3.19 \(\int \frac{(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=322 \[ \frac{2 \sqrt{c+d x} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^6}+\frac{2 (c+d x)^{3/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{3 d^6}-\frac{2 (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^6 \sqrt{c+d x}}-\frac{2 (b c-a d)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^6 (c+d x)^{3/2}}+\frac{2 b (c+d x)^{5/2} (2 a d D-5 b c D+b C d)}{5 d^6}+\frac{2 b^2 D (c+d x)^{7/2}}{7 d^6} \]

[Out]

(-2*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^6*(c + d*x)^(3/2)) -
(2*(b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 + 2*
A*d^3 - 5*c^3*D)))/(d^6*Sqrt[c + d*x]) + (2*(a^2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*
c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*Sqrt[
c + d*x])/d^6 + (2*(a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 1
0*c^2*D))*(c + d*x)^(3/2))/(3*d^6) + (2*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^
(5/2))/(5*d^6) + (2*b^2*D*(c + d*x)^(7/2))/(7*d^6)

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Rubi [A]  time = 0.545665, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031 \[ \frac{2 \sqrt{c+d x} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^6}+\frac{2 (c+d x)^{3/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{3 d^6}-\frac{2 (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^6 \sqrt{c+d x}}-\frac{2 (b c-a d)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^6 (c+d x)^{3/2}}+\frac{2 b (c+d x)^{5/2} (2 a d D-5 b c D+b C d)}{5 d^6}+\frac{2 b^2 D (c+d x)^{7/2}}{7 d^6} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]

[Out]

(-2*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^6*(c + d*x)^(3/2)) -
(2*(b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 + 2*
A*d^3 - 5*c^3*D)))/(d^6*Sqrt[c + d*x]) + (2*(a^2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*
c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*Sqrt[
c + d*x])/d^6 + (2*(a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 1
0*c^2*D))*(c + d*x)^(3/2))/(3*d^6) + (2*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^
(5/2))/(5*d^6) + (2*b^2*D*(c + d*x)^(7/2))/(7*d^6)

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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((b*x+a)**2*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.961516, size = 317, normalized size = 0.98 \[ \frac{-70 a^2 d^2 \left (d^3 \left (A+3 B x+x^2 (-(3 C+D x))\right )+2 c d^2 (B+3 x (D x-2 C))+16 c^3 D-8 c^2 d (C-3 D x)\right )+28 a b d \left (-2 c d^3 \left (5 A+x \left (-30 B+15 C x+4 D x^2\right )\right )+d^4 x \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+8 c^2 d^2 (5 B+3 x (2 D x-5 C))+128 c^4 D+c^3 (192 d D x-80 C d)\right )+2 b^2 \left (8 c^2 d^3 (35 A+x (2 x (21 C+5 D x)-105 B))-2 c d^4 x (x (105 B+x (28 C+15 D x))-210 A)+d^5 x^2 (105 A+x (35 B+3 x (7 C+5 D x)))-16 c^3 d^2 (35 B+6 x (5 D x-14 C))-1280 c^5 D+128 c^4 d (7 C-15 D x)\right )}{105 d^6 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]

[Out]

(-70*a^2*d^2*(16*c^3*D - 8*c^2*d*(C - 3*D*x) + 2*c*d^2*(B + 3*x*(-2*C + D*x)) +
d^3*(A + 3*B*x - x^2*(3*C + D*x))) + 28*a*b*d*(128*c^4*D + c^3*(-80*C*d + 192*d*
D*x) + 8*c^2*d^2*(5*B + 3*x*(-5*C + 2*D*x)) + d^4*x*(-15*A + x*(15*B + 5*C*x + 3
*D*x^2)) - 2*c*d^3*(5*A + x*(-30*B + 15*C*x + 4*D*x^2))) + 2*b^2*(-1280*c^5*D +
128*c^4*d*(7*C - 15*D*x) - 16*c^3*d^2*(35*B + 6*x*(-14*C + 5*D*x)) + d^5*x^2*(10
5*A + x*(35*B + 3*x*(7*C + 5*D*x))) + 8*c^2*d^3*(35*A + x*(-105*B + 2*x*(21*C +
5*D*x))) - 2*c*d^4*x*(-210*A + x*(105*B + x*(28*C + 15*D*x)))))/(105*d^6*(c + d*
x)^(3/2))

