3.1757 \(\int \frac{(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx\)

Optimal. Leaf size=208 \[ -\frac{2 (b c-a d)^2 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}+\frac{2 \sqrt{e+f x} (b c-a d)^2 (d e-c f)^2}{d^5}+\frac{2 (e+f x)^{3/2} (b c-a d)^2 (d e-c f)}{3 d^4}+\frac{2 (e+f x)^{5/2} (b c-a d)^2}{5 d^3}-\frac{2 b (e+f x)^{7/2} (-2 a d f+b c f+b d e)}{7 d^2 f^2}+\frac{2 b^2 (e+f x)^{9/2}}{9 d f^2} \]

[Out]

(2*(b*c - a*d)^2*(d*e - c*f)^2*Sqrt[e + f*x])/d^5 + (2*(b*c - a*d)^2*(d*e - c*f)
*(e + f*x)^(3/2))/(3*d^4) + (2*(b*c - a*d)^2*(e + f*x)^(5/2))/(5*d^3) - (2*b*(b*
d*e + b*c*f - 2*a*d*f)*(e + f*x)^(7/2))/(7*d^2*f^2) + (2*b^2*(e + f*x)^(9/2))/(9
*d*f^2) - (2*(b*c - a*d)^2*(d*e - c*f)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqr
t[d*e - c*f]])/d^(11/2)

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Rubi [A]  time = 0.442449, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 (b c-a d)^2 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}+\frac{2 \sqrt{e+f x} (b c-a d)^2 (d e-c f)^2}{d^5}+\frac{2 (e+f x)^{3/2} (b c-a d)^2 (d e-c f)}{3 d^4}+\frac{2 (e+f x)^{5/2} (b c-a d)^2}{5 d^3}-\frac{2 b (e+f x)^{7/2} (-2 a d f+b c f+b d e)}{7 d^2 f^2}+\frac{2 b^2 (e+f x)^{9/2}}{9 d f^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*c - a*d)^2*(d*e - c*f)^2*Sqrt[e + f*x])/d^5 + (2*(b*c - a*d)^2*(d*e - c*f)
*(e + f*x)^(3/2))/(3*d^4) + (2*(b*c - a*d)^2*(e + f*x)^(5/2))/(5*d^3) - (2*b*(b*
d*e + b*c*f - 2*a*d*f)*(e + f*x)^(7/2))/(7*d^2*f^2) + (2*b^2*(e + f*x)^(9/2))/(9
*d*f^2) - (2*(b*c - a*d)^2*(d*e - c*f)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqr
t[d*e - c*f]])/d^(11/2)

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Rubi in Sympy [A]  time = 57.3301, size = 192, normalized size = 0.92 \[ \frac{2 b^{2} \left (e + f x\right )^{\frac{9}{2}}}{9 d f^{2}} + \frac{2 b \left (e + f x\right )^{\frac{7}{2}} \left (2 a d f - b c f - b d e\right )}{7 d^{2} f^{2}} + \frac{2 \left (e + f x\right )^{\frac{5}{2}} \left (a d - b c\right )^{2}}{5 d^{3}} - \frac{2 \left (e + f x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2} \left (c f - d e\right )}{3 d^{4}} + \frac{2 \sqrt{e + f x} \left (a d - b c\right )^{2} \left (c f - d e\right )^{2}}{d^{5}} - \frac{2 \left (a d - b c\right )^{2} \left (c f - d e\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e + f*x)**(9/2)/(9*d*f**2) + 2*b*(e + f*x)**(7/2)*(2*a*d*f - b*c*f - b*d
*e)/(7*d**2*f**2) + 2*(e + f*x)**(5/2)*(a*d - b*c)**2/(5*d**3) - 2*(e + f*x)**(3
/2)*(a*d - b*c)**2*(c*f - d*e)/(3*d**4) + 2*sqrt(e + f*x)*(a*d - b*c)**2*(c*f -
d*e)**2/d**5 - 2*(a*d - b*c)**2*(c*f - d*e)**(5/2)*atan(sqrt(d)*sqrt(e + f*x)/sq
rt(c*f - d*e))/d**(11/2)

