3.2069 \(\int \frac{1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=519 \[ \frac{3003 c^5 d^5 e \sqrt{d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{3003 c^5 d^5 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{128 \left (c d^2-a e^2\right )^{15/2}}-\frac{1001 c^4 d^4 e}{128 \sqrt{d+e x} \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{1001 c^4 d^4 \sqrt{d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{429 c^3 d^3}{320 \sqrt{d+e x} \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{13 c d}{40 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{5 (d+e x)^{7/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

1/(5*(c*d^2 - a*e^2)*(d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2)) + (13*c*d)/(40*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2)) + (143*c^2*d^2)/(240*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)*(a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (429*c^3*d^3)/(320*(c*d^2 - a*e^2)^4
*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (1001*c^4*d^4*Sq
rt[d + e*x])/(320*(c*d^2 - a*e^2)^5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2
)) - (1001*c^4*d^4*e)/(128*(c*d^2 - a*e^2)^6*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2]) + (3003*c^5*d^5*e*Sqrt[d + e*x])/(128*(c*d^2 - a*e^2)^7*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3003*c^5*d^5*e^(3/2)*ArcTan[(Sqr
t[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d +
e*x])])/(128*(c*d^2 - a*e^2)^(15/2))

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Rubi [A]  time = 1.60534, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{3003 c^5 d^5 e \sqrt{d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{3003 c^5 d^5 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{128 \left (c d^2-a e^2\right )^{15/2}}-\frac{1001 c^4 d^4 e}{128 \sqrt{d+e x} \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{1001 c^4 d^4 \sqrt{d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{429 c^3 d^3}{320 \sqrt{d+e x} \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{13 c d}{40 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{5 (d+e x)^{7/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

1/(5*(c*d^2 - a*e^2)*(d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2)) + (13*c*d)/(40*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2)) + (143*c^2*d^2)/(240*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)*(a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (429*c^3*d^3)/(320*(c*d^2 - a*e^2)^4
*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (1001*c^4*d^4*Sq
rt[d + e*x])/(320*(c*d^2 - a*e^2)^5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2