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Maple [A]  time = 0.012, size = 505, normalized size = 1.6 \[ -{\frac{-30\,{b}^{2}D{x}^{5}{d}^{5}-42\,C{b}^{2}{d}^{5}{x}^{4}-84\,Dab{d}^{5}{x}^{4}+60\,D{b}^{2}c{d}^{4}{x}^{4}-70\,B{b}^{2}{d}^{5}{x}^{3}-140\,Cab{d}^{5}{x}^{3}+112\,C{b}^{2}c{d}^{4}{x}^{3}-70\,D{a}^{2}{d}^{5}{x}^{3}+224\,Dabc{d}^{4}{x}^{3}-160\,D{b}^{2}{c}^{2}{d}^{3}{x}^{3}-210\,A{b}^{2}{d}^{5}{x}^{2}-420\,Bab{d}^{5}{x}^{2}+420\,B{b}^{2}c{d}^{4}{x}^{2}-210\,C{a}^{2}{d}^{5}{x}^{2}+840\,Cabc{d}^{4}{x}^{2}-672\,C{b}^{2}{c}^{2}{d}^{3}{x}^{2}+420\,D{a}^{2}c{d}^{4}{x}^{2}-1344\,Dab{c}^{2}{d}^{3}{x}^{2}+960\,D{b}^{2}{c}^{3}{d}^{2}{x}^{2}+420\,Aab{d}^{5}x-840\,A{b}^{2}c{d}^{4}x+210\,B{a}^{2}{d}^{5}x-1680\,Babc{d}^{4}x+1680\,B{b}^{2}{c}^{2}{d}^{3}x-840\,C{a}^{2}c{d}^{4}x+3360\,Cab{c}^{2}{d}^{3}x-2688\,C{b}^{2}{c}^{3}{d}^{2}x+1680\,D{a}^{2}{c}^{2}{d}^{3}x-5376\,Dab{c}^{3}{d}^{2}x+3840\,D{b}^{2}{c}^{4}dx+70\,{a}^{2}A{d}^{5}+280\,Aabc{d}^{4}-560\,A{b}^{2}{c}^{2}{d}^{3}+140\,B{a}^{2}c{d}^{4}-1120\,Bab{c}^{2}{d}^{3}+1120\,B{b}^{2}{c}^{3}{d}^{2}-560\,C{a}^{2}{c}^{2}{d}^{3}+2240\,Cab{c}^{3}{d}^{2}-1792\,C{b}^{2}{c}^{4}d+1120\,D{a}^{2}{c}^{3}{d}^{2}-3584\,Dab{c}^{4}d+2560\,D{b}^{2}{c}^{5}}{105\,{d}^{6}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)

[Out]

-2/105/(d*x+c)^(3/2)*(-15*D*b^2*d^5*x^5-21*C*b^2*d^5*x^4-42*D*a*b*d^5*x^4+30*D*b
^2*c*d^4*x^4-35*B*b^2*d^5*x^3-70*C*a*b*d^5*x^3+56*C*b^2*c*d^4*x^3-35*D*a^2*d^5*x
^3+112*D*a*b*c*d^4*x^3-80*D*b^2*c^2*d^3*x^3-105*A*b^2*d^5*x^2-210*B*a*b*d^5*x^2+
210*B*b^2*c*d^4*x^2-105*C*a^2*d^5*x^2+420*C*a*b*c*d^4*x^2-336*C*b^2*c^2*d^3*x^2+
210*D*a^2*c*d^4*x^2-672*D*a*b*c^2*d^3*x^2+480*D*b^2*c^3*d^2*x^2+210*A*a*b*d^5*x-
420*A*b^2*c*d^4*x+105*B*a^2*d^5*x-840*B*a*b*c*d^4*x+840*B*b^2*c^2*d^3*x-420*C*a^
2*c*d^4*x+1680*C*a*b*c^2*d^3*x-1344*C*b^2*c^3*d^2*x+840*D*a^2*c^2*d^3*x-2688*D*a
*b*c^3*d^2*x+1920*D*b^2*c^4*d*x+35*A*a^2*d^5+140*A*a*b*c*d^4-280*A*b^2*c^2*d^3+7
0*B*a^2*c*d^4-560*B*a*b*c^2*d^3+560*B*b^2*c^3*d^2-280*C*a^2*c^2*d^3+1120*C*a*b*c
^3*d^2-896*C*b^2*c^4*d+560*D*a^2*c^3*d^2-1792*D*a*b*c^4*d+1280*D*b^2*c^5)/d^6