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Mathematica [A]  time = 0.593137, size = 295, normalized size = 1.42 \[ \frac{2 \sqrt{e+f x} \left (21 a^2 d^2 f^2 \left (15 c^2 f^2-5 c d f (7 e+f x)+d^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )+6 a b d f \left (-105 c^3 f^3+35 c^2 d f^2 (7 e+f x)-7 c d^2 f \left (23 e^2+11 e f x+3 f^2 x^2\right )+15 d^3 (e+f x)^3\right )+b^2 \left (315 c^4 f^4-105 c^3 d f^3 (7 e+f x)+21 c^2 d^2 f^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )-45 c d^3 f (e+f x)^3-5 d^4 (2 e-7 f x) (e+f x)^3\right )\right )}{315 d^5 f^2}-\frac{2 (b c-a d)^2 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e + f*x]*(21*a^2*d^2*f^2*(15*c^2*f^2 - 5*c*d*f*(7*e + f*x) + d^2*(23*e^2
+ 11*e*f*x + 3*f^2*x^2)) + 6*a*b*d*f*(-105*c^3*f^3 + 15*d^3*(e + f*x)^3 + 35*c^
2*d*f^2*(7*e + f*x) - 7*c*d^2*f*(23*e^2 + 11*e*f*x + 3*f^2*x^2)) + b^2*(315*c^4*
f^4 - 45*c*d^3*f*(e + f*x)^3 - 5*d^4*(2*e - 7*f*x)*(e + f*x)^3 - 105*c^3*d*f^3*(
7*e + f*x) + 21*c^2*d^2*f^2*(23*e^2 + 11*e*f*x + 3*f^2*x^2))))/(315*d^5*f^2) - (
2*(b*c - a*d)^2*(d*e - c*f)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f
]])/d^(11/2)

