3.456 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx\)

Optimal. Leaf size=498 \[ \frac{\left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}+\frac{\left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac{\left (-105 a^3 e^6+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac{\left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{9/2} d^{11/2} e^{9/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 x^5} \]

[Out]

((7*c^4*d^8 + 8*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 21*a^4*e^8)*(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^4*d^5*e^4*x^2)
- (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(6*d*x^6) - ((c/(a*e) - (3*e)/d
^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(20*x^5) + ((7*c^2*d^4 + 6*a*
c*d^2*e^2 - 21*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(160*a^2*
d^3*e^2*x^4) - ((35*c^3*d^6 + 33*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 - 105*a^3*e^6)
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(960*a^3*d^4*e^3*x^3) - ((c*d^2
- a*e^2)^3*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*ArcTan
h[(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^(9/2)*d^(11/2)*e^(9/2))

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Rubi [A]  time = 1.88759, antiderivative size = 498, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}+\frac{\left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac{\left (-105 a^3 e^6+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac{\left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{9/2} d^{11/2} e^{9/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}-\frac{\left (\frac{c}{a e}-\frac{3 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x]

[Out]

((7*c^4*d^8 + 8*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 21*a^4*e^8)*(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^4*d^5*e^4*x^2)
- (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(6*d*x^6) - ((c/(a*e) - (3*e)/d
^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(20*x^5) + ((7*c^2*d^4 + 6*a*
c*d^2*e^2 - 21*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(160*a^2*
d^3*e^2*x^4) - ((35*c^3*d^6 + 33*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 - 105*a^3*e^6)
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(960*a^3*d^4*e^3*x^3) - ((c*d^2
- a*e^2)^3*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*ArcTan
h[(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^(9/2)*d^(11/2)*e^(9/2))

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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)**(3/2)/x**7/(e*x+d),x)

[Out]

Timed out

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Mathematica [A]  time = 0.871886, size = 506, normalized size = 1.02 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (15 x^6 \log (x) \left (c d^2-a e^2\right )^3 \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right )-15 x^6 \left (c d^2-a e^2\right )^3 \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )-2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^5 e^5 \left (1280 d^5+128 d^4 e x-144 d^3 e^2 x^2+168 d^2 e^3 x^3-210 d e^4 x^4+315 e^5 x^5\right )+a^4 c d^2 e^4 x \left (1664 d^4+224 d^3 e x-264 d^2 e^2 x^2+336 d e^3 x^3-525 e^4 x^4\right )+6 a^3 c^2 d^4 e^3 x^2 \left (8 d^3+4 d^2 e x-6 d e^2 x^2+13 e^3 x^3\right )-2 a^2 c^3 d^6 e^2 x^3 \left (28 d^2+16 d e x-27 e^2 x^2\right )+5 a c^4 d^8 e x^4 (14 d+11 e x)-105 c^5 d^{10} x^5\right )\right )}{15360 a^{9/2} d^{11/2} e^{9/2} x^6 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x]

[Out]

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*S
qrt[d + e*x]*(-105*c^5*d^10*x^5 + 5*a*c^4*d^8*e*x^4*(14*d + 11*e*x) - 2*a^2*c^3*
d^6*e^2*x^3*(28*d^2 + 16*d*e*x - 27*e^2*x^2) + 6*a^3*c^2*d^4*e^3*x^2*(8*d^3 + 4*
d^2*e*x - 6*d*e^2*x^2 + 13*e^3*x^3) + a^4*c*d^2*e^4*x*(1664*d^4 + 224*d^3*e*x -
264*d^2*e^2*x^2 + 336*d*e^3*x^3 - 525*e^4*x^4) + a^5*e^5*(1280*d^5 + 128*d^4*e*x
- 144*d^3*e^2*x^2 + 168*d^2*e^3*x^3 - 210*d*e^4*x^4 + 315*e^5*x^5)) + 15*(c*d^2
- a*e^2)^3*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*x^6*L
og[x] - 15*(c*d^2 - a*e^2)^3*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 +
21*a^3*e^6)*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^(9/2)*d^(11/2)*e^(9/2)*x^6*Sqrt[(a*e + c*d
*x)*(d + e*x)])