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Maxima [A]  time = 1.36482, size = 531, normalized size = 1.65 \[ \frac{2 \,{\left (\frac{15 \,{\left (d x + c\right )}^{\frac{7}{2}} D b^{2} - 21 \,{\left (5 \, D b^{2} c -{\left (2 \, D a b + C b^{2}\right )} d\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 35 \,{\left (10 \, D b^{2} c^{2} - 4 \,{\left (2 \, D a b + C b^{2}\right )} c d +{\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} - 105 \,{\left (10 \, D b^{2} c^{3} - 6 \,{\left (2 \, D a b + C b^{2}\right )} c^{2} d + 3 \,{\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{2} -{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{3}\right )} \sqrt{d x + c}}{d^{5}} + \frac{35 \,{\left (D b^{2} c^{5} - A a^{2} d^{5} -{\left (2 \, D a b + C b^{2}\right )} c^{4} d +{\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} -{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} +{\left (B a^{2} + 2 \, A a b\right )} c d^{4} - 3 \,{\left (5 \, D b^{2} c^{4} - 4 \,{\left (2 \, D a b + C b^{2}\right )} c^{3} d + 3 \,{\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{2} - 2 \,{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{3} +{\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )}{\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{5}}\right )}}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)^2/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

2/105*((15*(d*x + c)^(7/2)*D*b^2 - 21*(5*D*b^2*c - (2*D*a*b + C*b^2)*d)*(d*x + c
)^(5/2) + 35*(10*D*b^2*c^2 - 4*(2*D*a*b + C*b^2)*c*d + (D*a^2 + 2*C*a*b + B*b^2)
*d^2)*(d*x + c)^(3/2) - 105*(10*D*b^2*c^3 - 6*(2*D*a*b + C*b^2)*c^2*d + 3*(D*a^2
+ 2*C*a*b + B*b^2)*c*d^2 - (C*a^2 + 2*B*a*b + A*b^2)*d^3)*sqrt(d*x + c))/d^5 +
35*(D*b^2*c^5 - A*a^2*d^5 - (2*D*a*b + C*b^2)*c^4*d + (D*a^2 + 2*C*a*b + B*b^2)*
c^3*d^2 - (C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 + (B*a^2 + 2*A*a*b)*c*d^4 - 3*(5*D*b
^2*c^4 - 4*(2*D*a*b + C*b^2)*c^3*d + 3*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^2 - 2*(C*
a^2 + 2*B*a*b + A*b^2)*c*d^3 + (B*a^2 + 2*A*a*b)*d^4)*(d*x + c))/((d*x + c)^(3/2
)*d^5))/d