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Maple [B]  time = 0.022, size = 972, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*f/d^2*a^2*c*e*(f*x+e)^(1/2)-4*f^2/d^4*a*b*c^3*(f*x+e)^(1/2)-4*f/d^4*b^2*c^3*e
*(f*x+e)^(1/2)-2*f^3/d^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d
)^(1/2))*a^2*c^3-2*f^3/d^5/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)
*d)^(1/2))*b^2*c^5-4/3/d^2*(f*x+e)^(3/2)*a*b*c*e-4/d^2*a*b*c*e^2*(f*x+e)^(1/2)+2
/d^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b^2*c^2*e^3
+2/d*a^2*e^2*(f*x+e)^(1/2)-12*f^2/d^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d
/((c*f-d*e)*d)^(1/2))*a*b*c^3*e+2/3/d^3*(f*x+e)^(3/2)*b^2*c^2*e+2/5/d^3*(f*x+e)^
(5/2)*b^2*c^2+2/3/d*(f*x+e)^(3/2)*a^2*e+2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/
2)*d/((c*f-d*e)*d)^(1/2))*a^2*e^3-6*f/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)
*d/((c*f-d*e)*d)^(1/2))*a^2*c*e^2-4/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d
/((c*f-d*e)*d)^(1/2))*a*b*c*e^3+8*f/d^3*a*b*c^2*e*(f*x+e)^(1/2)+4*f^3/d^4/((c*f-
d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a*b*c^4+6*f^2/d^4/((c*
f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b^2*c^4*e-6*f/d^3/((
c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b^2*c^3*e^2+6*f^2/
d^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a^2*c^2*e+2/
9*b^2*(f*x+e)^(9/2)/d/f^2+12*f/d^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((
c*f-d*e)*d)^(1/2))*a*b*c^2*e^2+2/5/d*(f*x+e)^(5/2)*a^2+4/3*f/d^3*(f*x+e)^(3/2)*a
*b*c^2+2/d^3*b^2*c^2*e^2*(f*x+e)^(1/2)-2/7/f^2/d*(f*x+e)^(7/2)*b^2*e+4/7/f/d*(f*
x+e)^(7/2)*a*b-2/7/f/d^2*(f*x+e)^(7/2)*b^2*c-2/3*f/d^2*(f*x+e)^(3/2)*a^2*c-2/3*f
/d^4*(f*x+e)^(3/2)*b^2*c^3+2*f^2/d^3*a^2*c^2*(f*x+e)^(1/2)-4/5/d^2*(f*x+e)^(5/2)
*a*b*c+2*f^2/d^5*b^2*c^4*(f*x+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((b*x + a)^2*(f*x + e)^(5/2)/(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230956, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/315*(315*((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e^2*f^2 - 2*(b^2*c^3*d - 2*a*
b*c^2*d^2 + a^2*c*d^3)*e*f^3 + (b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^4)*sqrt((
d*e - c*f)/d)*log((d*f*x + 2*d*e - c*f - 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/
(d*x + c)) + 2*(35*b^2*d^4*f^4*x^4 - 10*b^2*d^4*e^4 - 45*(b^2*c*d^3 - 2*a*b*d^4)
*e^3*f + 483*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e^2*f^2 - 735*(b^2*c^3*d - 2*
a*b*c^2*d^2 + a^2*c*d^3)*e*f^3 + 315*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^4 +
5*(19*b^2*d^4*e*f^3 - 9*(b^2*c*d^3 - 2*a*b*d^4)*f^4)*x^3 + 3*(25*b^2*d^4*e^2*f^
2 - 45*(b^2*c*d^3 - 2*a*b*d^4)*e*f^3 + 21*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*
f^4)*x^2 + (5*b^2*d^4*e^3*f - 135*(b^2*c*d^3 - 2*a*b*d^4)*e^2*f^2 + 231*(b^2*c^2
*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e*f^3 - 105*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3
)*f^4)*x)*sqrt(f*x + e))/(d^5*f^2), -2/315*(315*((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^
2*d^4)*e^2*f^2 - 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*e*f^3 + (b^2*c^4 - 2*
a*b*c^3*d + a^2*c^2*d^2)*f^4)*sqrt(-(d*e - c*f)/d)*arctan(sqrt(f*x + e)/sqrt(-(d
*e - c*f)/d)) - (35*b^2*d^4*f^4*x^4 - 10*b^2*d^4*e^4 - 45*(b^2*c*d^3 - 2*a*b*d^4
)*e^3*f + 483*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e^2*f^2 - 735*(b^2*c^3*d - 2
*a*b*c^2*d^2 + a^2*c*d^3)*e*f^3 + 315*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^4
+ 5*(19*b^2*d^4*e*f^3 - 9*(b^2*c*d^3 - 2*a*b*d^4)*f^4)*x^3 + 3*(25*b^2*d^4*e^2*f
^2 - 45*(b^2*c*d^3 - 2*a*b*d^4)*e*f^3 + 21*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)
*f^4)*x^2 + (5*b^2*d^4*e^3*f - 135*(b^2*c*d^3 - 2*a*b*d^4)*e^2*f^2 + 231*(b^2*c^
2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e*f^3 - 105*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^
3)*f^4)*x)*sqrt(f*x + e))/(d^5*f^2)]

