4*x^2/(a*e*d+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(3/2)*a^3-147/16*e^6/d^2/c^3*x^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
*a^2+35/64*e^9/d^5/c^5*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4-35/192*e^13
/d^5/c^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2
)*x*a^6+77/32*e^11/d^3/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(3/2)*x*a^5+35/8*e^10/d^4/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*
e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4+65/64*e^5*d^3/c/(-a^2*e^4+2*a*c*d^2*e
^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2-35/24*e^14/d^4/c^4/(-a
^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^6-11
5/4*e^4*d^6*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2)*x*a+265/48*e^7*d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)*x*a^3-415/64*e^9/d/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(
a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^4+265/6*e^8*d^2/c/(-a^2*e^4+2*a*c*d^2
*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+77/4*e^12/d^2/c^3/
(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^5
-35/8*e^7/d/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2)*a^3-185/48*e^9*d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(1/2)*a^4+1255/48*e^7*d^3/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^
2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+427/48*e^13/d^3/c^4/(-a^2*e^4+2*a*
c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6-3/4*e^3*d/c^2*x
/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+35/8*e^6/d^4/c^4*ln((1/2*a*e^2+1/2*c*
d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2
)*a^2-437/48*e^3*d^7*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*a+35/4*e^4/d^2/c^3*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a
+35/12*e^5/d/c^2*x^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+35/16*e^3*d^3/c/(
-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-35/8*e
^6/d^4/c^4*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-7/4*e^7/d^3/c^4*x/(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3-35/16*e^5/d^3/c^4/(a*e*d+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)*a^2-765/64*e*d^3/c*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-28
5/16*e^2*d^2/c*x^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+637/384*e^8/d^4/c^5/(
a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4-1165/192*e^6/d^2/c^4/(a*e*d+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(3/2)*a^3+253/48*e*d^9*c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/
(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-35/384*e^10/d^6/c^6/(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(3/2)*a^5-1249/128*e^2*d^2/c^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
3/2)*a-35/24*e^3*d/c*x^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-185/384*e^8/c^3
/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4-16
5/16*e^5*d^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(1/2)*a^2+35/8*e^2*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*x+237/16*e^4/c^2*x^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a
-437/384*e^2*d^6/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(3/2)*a+35/16*e^7/d^5/c^5/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3

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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)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.45203, 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)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(105*(a^2*c^2*d^4*e^3 - 2*a^3*c*d^2*e^5 + a^4*e^7 + (c^4*d^6*e - 2*a*c^3*d
^4*e^3 + a^2*c^2*d^2*e^5)*x^2 + 2*(a*c^3*d^5*e^2 - 2*a^2*c^2*d^3*e^4 + a^3*c*d*e
^6)*x)*sqrt(e/(c*d))*log(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 + 4*(2*c^2*d^2*e*x + c^2*d^3 + a*c*d*e^2)*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d))) + 4*(6*c^3*d^3*e^3*x^3 - 8*c^3
*d^6 - 56*a*c^2*d^4*e^2 + 175*a^2*c*d^2*e^4 - 105*a^3*e^6 + 3*(13*c^3*d^4*e^2 -
7*a*c^2*d^2*e^4)*x^2 - 2*(40*c^3*d^5*e - 119*a*c^2*d^3*e^3 + 70*a^2*c*d*e^5)*x)*
sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^
2*c^4*d^4*e^2), 1/24*(105*(a^2*c^2*d^4*e^3 - 2*a^3*c*d^2*e^5 + a^4*e^7 + (c^4*d^
6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^2 + 2*(a*c^3*d^5*e^2 - 2*a^2*c^2*d^3*
e^4 + a^3*c*d*e^6)*x)*sqrt(-e/(c*d))*arctan(1/2*(2*c*d*e*x + c*d^2 + a*e^2)/(sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*sqrt(-e/(c*d)))) + 2*(6*c^3*d^3*e^3
*x^3 - 8*c^3*d^6 - 56*a*c^2*d^4*e^2 + 175*a^2*c*d^2*e^4 - 105*a^3*e^6 + 3*(13*c^
3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 - 2*(40*c^3*d^5*e - 119*a*c^2*d^3*e^3 + 70*a^2*
c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*x^2 + 2*a*c^5*
d^5*e*x + a^2*c^4*d^4*e^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((e*x+d)**6/(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 = 0.269172, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="giac")

[Out]

Done