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

Optimal. Leaf size=322 \[ \frac{2 (c+d x)^{3/2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )}{3 d^6}+\frac{2 (c+d x)^{5/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{5 d^6}+\frac{2 \sqrt{c+d x} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right )\right )}{d^6}-\frac{2 (b c-a d)^2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^6 \sqrt{c+d x}}+\frac{2 b (c+d x)^{7/2} (2 a d D-5 b c D+b C d)}{7 d^6}+\frac{2 b^2 D (c+d x)^{9/2}}{9 d^6} \]

[Out]

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

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Rubi [A]  time = 0.258515, 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, Rules used = {1620} \[ \frac{2 (c+d x)^{3/2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )}{3 d^6}+\frac{2 (c+d x)^{5/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{5 d^6}+\frac{2 \sqrt{c+d x} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right )\right )}{d^6}-\frac{2 (b c-a d)^2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^6 \sqrt{c+d x}}+\frac{2 b (c+d x)^{7/2} (2 a d D-5 b c D+b C d)}{7 d^6}+\frac{2 b^2 D (c+d x)^{9/2}}{9 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.635643, size = 287, normalized size = 0.89 \[ \frac{2 \left (105 (c+d x)^2 \left (a^2 d^2 (C d-3 c D)+2 a b d \left (B d^2+6 c^2 D-3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )+63 (c+d x)^3 \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (B d^2+10 c^2 D-4 c C d\right )\right )+315 (c+d x) (b c-a d) \left (b \left (-2 A d^3+3 B c d^2-4 c^2 C d+5 c^3 D\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )+315 (b c-a d)^2 \left (-A d^3+B c d^2-c^2 C d+c^3 D\right )+45 b (c+d x)^4 (2 a d D-5 b c D+b C d)+35 b^2 D (c+d x)^5\right )}{315 d^6 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(315*(b*c - a*d)^2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D) + 315*(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))*(c + d*x) + 105*(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))*(c + d*x)^2 + 63*(a^2*d^2*D + 2*a*b*
d*(C*d - 4*c*D) + b^2*(-4*c*C*d + B*d^2 + 10*c^2*D))*(c + d*x)^3 + 45*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^
4 + 35*b^2*D*(c + d*x)^5))/(315*d^6*Sqrt[c + d*x])

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Maple [A]  time = 0.009, size = 505, normalized size = 1.6 \begin{align*} -{\frac{-70\,{b}^{2}D{x}^{5}{d}^{5}-90\,C{b}^{2}{d}^{5}{x}^{4}-180\,Dab{d}^{5}{x}^{4}+100\,D{b}^{2}c{d}^{4}{x}^{4}-126\,B{b}^{2}{d}^{5}{x}^{3}-252\,Cab{d}^{5}{x}^{3}+144\,C{b}^{2}c{d}^{4}{x}^{3}-126\,D{a}^{2}{d}^{5}{x}^{3}+288\,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}+252\,B{b}^{2}c{d}^{4}{x}^{2}-210\,C{a}^{2}{d}^{5}{x}^{2}+504\,Cabc{d}^{4}{x}^{2}-288\,C{b}^{2}{c}^{2}{d}^{3}{x}^{2}+252\,D{a}^{2}c{d}^{4}{x}^{2}-576\,Dab{c}^{2}{d}^{3}{x}^{2}+320\,D{b}^{2}{c}^{3}{d}^{2}{x}^{2}-1260\,Aab{d}^{5}x+840\,A{b}^{2}c{d}^{4}x-630\,B{a}^{2}{d}^{5}x+1680\,Babc{d}^{4}x-1008\,B{b}^{2}{c}^{2}{d}^{3}x+840\,C{a}^{2}c{d}^{4}x-2016\,Cab{c}^{2}{d}^{3}x+1152\,C{b}^{2}{c}^{3}{d}^{2}x-1008\,D{a}^{2}{c}^{2}{d}^{3}x+2304\,Dab{c}^{3}{d}^{2}x-1280\,D{b}^{2}{c}^{4}dx+630\,{a}^{2}A{d}^{5}-2520\,Aabc{d}^{4}+1680\,A{b}^{2}{c}^{2}{d}^{3}-1260\,B{a}^{2}c{d}^{4}+3360\,Bab{c}^{2}{d}^{3}-2016\,B{b}^{2}{c}^{3}{d}^{2}+1680\,C{a}^{2}{c}^{2}{d}^{3}-4032\,Cab{c}^{3}{d}^{2}+2304\,C{b}^{2}{c}^{4}d-2016\,D{a}^{2}{c}^{3}{d}^{2}+4608\,Dab{c}^{4}d-2560\,D{b}^{2}{c}^{5}}{315\,{d}^{6}}{\frac{1}{\sqrt{dx+c}}}} \end{align*}

