3.1 \(\int \frac{(a+b x)^3 (A+B x+C x^2+D x^3)}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=436 \[ \frac{2 (c+d x)^{7/2} \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )}{7 d^7}-\frac{2 (c+d x)^{5/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right )\right )}{5 d^7}+\frac{2 b (c+d x)^{9/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{9 d^7}-\frac{2 (c+d x)^{3/2} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right )\right )}{3 d^7}-\frac{2 \sqrt{c+d x} (b c-a d)^3 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^7}+\frac{2 b^2 (c+d x)^{11/2} (3 a d D-6 b c D+b C d)}{11 d^7}+\frac{2 b^3 D (c+d x)^{13/2}}{13 d^7} \]

[Out]

(-2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^7 - (2*(b*c - a*d)^2*(a*d*(2*c*C*d - B*
d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*(c + d*x)^(3/2))/(3*d^7) - (2*(b*c - a*d)*(a^2
*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))
*(c + d*x)^(5/2))/(5*d^7) + (2*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2*D)
+ b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(7/2))/(7*d^7) + (2*b*(3*a^2*d^2*D + 3*a*b*d*(C*
d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(9/2))/(9*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(
c + d*x)^(11/2))/(11*d^7) + (2*b^3*D*(c + d*x)^(13/2))/(13*d^7)

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Rubi [A]  time = 0.407548, antiderivative size = 436, 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, Rules used = {1620} \[ \frac{2 (c+d x)^{7/2} \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )}{7 d^7}-\frac{2 (c+d x)^{5/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right )\right )}{5 d^7}+\frac{2 b (c+d x)^{9/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{9 d^7}-\frac{2 (c+d x)^{3/2} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right )\right )}{3 d^7}-\frac{2 \sqrt{c+d x} (b c-a d)^3 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^7}+\frac{2 b^2 (c+d x)^{11/2} (3 a d D-6 b c D+b C d)}{11 d^7}+\frac{2 b^3 D (c+d x)^{13/2}}{13 d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^7 - (2*(b*c - a*d)^2*(a*d*(2*c*C*d - B*
d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*(c + d*x)^(3/2))/(3*d^7) - (2*(b*c - a*d)*(a^2
*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))
*(c + d*x)^(5/2))/(5*d^7) + (2*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2*D)
+ b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(7/2))/(7*d^7) + (2*b*(3*a^2*d^2*D + 3*a*b*d*(C*
d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(9/2))/(9*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(
c + d*x)^(11/2))/(11*d^7) + (2*b^3*D*(c + d*x)^(13/2))/(13*d^7)

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \frac{(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt{c+d x}} \, dx &=\int \left (\frac{(-b c+a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^6 \sqrt{c+d x}}+\frac{(b c-a d)^2 \left (-a d \left (2 c C d-B d^2-3 c^2 D\right )+b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) \sqrt{c+d x}}{d^6}+\frac{(b c-a d) \left (-a^2 d^2 (C d-3 c D)+a b d \left (8 c C d-3 B d^2-15 c^2 D\right )-b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{3/2}}{d^6}+\frac{\left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{5/2}}{d^6}+\frac{b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{7/2}}{d^6}+\frac{b^2 (b C d-6 b c D+3 a d D) (c+d x)^{9/2}}{d^6}+\frac{b^3 D (c+d x)^{11/2}}{d^6}\right ) \, dx\\ &=-\frac{2 (b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt{c+d x}}{d^7}-\frac{2 (b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) (c+d x)^{3/2}}{3 d^7}-\frac{2 (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{5/2}}{5 d^7}+\frac{2 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{7/2}}{7 d^7}+\frac{2 b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{9/2}}{9 d^7}+\frac{2 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{11/2}}{11 d^7}+\frac{2 b^3 D (c+d x)^{13/2}}{13 d^7}\\ \end{align*}