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Sympy [A]  time = 138.01, size = 493, normalized size = 2.37 \[ \frac{2 b^{2} \left (e + f x\right )^{\frac{9}{2}}}{9 d f^{2}} + \frac{\left (e + f x\right )^{\frac{7}{2}} \left (4 a b d f - 2 b^{2} c f - 2 b^{2} d e\right )}{7 d^{2} f^{2}} + \frac{\left (e + f x\right )^{\frac{5}{2}} \left (2 a^{2} d^{2} - 4 a b c d + 2 b^{2} c^{2}\right )}{5 d^{3}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (- 2 a^{2} c d^{2} f + 2 a^{2} d^{3} e + 4 a b c^{2} d f - 4 a b c d^{2} e - 2 b^{2} c^{3} f + 2 b^{2} c^{2} d e\right )}{3 d^{4}} + \frac{\sqrt{e + f x} \left (2 a^{2} c^{2} d^{2} f^{2} - 4 a^{2} c d^{3} e f + 2 a^{2} d^{4} e^{2} - 4 a b c^{3} d f^{2} + 8 a b c^{2} d^{2} e f - 4 a b c d^{3} e^{2} + 2 b^{2} c^{4} f^{2} - 4 b^{2} c^{3} d e f + 2 b^{2} c^{2} d^{2} e^{2}\right )}{d^{5}} - \frac{2 \left (a d - b c\right )^{2} \left (c f - d e\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}}} & \text{for}\: \frac{c f - d e}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: e + f x > \frac{- c f + d e}{d} \wedge \frac{c f - d e}{d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: \frac{c f - d e}{d} < 0 \wedge e + f x < \frac{- c f + d e}{d} \end{cases}\right )}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e + f*x)**(9/2)/(9*d*f**2) + (e + f*x)**(7/2)*(4*a*b*d*f - 2*b**2*c*f -
2*b**2*d*e)/(7*d**2*f**2) + (e + f*x)**(5/2)*(2*a**2*d**2 - 4*a*b*c*d + 2*b**2*c
**2)/(5*d**3) + (e + f*x)**(3/2)*(-2*a**2*c*d**2*f + 2*a**2*d**3*e + 4*a*b*c**2*
d*f - 4*a*b*c*d**2*e - 2*b**2*c**3*f + 2*b**2*c**2*d*e)/(3*d**4) + sqrt(e + f*x)
*(2*a**2*c**2*d**2*f**2 - 4*a**2*c*d**3*e*f + 2*a**2*d**4*e**2 - 4*a*b*c**3*d*f*
*2 + 8*a*b*c**2*d**2*e*f - 4*a*b*c*d**3*e**2 + 2*b**2*c**4*f**2 - 4*b**2*c**3*d*
e*f + 2*b**2*c**2*d**2*e**2)/d**5 - 2*(a*d - b*c)**2*(c*f - d*e)**3*Piecewise((a
tan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d*sqrt((c*f - d*e)/d)), (c*f - d*e)/d >
0), (-acoth(sqrt(e + f*x)/sqrt((-c*f + d*e)/d))/(d*sqrt((-c*f + d*e)/d)), ((c*f
- d*e)/d < 0) & (e + f*x > (-c*f + d*e)/d)), (-atanh(sqrt(e + f*x)/sqrt((-c*f +
d*e)/d))/(d*sqrt((-c*f + d*e)/d)), ((c*f - d*e)/d < 0) & (e + f*x < (-c*f + d*e)
/d)))/d**5