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)^(3/2),x)

[Out]

-2/315/(d*x+c)^(1/2)*(-35*D*b^2*d^5*x^5-45*C*b^2*d^5*x^4-90*D*a*b*d^5*x^4+50*D*b^2*c*d^4*x^4-63*B*b^2*d^5*x^3-
126*C*a*b*d^5*x^3+72*C*b^2*c*d^4*x^3-63*D*a^2*d^5*x^3+144*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+126*B*b^2*c*d^4*x^2-105*C*a^2*d^5*x^2+252*C*a*b*c*d^4*x^2-144*C*b^2*c^2*d^3*x^2+126*D*a^2
*c*d^4*x^2-288*D*a*b*c^2*d^3*x^2+160*D*b^2*c^3*d^2*x^2-630*A*a*b*d^5*x+420*A*b^2*c*d^4*x-315*B*a^2*d^5*x+840*B
*a*b*c*d^4*x-504*B*b^2*c^2*d^3*x+420*C*a^2*c*d^4*x-1008*C*a*b*c^2*d^3*x+576*C*b^2*c^3*d^2*x-504*D*a^2*c^2*d^3*
x+1152*D*a*b*c^3*d^2*x-640*D*b^2*c^4*d*x+315*A*a^2*d^5-1260*A*a*b*c*d^4+840*A*b^2*c^2*d^3-630*B*a^2*c*d^4+1680
*B*a*b*c^2*d^3-1008*B*b^2*c^3*d^2+840*C*a^2*c^2*d^3-2016*C*a*b*c^3*d^2+1152*C*b^2*c^4*d-1008*D*a^2*c^3*d^2+230
4*D*a*b*c^4*d-1280*D*b^2*c^5)/d^6

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Maxima [A]  time = 2.37962, size = 533, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (d x + c\right )}^{\frac{9}{2}} D b^{2} - 45 \,{\left (5 \, D b^{2} c -{\left (2 \, D a b + C b^{2}\right )} d\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 63 \,{\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{5}{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 )}{\left (d x + c\right )}^{\frac{3}{2}} + 315 \,{\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 )} \sqrt{d x + c}}{d^{5}} + \frac{315 \,{\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}\right )}}{\sqrt{d x + c} d^{5}}\right )}}{315 \, d} \end{align*}

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)^(3/2),x, algorithm="maxima")

[Out]

2/315*((35*(d*x + c)^(9/2)*D*b^2 - 45*(5*D*b^2*c - (2*D*a*b + C*b^2)*d)*(d*x + c)^(7/2) + 63*(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)^(5/2) - 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)*(d*x + c)^(3/2) + 315*(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)*sqrt(d*x + c))/d^5 + 315*(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)/(sqrt(d*x + c)*d^5))/d

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

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 107.427, size = 435, normalized size = 1.35 \begin{align*} \frac{2 D b^{2} \left (c + d x\right )^{\frac{9}{2}}}{9 d^{6}} + \frac{\left (c + d x\right )^{\frac{7}{2}} \left (2 C b^{2} d + 4 D a b d - 10 D b^{2} c\right )}{7 d^{6}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (2 B b^{2} d^{2} + 4 C a b d^{2} - 8 C b^{2} c d + 2 D a^{2} d^{2} - 16 D a b c d + 20 D b^{2} c^{2}\right )}{5 d^{6}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (2 A b^{2} d^{3} + 4 B a b d^{3} - 6 B b^{2} c d^{2} + 2 C a^{2} d^{3} - 12 C a b c d^{2} + 12 C b^{2} c^{2} d - 6 D a^{2} c d^{2} + 24 D a b c^{2} d - 20 D b^{2} c^{3}\right )}{3 d^{6}} + \frac{\sqrt{c + d x} \left (4 A a b d^{4} - 4 A b^{2} c d^{3} + 2 B a^{2} d^{4} - 8 B a b c d^{3} + 6 B b^{2} c^{2} d^{2} - 4 C a^{2} c d^{3} + 12 C a b c^{2} d^{2} - 8 C b^{2} c^{3} d + 6 D a^{2} c^{2} d^{2} - 16 D a b c^{3} d + 10 D b^{2} c^{4}\right )}{d^{6}} + \frac{2 \left (a d - b c\right )^{2} \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{6} \sqrt{c + d x}} \end{align*}