)) - (1001*c^4*d^4*e)/(128*(c*d^2 - a*e^2)^6*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2]) + (3003*c^5*d^5*e*Sqrt[d + e*x])/(128*(c*d^2 - a*e^2)^7*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3003*c^5*d^5*e^(3/2)*ArcTan[(Sqr
t[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d +
e*x])])/(128*(c*d^2 - a*e^2)^(15/2))

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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(1/(e*x+d)**(7/2)/(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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Mathematica [A]  time = 2.58092, size = 320, normalized size = 0.62 \[ \frac{(d+e x)^{5/2} \left (\frac{3003 c^5 d^5 e^{3/2} (a e+c d x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{\left (a e^2-c d^2\right )^{15/2}}+\frac{(a e+c d x)^3 \left (-\frac{23040 c^5 d^5 e}{a e+c d x}+\frac{1280 c^5 d^5 \left (c d^2-a e^2\right )}{(a e+c d x)^2}+\frac{8270 c^3 d^3 e^2 \left (a e^2-c d^2\right )}{(d+e x)^2}-\frac{3544 c^2 d^2 e^2 \left (c d^2-a e^2\right )^2}{(d+e x)^3}+\frac{1392 c d e^2 \left (a e^2-c d^2\right )^3}{(d+e x)^4}-\frac{384 e^2 \left (c d^2-a e^2\right )^4}{(d+e x)^5}-\frac{22005 c^4 d^4 e^2}{d+e x}\right )}{15 \left (a e^2-c d^2\right )^7}\right )}{128 ((d+e x) (a e+c d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

((d + e*x)^(5/2)*(((a*e + c*d*x)^3*((1280*c^5*d^5*(c*d^2 - a*e^2))/(a*e + c*d*x)
^2 - (23040*c^5*d^5*e)/(a*e + c*d*x) - (384*e^2*(c*d^2 - a*e^2)^4)/(d + e*x)^5 +
(1392*c*d*e^2*(-(c*d^2) + a*e^2)^3)/(d + e*x)^4 - (3544*c^2*d^2*e^2*(c*d^2 - a*
e^2)^2)/(d + e*x)^3 + (8270*c^3*d^3*e^2*(-(c*d^2) + a*e^2))/(d + e*x)^2 - (22005
*c^4*d^4*e^2)/(d + e*x)))/(15*(-(c*d^2) + a*e^2)^7) + (3003*c^5*d^5*e^(3/2)*(a*e
+ c*d*x)^(5/2)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d^2) + a*e^2]])/(-(
c*d^2) + a*e^2)^(15/2)))/(128*((a*e + c*d*x)*(d + e*x))^(5/2))

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Maple [B]  time = 0.071, size = 1553, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(7/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

1/1920*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(1280*((a*e^2-c*d^2)*e)^(1/2)*c^6
*d^12-384*((a*e^2-c*d^2)*e)^(1/2)*a^6*e^12-24320*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d
^10*e^2-16640*((a*e^2-c*d^2)*e)^(1/2)*x*c^6*d^11*e-45045*((a*e^2-c*d^2)*e)^(1/2)
*x^6*c^6*d^6*e^6-210210*((a*e^2-c*d^2)*e)^(1/2)*x^5*c^6*d^7*e^5-384384*((a*e^2-c
*d^2)*e)^(1/2)*x^4*c^6*d^8*e^4-338910*((a*e^2-c*d^2)*e)^(1/2)*x^3*c^6*d^9*e^3+45
045*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^5*a*c^5*d^5*e^8*(c*d*
x+a*e)^(1/2)+225225*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^4*a*c
^5*d^6*e^7*(c*d*x+a*e)^(1/2)+450450*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e
)^(1/2))*x^3*a*c^5*d^7*e^6*(c*d*x+a*e)^(1/2)+450450*arctanh(e*(c*d*x+a*e)^(1/2)/
((a*e^2-c*d^2)*e)^(1/2))*x^2*a*c^5*d^8*e^5*(c*d*x+a*e)^(1/2)+225225*arctanh(e*(c