Mathematica [A]  time = 1.23082, size = 391, normalized size = 0.9 \[ \frac{2 \sqrt{c+d x} \left (6435 (c+d x)^3 \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D+3 a b^2 d \left (B d^2+10 c^2 D-4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )-9009 (c+d x)^2 (b c-a d) \left (a^2 d^2 (C d-3 c D)+a b d \left (3 B d^2+15 c^2 D-8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right )\right )+5005 b (c+d x)^4 \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (B d^2+15 c^2 D-5 c C d\right )\right )-15015 (c+d x) (b c-a d)^2 \left (b \left (-3 A d^3+4 B c d^2-5 c^2 C d+6 c^3 D\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )+45045 (b c-a d)^3 \left (-A d^3+B c d^2-c^2 C d+c^3 D\right )+4095 b^2 (c+d x)^5 (3 a d D-6 b c D+b C d)+3465 b^3 D (c+d x)^6\right )}{45045 d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(45045*(b*c - a*d)^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D) - 15015*(b*c - a*d)^2*(-(a*d*(-2*
c*C*d + B*d^2 + 3*c^2*D)) + b*(-5*c^2*C*d + 4*B*c*d^2 - 3*A*d^3 + 6*c^3*D))*(c + d*x) - 9009*(b*c - a*d)*(a^2*
d^2*(C*d - 3*c*D) + a*b*d*(-8*c*C*d + 3*B*d^2 + 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))
*(c + d*x)^2 + 6435*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) + 3*a*b^2*d*(-4*c*C*d + B*d^2 + 10*c^2*D) + b^3*(10
*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^3 + 5005*b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D) + b^2*(-5*
c*C*d + B*d^2 + 15*c^2*D))*(c + d*x)^4 + 4095*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^5 + 3465*b^3*D*(c + d*
x)^6))/(45045*d^7)

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Maple [B]  time = 0.008, size = 841, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x)

[Out]

2/45045*(d*x+c)^(1/2)*(3465*D*b^3*d^6*x^6+4095*C*b^3*d^6*x^5+12285*D*a*b^2*d^6*x^5-3780*D*b^3*c*d^5*x^5+5005*B
*b^3*d^6*x^4+15015*C*a*b^2*d^6*x^4-4550*C*b^3*c*d^5*x^4+15015*D*a^2*b*d^6*x^4-13650*D*a*b^2*c*d^5*x^4+4200*D*b
^3*c^2*d^4*x^4+6435*A*b^3*d^6*x^3+19305*B*a*b^2*d^6*x^3-5720*B*b^3*c*d^5*x^3+19305*C*a^2*b*d^6*x^3-17160*C*a*b
^2*c*d^5*x^3+5200*C*b^3*c^2*d^4*x^3+6435*D*a^3*d^6*x^3-17160*D*a^2*b*c*d^5*x^3+15600*D*a*b^2*c^2*d^4*x^3-4800*
D*b^3*c^3*d^3*x^3+27027*A*a*b^2*d^6*x^2-7722*A*b^3*c*d^5*x^2+27027*B*a^2*b*d^6*x^2-23166*B*a*b^2*c*d^5*x^2+686
4*B*b^3*c^2*d^4*x^2+9009*C*a^3*d^6*x^2-23166*C*a^2*b*c*d^5*x^2+20592*C*a*b^2*c^2*d^4*x^2-6240*C*b^3*c^3*d^3*x^
2-7722*D*a^3*c*d^5*x^2+20592*D*a^2*b*c^2*d^4*x^2-18720*D*a*b^2*c^3*d^3*x^2+5760*D*b^3*c^4*d^2*x^2+45045*A*a^2*
b*d^6*x-36036*A*a*b^2*c*d^5*x+10296*A*b^3*c^2*d^4*x+15015*B*a^3*d^6*x-36036*B*a^2*b*c*d^5*x+30888*B*a*b^2*c^2*
d^4*x-9152*B*b^3*c^3*d^3*x-12012*C*a^3*c*d^5*x+30888*C*a^2*b*c^2*d^4*x-27456*C*a*b^2*c^3*d^3*x+8320*C*b^3*c^4*
d^2*x+10296*D*a^3*c^2*d^4*x-27456*D*a^2*b*c^3*d^3*x+24960*D*a*b^2*c^4*d^2*x-7680*D*b^3*c^5*d*x+45045*A*a^3*d^6
-90090*A*a^2*b*c*d^5+72072*A*a*b^2*c^2*d^4-20592*A*b^3*c^3*d^3-30030*B*a^3*c*d^5+72072*B*a^2*b*c^2*d^4-61776*B
*a*b^2*c^3*d^3+18304*B*b^3*c^4*d^2+24024*C*a^3*c^2*d^4-61776*C*a^2*b*c^3*d^3+54912*C*a*b^2*c^4*d^2-16640*C*b^3
*c^5*d-20592*D*a^3*c^3*d^3+54912*D*a^2*b*c^4*d^2-49920*D*a*b^2*c^5*d+15360*D*b^3*c^6)/d^7