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GIAC/XCAS [A]  time = 0.231479, size = 909, normalized size = 4.37 \[ -\frac{2 \,{\left (b^{2} c^{5} f^{3} - 2 \, a b c^{4} d f^{3} + a^{2} c^{3} d^{2} f^{3} - 3 \, b^{2} c^{4} d f^{2} e + 6 \, a b c^{3} d^{2} f^{2} e - 3 \, a^{2} c^{2} d^{3} f^{2} e + 3 \, b^{2} c^{3} d^{2} f e^{2} - 6 \, a b c^{2} d^{3} f e^{2} + 3 \, a^{2} c d^{4} f e^{2} - b^{2} c^{2} d^{3} e^{3} + 2 \, a b c d^{4} e^{3} - a^{2} d^{5} e^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{5}} + \frac{2 \,{\left (35 \,{\left (f x + e\right )}^{\frac{9}{2}} b^{2} d^{8} f^{16} - 45 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{2} c d^{7} f^{17} + 90 \,{\left (f x + e\right )}^{\frac{7}{2}} a b d^{8} f^{17} + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{2} c^{2} d^{6} f^{18} - 126 \,{\left (f x + e\right )}^{\frac{5}{2}} a b c d^{7} f^{18} + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} a^{2} d^{8} f^{18} - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{2} c^{3} d^{5} f^{19} + 210 \,{\left (f x + e\right )}^{\frac{3}{2}} a b c^{2} d^{6} f^{19} - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{2} c d^{7} f^{19} + 315 \, \sqrt{f x + e} b^{2} c^{4} d^{4} f^{20} - 630 \, \sqrt{f x + e} a b c^{3} d^{5} f^{20} + 315 \, \sqrt{f x + e} a^{2} c^{2} d^{6} f^{20} - 45 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{2} d^{8} f^{16} e + 105 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{2} c^{2} d^{6} f^{18} e - 210 \,{\left (f x + e\right )}^{\frac{3}{2}} a b c d^{7} f^{18} e + 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{2} d^{8} f^{18} e - 630 \, \sqrt{f x + e} b^{2} c^{3} d^{5} f^{19} e + 1260 \, \sqrt{f x + e} a b c^{2} d^{6} f^{19} e - 630 \, \sqrt{f x + e} a^{2} c d^{7} f^{19} e + 315 \, \sqrt{f x + e} b^{2} c^{2} d^{6} f^{18} e^{2} - 630 \, \sqrt{f x + e} a b c d^{7} f^{18} e^{2} + 315 \, \sqrt{f x + e} a^{2} d^{8} f^{18} e^{2}\right )}}{315 \, d^{9} f^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b^2*c^5*f^3 - 2*a*b*c^4*d*f^3 + a^2*c^3*d^2*f^3 - 3*b^2*c^4*d*f^2*e + 6*a*b*
c^3*d^2*f^2*e - 3*a^2*c^2*d^3*f^2*e + 3*b^2*c^3*d^2*f*e^2 - 6*a*b*c^2*d^3*f*e^2
+ 3*a^2*c*d^4*f*e^2 - b^2*c^2*d^3*e^3 + 2*a*b*c*d^4*e^3 - a^2*d^5*e^3)*arctan(sq
rt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^5) + 2/315*(35*(f*x +
e)^(9/2)*b^2*d^8*f^16 - 45*(f*x + e)^(7/2)*b^2*c*d^7*f^17 + 90*(f*x + e)^(7/2)*a
*b*d^8*f^17 + 63*(f*x + e)^(5/2)*b^2*c^2*d^6*f^18 - 126*(f*x + e)^(5/2)*a*b*c*d^
7*f^18 + 63*(f*x + e)^(5/2)*a^2*d^8*f^18 - 105*(f*x + e)^(3/2)*b^2*c^3*d^5*f^19
+ 210*(f*x + e)^(3/2)*a*b*c^2*d^6*f^19 - 105*(f*x + e)^(3/2)*a^2*c*d^7*f^19 + 31
5*sqrt(f*x + e)*b^2*c^4*d^4*f^20 - 630*sqrt(f*x + e)*a*b*c^3*d^5*f^20 + 315*sqrt
(f*x + e)*a^2*c^2*d^6*f^20 - 45*(f*x + e)^(7/2)*b^2*d^8*f^16*e + 105*(f*x + e)^(
3/2)*b^2*c^2*d^6*f^18*e - 210*(f*x + e)^(3/2)*a*b*c*d^7*f^18*e + 105*(f*x + e)^(
3/2)*a^2*d^8*f^18*e - 630*sqrt(f*x + e)*b^2*c^3*d^5*f^19*e + 1260*sqrt(f*x + e)*
a*b*c^2*d^6*f^19*e - 630*sqrt(f*x + e)*a^2*c*d^7*f^19*e + 315*sqrt(f*x + e)*b^2*
c^2*d^6*f^18*e^2 - 630*sqrt(f*x + e)*a*b*c*d^7*f^18*e^2 + 315*sqrt(f*x + e)*a^2*
d^8*f^18*e^2)/(d^9*f^18)