*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x*a*c^5*d^9*e^4*(c*d*x+a*e)^(1/2)+2928*
((a*e^2-c*d^2)*e)^(1/2)*a^5*c*d^2*e^10-10024*((a*e^2-c*d^2)*e)^(1/2)*a^4*c^2*d^4
*e^8+21070*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^3*d^6*e^6-35595*((a*e^2-c*d^2)*e)^(1/2)
*a^2*c^4*d^8*e^4-137995*((a*e^2-c*d^2)*e)^(1/2)*x^2*c^6*d^10*e^2+624*((a*e^2-c*d
^2)*e)^(1/2)*x*a^5*c*d*e^11-5408*((a*e^2-c*d^2)*e)^(1/2)*x*a^4*c^2*d^3*e^9+23114
*((a*e^2-c*d^2)*e)^(1/2)*x*a^3*c^3*d^5*e^7-79170*((a*e^2-c*d^2)*e)^(1/2)*x*a^2*c
^4*d^7*e^5-60060*((a*e^2-c*d^2)*e)^(1/2)*x^5*a*c^5*d^5*e^7+45045*arctanh(e*(c*d*
x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^6*c^6*d^6*e^7*(c*d*x+a*e)^(1/2)+225225*a
rctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^5*c^6*d^7*e^6*(c*d*x+a*e)^
(1/2)+450450*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^4*c^6*d^8*e^
5*(c*d*x+a*e)^(1/2)+450450*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*
x^3*c^6*d^9*e^4*(c*d*x+a*e)^(1/2)+225225*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d
^2)*e)^(1/2))*x^2*c^6*d^10*e^3*(c*d*x+a*e)^(1/2)+45045*arctanh(e*(c*d*x+a*e)^(1/
2)/((a*e^2-c*d^2)*e)^(1/2))*x*c^6*d^11*e^2*(c*d*x+a*e)^(1/2)+45045*arctanh(e*(c*
d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^5*d^10*e^3*(c*d*x+a*e)^(1/2)-9009*((
a*e^2-c*d^2)*e)^(1/2)*x^4*a^2*c^4*d^4*e^8+2574*((a*e^2-c*d^2)*e)^(1/2)*x^3*a^3*c
^3*d^3*e^9-43758*((a*e^2-c*d^2)*e)^(1/2)*x^3*a^2*c^4*d^5*e^7-1144*((a*e^2-c*d^2)
*e)^(1/2)*x^2*a^4*c^2*d^2*e^10+12298*((a*e^2-c*d^2)*e)^(1/2)*x^2*a^3*c^3*d^4*e^8
-84084*((a*e^2-c*d^2)*e)^(1/2)*x^2*a^2*c^4*d^6*e^6-282282*((a*e^2-c*d^2)*e)^(1/2
)*x^4*a*c^5*d^6*e^6-520806*((a*e^2-c*d^2)*e)^(1/2)*x^3*a*c^5*d^7*e^5-464750*((a*
e^2-c*d^2)*e)^(1/2)*x^2*a*c^5*d^8*e^4-192790*((a*e^2-c*d^2)*e)^(1/2)*x*a*c^5*d^9
*e^3)/(e*x+d)^(11/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^7/((a*e^2-c*d^2)*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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277073, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[1/3840*(45045*(c^7*d^7*e^7*x^8 + a^2*c^5*d^11*e^3 + 2*(3*c^7*d^8*e^6 + a*c^6*d^
6*e^8)*x^7 + (15*c^7*d^9*e^5 + 12*a*c^6*d^7*e^7 + a^2*c^5*d^5*e^9)*x^6 + 2*(10*c
^7*d^10*e^4 + 15*a*c^6*d^8*e^6 + 3*a^2*c^5*d^6*e^8)*x^5 + 5*(3*c^7*d^11*e^3 + 8*
a*c^6*d^9*e^5 + 3*a^2*c^5*d^7*e^7)*x^4 + 2*(3*c^7*d^12*e^2 + 15*a*c^6*d^10*e^4 +
10*a^2*c^5*d^8*e^6)*x^3 + (c^7*d^13*e + 12*a*c^6*d^11*e^3 + 15*a^2*c^5*d^9*e^5)
*x^2 + 2*(a*c^6*d^12*e^2 + 3*a^2*c^5*d^10*e^4)*x)*sqrt(-e/(c*d^2 - a*e^2))*log(-
(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2 +
2*d*e*x + d^2)) + 2*(45045*c^6*d^6*e^6*x^6 - 1280*c^6*d^12 + 24320*a*c^5*d^10*e^
2 + 35595*a^2*c^4*d^8*e^4 - 21070*a^3*c^3*d^6*e^6 + 10024*a^4*c^2*d^4*e^8 - 2928