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Fricas [A]  time = 0.222851, size = 536, normalized size = 1.66 \[ \frac{2 \,{\left (15 \, D b^{2} d^{5} x^{5} - 1280 \, D b^{2} c^{5} - 35 \, A a^{2} d^{5} + 896 \,{\left (2 \, D a b + C b^{2}\right )} c^{4} d - 560 \,{\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} + 280 \,{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} - 70 \,{\left (B a^{2} + 2 \, A a b\right )} c d^{4} - 3 \,{\left (10 \, D b^{2} c d^{4} - 7 \,{\left (2 \, D a b + C b^{2}\right )} d^{5}\right )} x^{4} +{\left (80 \, D b^{2} c^{2} d^{3} - 56 \,{\left (2 \, D a b + C b^{2}\right )} c d^{4} + 35 \,{\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{5}\right )} x^{3} - 3 \,{\left (160 \, D b^{2} c^{3} d^{2} - 112 \,{\left (2 \, D a b + C b^{2}\right )} c^{2} d^{3} + 70 \,{\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{4} - 35 \,{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{5}\right )} x^{2} - 3 \,{\left (640 \, D b^{2} c^{4} d - 448 \,{\left (2 \, D a b + C b^{2}\right )} c^{3} d^{2} + 280 \,{\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{3} - 140 \,{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{4} + 35 \,{\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} x\right )}}{105 \,{\left (d^{7} x + c d^{6}\right )} \sqrt{d x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)^2/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

2/105*(15*D*b^2*d^5*x^5 - 1280*D*b^2*c^5 - 35*A*a^2*d^5 + 896*(2*D*a*b + C*b^2)*
c^4*d - 560*(D*a^2 + 2*C*a*b + B*b^2)*c^3*d^2 + 280*(C*a^2 + 2*B*a*b + A*b^2)*c^
2*d^3 - 70*(B*a^2 + 2*A*a*b)*c*d^4 - 3*(10*D*b^2*c*d^4 - 7*(2*D*a*b + C*b^2)*d^5
)*x^4 + (80*D*b^2*c^2*d^3 - 56*(2*D*a*b + C*b^2)*c*d^4 + 35*(D*a^2 + 2*C*a*b + B
*b^2)*d^5)*x^3 - 3*(160*D*b^2*c^3*d^2 - 112*(2*D*a*b + C*b^2)*c^2*d^3 + 70*(D*a^
2 + 2*C*a*b + B*b^2)*c*d^4 - 35*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*x^2 - 3*(640*D*b^
2*c^4*d - 448*(2*D*a*b + C*b^2)*c^3*d^2 + 280*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^3
- 140*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 35*(B*a^2 + 2*A*a*b)*d^5)*x)/((d^7*x + c
*d^6)*sqrt(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{2} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)

[Out]

Integral((a + b*x)**2*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**(5/2), x)

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GIAC/XCAS [A]  time = 0.218949, size = 840, normalized size = 2.61 \[ -\frac{2 \,{\left (15 \,{\left (d x + c\right )} D b^{2} c^{4} - D b^{2} c^{5} - 24 \,{\left (d x + c\right )} D a b c^{3} d - 12 \,{\left (d x + c\right )} C b^{2} c^{3} d + 2 \, D a b c^{4} d + C b^{2} c^{4} d + 9 \,{\left (d x + c\right )} D a^{2} c^{2} d^{2} + 18 \,{\left (d x + c\right )} C a b c^{2} d^{2} + 9 \,{\left (d x + c\right )} B b^{2} c^{2} d^{2} - D a^{2} c^{3} d^{2} - 2 \, C a b c^{3} d^{2} - B b^{2} c^{3} d^{2} - 6 \,{\left (d x + c\right )} C a^{2} c d^{3} - 12 \,{\left (d x + c\right )} B a b c d^{3} - 6 \,{\left (d x + c\right )} A b^{2} c d^{3} + C a^{2} c^{2} d^{3} + 2 \, B a b c^{2} d^{3} + A b^{2} c^{2} d^{3} + 3 \,{\left (d x + c\right )} B a^{2} d^{4} + 6 \,{\left (d x + c\right )} A a b d^{4} - B a^{2} c d^{4} - 2 \, A a b c d^{4} + A a^{2} d^{5}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{6}} + \frac{2 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} D b^{2} d^{36} - 105 \,{\left (d x + c\right )}^{\frac{5}{2}} D b^{2} c d^{36} + 350 \,{\left (d x + c\right )}^{\frac{3}{2}} D b^{2} c^{2} d^{36} - 1050 \, \sqrt{d x + c} D b^{2} c^{3} d^{36} + 42 \,{\left (d x + c\right )}^{\frac{5}{2}} D a b d^{37} + 21 \,{\left (d x + c\right )}^{\frac{5}{2}} C b^{2} d^{37} - 280 \,{\left (d x + c\right )}^{\frac{3}{2}} D a b c d^{37} - 140 \,{\left (d x + c\right )}^{\frac{3}{2}} C b^{2} c d^{37} + 1260 \, \sqrt{d x + c} D a b c^{2} d^{37} + 630 \, \sqrt{d x + c} C b^{2} c^{2} d^{37} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} D a^{2} d^{38} + 70 \,{\left (d x + c\right )}^{\frac{3}{2}} C a b d^{38} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} B b^{2} d^{38} - 315 \, \sqrt{d x + c} D a^{2} c d^{38} - 630 \, \sqrt{d x + c} C a b c d^{38} - 315 \, \sqrt{d x + c} B b^{2} c d^{38} + 105 \, \sqrt{d x + c} C a^{2} d^{39} + 210 \, \sqrt{d x + c} B a b d^{39} + 105 \, \sqrt{d x + c} A b^{2} d^{39}\right )}}{105 \, d^{42}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)^2/(d*x + c)^(5/2),x, algorithm="giac")