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Maple [B]  time = 0.058, size = 3387, normalized size = 6.8 \[ \text{output too large to display} \]

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)^(3/2)/x^7/(e*x+d),x)

[Out]

1/8/d^4*e^5*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-21/512/d^6*a*e^7*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/8/d^4*c*e^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2)+19/60/d^3/a/x^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-23/96/d/a^3*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^3+1/64*c^3/a^2*e*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)-1/3/d^7*e^6*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-533/
1536/d^7*e^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/16/d^8*e^11*a^3/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)-3/16/d^4*e^7*a*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)+41/1536*d/a^5/e^4/x*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^4-7/1536*d^3/a^6/e^6/x*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(5/2)*c^5-7/768*d^2/a^5/e^5/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
5/2)*c^4+3/512*d^5/a^3/e^2/(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^5-7/1024*d^7/a^4/e^4/(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+877/1536/d^5/a^2*e^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)
*c+15/512*d^3/a^4/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^5+7/512*d^5/a^
5/e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^6+29/192/d/a^3/e^2/x^3*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2+7/192*d/a^4/e^4/x^3*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(5/2)*c^3-257/768/d^2/a^3*e*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/
2)*x-91/384/d^2/a^3/e/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2+1/256*d/a*
e^2/(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^3-21/512/d^3/a*e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)*x*c^2-11/48/d^2/a^2/e/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-21/51
2/d^3*a*e^6/(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+7/60/d/a^2/e^2/x^5*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(5/2)*c+3/256/d/a^2*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^3-4
1/1536*d^2/a^5/e^3*c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-43/96/d^4/a^2*e
/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-1045/1536/d^6/a*e^5*c*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-877/1536/d^4/a^2*e^3*c^2*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(3/2)*x+7/1536*d^4/a^6/e^5*c^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
*x+109/768/d/a^4/e^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3+1/16/d^8*e^11
*a^3/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)+3/16/d^4*e^7*a*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)+3/16/d^6*e^9*a^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)+1/16/d^2*e^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)-7/96/a^3/e^3/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2-109/768/a
^4/e*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-1/12/a^4/e^3/x^2*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3+3/1024*d^3/a^2/(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+65/192/
d^3/a^2/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c+7/256*d/a^3*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^4+15/1024/d*e^4/(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+257/768/
d^3/a^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2-149/512/d^5*e^6*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c-1/6/d^2/a/e/x^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(5/2)+1/64*d^4*c^5/a^4/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-235/384/d
^5/a*e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c-15/512/d^2/a*e^3*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2+7/1536*d^5/a^6/e^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(3/2)*c^6+13/512*d^2/a^3/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^4+7/51
2*d^6/a^5/e^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^6-491/768/d^6/a*e^3/x^2*
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+21/1024/d^5*a^2*e^8/(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
)+1045/1536/d^7/a*e^4/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+107/192/d^5/a*e^
2/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-43/96/d^4/a*e/x^4*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(5/2)-703/1536/d^3/a^2*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
3/2)*c^2-131/1536*d/a^4/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^4-5/384*d^
3/a^5/e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^5-1/4/d^7*e^8*a*(c*d*e*(x+d/
e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-1/8/d^8*e^9*a^2/c*(c*d*e*(x+d/e)^2+(a*e^2-c*
d^2)*(x+d/e))^(1/2)-3/16/d^6*e^9*a^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)+1/4/d^5*e
^6*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-1/16/d^2*e^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)+1/8/d^8*e^9*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)+1/4/d^7*e^8*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x

_______________________________________________________________________________________

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)^(3/2)/((e*x + d)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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)^(3/2)/((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)**(3/2)/x**7/(e*x+d),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 50.9192, 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)^(3/2)/((e*x + d)*x^7),x, algorithm="giac")

[Out]

Done