*a^5*c*d^2*e^10 + 384*a^6*e^12 + 30030*(7*c^6*d^7*e^5 + 2*a*c^5*d^5*e^7)*x^5 + 3
003*(128*c^6*d^8*e^4 + 94*a*c^5*d^6*e^6 + 3*a^2*c^4*d^4*e^8)*x^4 + 858*(395*c^6*
d^9*e^3 + 607*a*c^5*d^7*e^5 + 51*a^2*c^4*d^5*e^7 - 3*a^3*c^3*d^3*e^9)*x^3 + 143*
(965*c^6*d^10*e^2 + 3250*a*c^5*d^8*e^4 + 588*a^2*c^4*d^6*e^6 - 86*a^3*c^3*d^4*e^
8 + 8*a^4*c^2*d^2*e^10)*x^2 + 26*(640*c^6*d^11*e + 7415*a*c^5*d^9*e^3 + 3045*a^2
*c^4*d^7*e^5 - 889*a^3*c^3*d^5*e^7 + 208*a^4*c^2*d^3*e^9 - 24*a^5*c*d*e^11)*x)*s
qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^7*d^20*e^2 - 7*
a^3*c^6*d^18*e^4 + 21*a^4*c^5*d^16*e^6 - 35*a^5*c^4*d^14*e^8 + 35*a^6*c^3*d^12*e
^10 - 21*a^7*c^2*d^10*e^12 + 7*a^8*c*d^8*e^14 - a^9*d^6*e^16 + (c^9*d^16*e^6 - 7
*a*c^8*d^14*e^8 + 21*a^2*c^7*d^12*e^10 - 35*a^3*c^6*d^10*e^12 + 35*a^4*c^5*d^8*e
^14 - 21*a^5*c^4*d^6*e^16 + 7*a^6*c^3*d^4*e^18 - a^7*c^2*d^2*e^20)*x^8 + 2*(3*c^
9*d^17*e^5 - 20*a*c^8*d^15*e^7 + 56*a^2*c^7*d^13*e^9 - 84*a^3*c^6*d^11*e^11 + 70
*a^4*c^5*d^9*e^13 - 28*a^5*c^4*d^7*e^15 + 4*a^7*c^2*d^3*e^19 - a^8*c*d*e^21)*x^7
+ (15*c^9*d^18*e^4 - 93*a*c^8*d^16*e^6 + 232*a^2*c^7*d^14*e^8 - 280*a^3*c^6*d^1
2*e^10 + 126*a^4*c^5*d^10*e^12 + 70*a^5*c^4*d^8*e^14 - 112*a^6*c^3*d^6*e^16 + 48
*a^7*c^2*d^4*e^18 - 5*a^8*c*d^2*e^20 - a^9*e^22)*x^6 + 2*(10*c^9*d^19*e^3 - 55*a
*c^8*d^17*e^5 + 108*a^2*c^7*d^15*e^7 - 56*a^3*c^6*d^13*e^9 - 112*a^4*c^5*d^11*e^
11 + 210*a^5*c^4*d^9*e^13 - 140*a^6*c^3*d^7*e^15 + 32*a^7*c^2*d^5*e^17 + 6*a^8*c
*d^3*e^19 - 3*a^9*d*e^21)*x^5 + 5*(3*c^9*d^20*e^2 - 13*a*c^8*d^18*e^4 + 10*a^2*c
^7*d^16*e^6 + 42*a^3*c^6*d^14*e^8 - 112*a^4*c^5*d^12*e^10 + 112*a^5*c^4*d^10*e^1
2 - 42*a^6*c^3*d^8*e^14 - 10*a^7*c^2*d^6*e^16 + 13*a^8*c*d^4*e^18 - 3*a^9*d^2*e^
20)*x^4 + 2*(3*c^9*d^21*e - 6*a*c^8*d^19*e^3 - 32*a^2*c^7*d^17*e^5 + 140*a^3*c^6
*d^15*e^7 - 210*a^4*c^5*d^13*e^9 + 112*a^5*c^4*d^11*e^11 + 56*a^6*c^3*d^9*e^13 -
108*a^7*c^2*d^7*e^15 + 55*a^8*c*d^5*e^17 - 10*a^9*d^3*e^19)*x^3 + (c^9*d^22 + 5
*a*c^8*d^20*e^2 - 48*a^2*c^7*d^18*e^4 + 112*a^3*c^6*d^16*e^6 - 70*a^4*c^5*d^14*e
^8 - 126*a^5*c^4*d^12*e^10 + 280*a^6*c^3*d^10*e^12 - 232*a^7*c^2*d^8*e^14 + 93*a
^8*c*d^6*e^16 - 15*a^9*d^4*e^18)*x^2 + 2*(a*c^8*d^21*e - 4*a^2*c^7*d^19*e^3 + 28
*a^4*c^5*d^15*e^7 - 70*a^5*c^4*d^13*e^9 + 84*a^6*c^3*d^11*e^11 - 56*a^7*c^2*d^9*
e^13 + 20*a^8*c*d^7*e^15 - 3*a^9*d^5*e^17)*x), -1/1920*(45045*(c^7*d^7*e^7*x^8 +
a^2*c^5*d^11*e^3 + 2*(3*c^7*d^8*e^6 + a*c^6*d^6*e^8)*x^7 + (15*c^7*d^9*e^5 + 12
*a*c^6*d^7*e^7 + a^2*c^5*d^5*e^9)*x^6 + 2*(10*c^7*d^10*e^4 + 15*a*c^6*d^8*e^6 +
3*a^2*c^5*d^6*e^8)*x^5 + 5*(3*c^7*d^11*e^3 + 8*a*c^6*d^9*e^5 + 3*a^2*c^5*d^7*e^7
)*x^4 + 2*(3*c^7*d^12*e^2 + 15*a*c^6*d^10*e^4 + 10*a^2*c^5*d^8*e^6)*x^3 + (c^7*d
^13*e + 12*a*c^6*d^11*e^3 + 15*a^2*c^5*d^9*e^5)*x^2 + 2*(a*c^6*d^12*e^2 + 3*a^2*