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)**(3/2),x)

[Out]

2*D*b**2*(c + d*x)**(9/2)/(9*d**6) + (c + d*x)**(7/2)*(2*C*b**2*d + 4*D*a*b*d - 10*D*b**2*c)/(7*d**6) + (c + d
*x)**(5/2)*(2*B*b**2*d**2 + 4*C*a*b*d**2 - 8*C*b**2*c*d + 2*D*a**2*d**2 - 16*D*a*b*c*d + 20*D*b**2*c**2)/(5*d*
*6) + (c + d*x)**(3/2)*(2*A*b**2*d**3 + 4*B*a*b*d**3 - 6*B*b**2*c*d**2 + 2*C*a**2*d**3 - 12*C*a*b*c*d**2 + 12*
C*b**2*c**2*d - 6*D*a**2*c*d**2 + 24*D*a*b*c**2*d - 20*D*b**2*c**3)/(3*d**6) + sqrt(c + d*x)*(4*A*a*b*d**4 - 4
*A*b**2*c*d**3 + 2*B*a**2*d**4 - 8*B*a*b*c*d**3 + 6*B*b**2*c**2*d**2 - 4*C*a**2*c*d**3 + 12*C*a*b*c**2*d**2 -
8*C*b**2*c**3*d + 6*D*a**2*c**2*d**2 - 16*D*a*b*c**3*d + 10*D*b**2*c**4)/d**6 + 2*(a*d - b*c)**2*(-A*d**3 + B*
c*d**2 - C*c**2*d + D*c**3)/(d**6*sqrt(c + d*x))

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Giac [B]  time = 2.57663, size = 879, normalized size = 2.73 \begin{align*} \frac{2 \,{\left (D b^{2} c^{5} - 2 \, D a b c^{4} d - C b^{2} c^{4} d + D a^{2} c^{3} d^{2} + 2 \, C a b c^{3} d^{2} + B b^{2} c^{3} d^{2} - C a^{2} c^{2} d^{3} - 2 \, B a b c^{2} d^{3} - A b^{2} c^{2} d^{3} + B a^{2} c d^{4} + 2 \, A a b c d^{4} - A a^{2} d^{5}\right )}}{\sqrt{d x + c} d^{6}} + \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} D b^{2} d^{48} - 225 \,{\left (d x + c\right )}^{\frac{7}{2}} D b^{2} c d^{48} + 630 \,{\left (d x + c\right )}^{\frac{5}{2}} D b^{2} c^{2} d^{48} - 1050 \,{\left (d x + c\right )}^{\frac{3}{2}} D b^{2} c^{3} d^{48} + 1575 \, \sqrt{d x + c} D b^{2} c^{4} d^{48} + 90 \,{\left (d x + c\right )}^{\frac{7}{2}} D a b d^{49} + 45 \,{\left (d x + c\right )}^{\frac{7}{2}} C b^{2} d^{49} - 504 \,{\left (d x + c\right )}^{\frac{5}{2}} D a b c d^{49} - 252 \,{\left (d x + c\right )}^{\frac{5}{2}} C b^{2} c d^{49} + 1260 \,{\left (d x + c\right )}^{\frac{3}{2}} D a b c^{2} d^{49} + 630 \,{\left (d x + c\right )}^{\frac{3}{2}} C b^{2} c^{2} d^{49} - 2520 \, \sqrt{d x + c} D a b c^{3} d^{49} - 1260 \, \sqrt{d x + c} C b^{2} c^{3} d^{49} + 63 \,{\left (d x + c\right )}^{\frac{5}{2}} D a^{2} d^{50} + 126 \,{\left (d x + c\right )}^{\frac{5}{2}} C a b d^{50} + 63 \,{\left (d x + c\right )}^{\frac{5}{2}} B b^{2} d^{50} - 315 \,{\left (d x + c\right )}^{\frac{3}{2}} D a^{2} c d^{50} - 630 \,{\left (d x + c\right )}^{\frac{3}{2}} C a b c d^{50} - 315 \,{\left (d x + c\right )}^{\frac{3}{2}} B b^{2} c d^{50} + 945 \, \sqrt{d x + c} D a^{2} c^{2} d^{50} + 1890 \, \sqrt{d x + c} C a b c^{2} d^{50} + 945 \, \sqrt{d x + c} B b^{2} c^{2} d^{50} + 105 \,{\left (d x + c\right )}^{\frac{3}{2}} C a^{2} d^{51} + 210 \,{\left (d x + c\right )}^{\frac{3}{2}} B a b d^{51} + 105 \,{\left (d x + c\right )}^{\frac{3}{2}} A b^{2} d^{51} - 630 \, \sqrt{d x + c} C a^{2} c d^{51} - 1260 \, \sqrt{d x + c} B a b c d^{51} - 630 \, \sqrt{d x + c} A b^{2} c d^{51} + 315 \, \sqrt{d x + c} B a^{2} d^{52} + 630 \, \sqrt{d x + c} A a b d^{52}\right )}}{315 \, d^{54}} \end{align*}