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Maxima [A]  time = 2.79667, size = 838, normalized size = 1.92 \begin{align*} \frac{2 \,{\left (3465 \,{\left (d x + c\right )}^{\frac{13}{2}} D b^{3} - 4095 \,{\left (6 \, D b^{3} c -{\left (3 \, D a b^{2} + C b^{3}\right )} d\right )}{\left (d x + c\right )}^{\frac{11}{2}} + 5005 \,{\left (15 \, D b^{3} c^{2} - 5 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c d +{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )}{\left (d x + c\right )}^{\frac{9}{2}} - 6435 \,{\left (20 \, D b^{3} c^{3} - 10 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d + 4 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{2} -{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 9009 \,{\left (15 \, D b^{3} c^{4} - 10 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d + 6 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{2} - 3 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )}{\left (d x + c\right )}^{\frac{5}{2}} - 15015 \,{\left (6 \, D b^{3} c^{5} - 5 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d + 4 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} - 3 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} + 2 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{4} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 45045 \,{\left (D b^{3} c^{6} + A a^{3} d^{6} -{\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d +{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} -{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} -{\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5}\right )} \sqrt{d x + c}\right )}}{45045 \, d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3465*(d*x + c)^(13/2)*D*b^3 - 4095*(6*D*b^3*c - (3*D*a*b^2 + C*b^3)*d)*(d*x + c)^(11/2) + 5005*(15*D*
b^3*c^2 - 5*(3*D*a*b^2 + C*b^3)*c*d + (3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^2)*(d*x + c)^(9/2) - 6435*(20*D*b^3*c^
3 - 10*(3*D*a*b^2 + C*b^3)*c^2*d + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^2 - (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 +
A*b^3)*d^3)*(d*x + c)^(7/2) + 9009*(15*D*b^3*c^4 - 10*(3*D*a*b^2 + C*b^3)*c^3*d + 6*(3*D*a^2*b + 3*C*a*b^2 + B
*b^3)*c^2*d^2 - 3*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c*d^3 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^4)*(d*x +
c)^(5/2) - 15015*(6*D*b^3*c^5 - 5*(3*D*a*b^2 + C*b^3)*c^4*d + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^3*d^2 - 3*(D
*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^3 + 2*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^4 - (B*a^3 + 3*A*a^2*b)*
d^5)*(d*x + c)^(3/2) + 45045*(D*b^3*c^6 + A*a^3*d^6 - (3*D*a*b^2 + C*b^3)*c^5*d + (3*D*a^2*b + 3*C*a*b^2 + B*b
^3)*c^4*d^2 - (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^4 - (B*a
^3 + 3*A*a^2*b)*c*d^5)*sqrt(d*x + c))/d^7

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.86737, size = 1153, normalized size = 2.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/45045*(45045*sqrt(d*x + c)*A*a^3 + 15015*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B*a^3/d + 45045*((d*x + c)^(3
/2) - 3*sqrt(d*x + c)*c)*A*a^2*b/d + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*C*
a^3/d^2 + 9009*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*B*a^2*b/d^2 + 9009*(3*(d*x +
c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*A*a*b^2/d^2 + 1287*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(
5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*D*a^3/d^3 + 3861*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/
2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*C*a^2*b/d^3 + 3861*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/
2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*B*a*b^2/d^3 + 1287*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/
2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*A*b^3/d^3 + 429*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2
)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*D*a^2*b/d^4 + 429*(35*(d*x +
c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*
C*a*b^2/d^4 + 143*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*
c^3 + 315*sqrt(d*x + c)*c^4)*B*b^3/d^4 + 195*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2
)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*D*a*b^2/d^5 + 65*(63*(d*x
+ c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/
2)*c^4 - 693*sqrt(d*x + c)*c^5)*C*b^3/d^5 + 15*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c
)^(9/2)*c^2 - 8580*(d*x + c)^(7/2)*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3003*sqrt(d*x +
c)*c^6)*D*b^3/d^6)/d