c^5*d^10*e^4)*x)*sqrt(e/(c*d^2 - a*e^2))*arctan(sqrt(e*x + d)/(sqrt(c*d*e*x^2 +
a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d^2 - a*e^2)))) - (45045*c^6*d^6*e^6*x^6 -
1280*c^6*d^12 + 24320*a*c^5*d^10*e^2 + 35595*a^2*c^4*d^8*e^4 - 21070*a^3*c^3*d^6
*e^6 + 10024*a^4*c^2*d^4*e^8 - 2928*a^5*c*d^2*e^10 + 384*a^6*e^12 + 30030*(7*c^6
*d^7*e^5 + 2*a*c^5*d^5*e^7)*x^5 + 3003*(128*c^6*d^8*e^4 + 94*a*c^5*d^6*e^6 + 3*a
^2*c^4*d^4*e^8)*x^4 + 858*(395*c^6*d^9*e^3 + 607*a*c^5*d^7*e^5 + 51*a^2*c^4*d^5*
e^7 - 3*a^3*c^3*d^3*e^9)*x^3 + 143*(965*c^6*d^10*e^2 + 3250*a*c^5*d^8*e^4 + 588*
a^2*c^4*d^6*e^6 - 86*a^3*c^3*d^4*e^8 + 8*a^4*c^2*d^2*e^10)*x^2 + 26*(640*c^6*d^1
1*e + 7415*a*c^5*d^9*e^3 + 3045*a^2*c^4*d^7*e^5 - 889*a^3*c^3*d^5*e^7 + 208*a^4*
c^2*d^3*e^9 - 24*a^5*c*d*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq
rt(e*x + d))/(a^2*c^7*d^20*e^2 - 7*a^3*c^6*d^18*e^4 + 21*a^4*c^5*d^16*e^6 - 35*a
^5*c^4*d^14*e^8 + 35*a^6*c^3*d^12*e^10 - 21*a^7*c^2*d^10*e^12 + 7*a^8*c*d^8*e^14
- a^9*d^6*e^16 + (c^9*d^16*e^6 - 7*a*c^8*d^14*e^8 + 21*a^2*c^7*d^12*e^10 - 35*a
^3*c^6*d^10*e^12 + 35*a^4*c^5*d^8*e^14 - 21*a^5*c^4*d^6*e^16 + 7*a^6*c^3*d^4*e^1
8 - a^7*c^2*d^2*e^20)*x^8 + 2*(3*c^9*d^17*e^5 - 20*a*c^8*d^15*e^7 + 56*a^2*c^7*d
^13*e^9 - 84*a^3*c^6*d^11*e^11 + 70*a^4*c^5*d^9*e^13 - 28*a^5*c^4*d^7*e^15 + 4*a
^7*c^2*d^3*e^19 - a^8*c*d*e^21)*x^7 + (15*c^9*d^18*e^4 - 93*a*c^8*d^16*e^6 + 232
*a^2*c^7*d^14*e^8 - 280*a^3*c^6*d^12*e^10 + 126*a^4*c^5*d^10*e^12 + 70*a^5*c^4*d
^8*e^14 - 112*a^6*c^3*d^6*e^16 + 48*a^7*c^2*d^4*e^18 - 5*a^8*c*d^2*e^20 - a^9*e^
22)*x^6 + 2*(10*c^9*d^19*e^3 - 55*a*c^8*d^17*e^5 + 108*a^2*c^7*d^15*e^7 - 56*a^3
*c^6*d^13*e^9 - 112*a^4*c^5*d^11*e^11 + 210*a^5*c^4*d^9*e^13 - 140*a^6*c^3*d^7*e
^15 + 32*a^7*c^2*d^5*e^17 + 6*a^8*c*d^3*e^19 - 3*a^9*d*e^21)*x^5 + 5*(3*c^9*d^20
*e^2 - 13*a*c^8*d^18*e^4 + 10*a^2*c^7*d^16*e^6 + 42*a^3*c^6*d^14*e^8 - 112*a^4*c
^5*d^12*e^10 + 112*a^5*c^4*d^10*e^12 - 42*a^6*c^3*d^8*e^14 - 10*a^7*c^2*d^6*e^16
+ 13*a^8*c*d^4*e^18 - 3*a^9*d^2*e^20)*x^4 + 2*(3*c^9*d^21*e - 6*a*c^8*d^19*e^3
- 32*a^2*c^7*d^17*e^5 + 140*a^3*c^6*d^15*e^7 - 210*a^4*c^5*d^13*e^9 + 112*a^5*c^
4*d^11*e^11 + 56*a^6*c^3*d^9*e^13 - 108*a^7*c^2*d^7*e^15 + 55*a^8*c*d^5*e^17 - 1
0*a^9*d^3*e^19)*x^3 + (c^9*d^22 + 5*a*c^8*d^20*e^2 - 48*a^2*c^7*d^18*e^4 + 112*a
^3*c^6*d^16*e^6 - 70*a^4*c^5*d^14*e^8 - 126*a^5*c^4*d^12*e^10 + 280*a^6*c^3*d^10
*e^12 - 232*a^7*c^2*d^8*e^14 + 93*a^8*c*d^6*e^16 - 15*a^9*d^4*e^18)*x^2 + 2*(a*c
^8*d^21*e - 4*a^2*c^7*d^19*e^3 + 28*a^4*c^5*d^15*e^7 - 70*a^5*c^4*d^13*e^9 + 84*
a^6*c^3*d^11*e^11 - 56*a^7*c^2*d^9*e^13 + 20*a^8*c*d^7*e^15 - 3*a^9*d^5*e^17)*x)
]

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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(1/(e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, undef, undef, undef, undef, 2]