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)^(3/2),x, algorithm="giac")

[Out]

2*(D*b^2*c^5 - 2*D*a*b*c^4*d - C*b^2*c^4*d + D*a^2*c^3*d^2 + 2*C*a*b*c^3*d^2 + B*b^2*c^3*d^2 - C*a^2*c^2*d^3 -
2*B*a*b*c^2*d^3 - A*b^2*c^2*d^3 + B*a^2*c*d^4 + 2*A*a*b*c*d^4 - A*a^2*d^5)/(sqrt(d*x + c)*d^6) + 2/315*(35*(d
*x + c)^(9/2)*D*b^2*d^48 - 225*(d*x + c)^(7/2)*D*b^2*c*d^48 + 630*(d*x + c)^(5/2)*D*b^2*c^2*d^48 - 1050*(d*x +
c)^(3/2)*D*b^2*c^3*d^48 + 1575*sqrt(d*x + c)*D*b^2*c^4*d^48 + 90*(d*x + c)^(7/2)*D*a*b*d^49 + 45*(d*x + c)^(7
/2)*C*b^2*d^49 - 504*(d*x + c)^(5/2)*D*a*b*c*d^49 - 252*(d*x + c)^(5/2)*C*b^2*c*d^49 + 1260*(d*x + c)^(3/2)*D*
a*b*c^2*d^49 + 630*(d*x + c)^(3/2)*C*b^2*c^2*d^49 - 2520*sqrt(d*x + c)*D*a*b*c^3*d^49 - 1260*sqrt(d*x + c)*C*b
^2*c^3*d^49 + 63*(d*x + c)^(5/2)*D*a^2*d^50 + 126*(d*x + c)^(5/2)*C*a*b*d^50 + 63*(d*x + c)^(5/2)*B*b^2*d^50 -
315*(d*x + c)^(3/2)*D*a^2*c*d^50 - 630*(d*x + c)^(3/2)*C*a*b*c*d^50 - 315*(d*x + c)^(3/2)*B*b^2*c*d^50 + 945*
sqrt(d*x + c)*D*a^2*c^2*d^50 + 1890*sqrt(d*x + c)*C*a*b*c^2*d^50 + 945*sqrt(d*x + c)*B*b^2*c^2*d^50 + 105*(d*x
+ c)^(3/2)*C*a^2*d^51 + 210*(d*x + c)^(3/2)*B*a*b*d^51 + 105*(d*x + c)^(3/2)*A*b^2*d^51 - 630*sqrt(d*x + c)*C
*a^2*c*d^51 - 1260*sqrt(d*x + c)*B*a*b*c*d^51 - 630*sqrt(d*x + c)*A*b^2*c*d^51 + 315*sqrt(d*x + c)*B*a^2*d^52
+ 630*sqrt(d*x + c)*A*a*b*d^52)/d^54