[Out]

-2/3*(15*(d*x + c)*D*b^2*c^4 - D*b^2*c^5 - 24*(d*x + c)*D*a*b*c^3*d - 12*(d*x +
c)*C*b^2*c^3*d + 2*D*a*b*c^4*d + C*b^2*c^4*d + 9*(d*x + c)*D*a^2*c^2*d^2 + 18*(d
*x + c)*C*a*b*c^2*d^2 + 9*(d*x + c)*B*b^2*c^2*d^2 - D*a^2*c^3*d^2 - 2*C*a*b*c^3*
d^2 - B*b^2*c^3*d^2 - 6*(d*x + c)*C*a^2*c*d^3 - 12*(d*x + c)*B*a*b*c*d^3 - 6*(d*
x + c)*A*b^2*c*d^3 + C*a^2*c^2*d^3 + 2*B*a*b*c^2*d^3 + A*b^2*c^2*d^3 + 3*(d*x +
c)*B*a^2*d^4 + 6*(d*x + c)*A*a*b*d^4 - B*a^2*c*d^4 - 2*A*a*b*c*d^4 + A*a^2*d^5)/
((d*x + c)^(3/2)*d^6) + 2/105*(15*(d*x + c)^(7/2)*D*b^2*d^36 - 105*(d*x + c)^(5/
2)*D*b^2*c*d^36 + 350*(d*x + c)^(3/2)*D*b^2*c^2*d^36 - 1050*sqrt(d*x + c)*D*b^2*
c^3*d^36 + 42*(d*x + c)^(5/2)*D*a*b*d^37 + 21*(d*x + c)^(5/2)*C*b^2*d^37 - 280*(
d*x + c)^(3/2)*D*a*b*c*d^37 - 140*(d*x + c)^(3/2)*C*b^2*c*d^37 + 1260*sqrt(d*x +
c)*D*a*b*c^2*d^37 + 630*sqrt(d*x + c)*C*b^2*c^2*d^37 + 35*(d*x + c)^(3/2)*D*a^2
*d^38 + 70*(d*x + c)^(3/2)*C*a*b*d^38 + 35*(d*x + c)^(3/2)*B*b^2*d^38 - 315*sqrt
(d*x + c)*D*a^2*c*d^38 - 630*sqrt(d*x + c)*C*a*b*c*d^38 - 315*sqrt(d*x + c)*B*b^
2*c*d^38 + 105*sqrt(d*x + c)*C*a^2*d^39 + 210*sqrt(d*x + c)*B*a*b*d^39 + 105*sqr
t(d*x + c)*A*b^2*d^39)/d^42