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

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

[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 - 10*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 = 0.242589, 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} \[ \frac{2 \sqrt{c+d x} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )}{d^6}+\frac{2 (c+d x)^{3/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{3 d^6}-\frac{2 (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right )\right )}{d^6 \sqrt{c+d x}}-\frac{2 (b c-a d)^2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{3 d^6 (c+d x)^{3/2}}+\frac{2 b (c+d x)^{5/2} (2 a d D-5 b c D+b C d)}{5 d^6}+\frac{2 b^2 D (c+d x)^{7/2}}{7 d^6} \]

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 - 10*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)

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)^{5/2}} \, 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)^{5/2}}+\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 (c+d x)^{3/2}}+\frac{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 )}{d^5 \sqrt{c+d x}}+\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 ) \sqrt{c+d x}}{d^5}+\frac{b (b C d-5 b c D+2 a d D) (c+d x)^{3/2}}{d^5}+\frac{b^2 D (c+d x)^{5/2}}{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 )}{3 d^6 (c+d x)^{3/2}}-\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 )}{d^6 \sqrt{c+d x}}+\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 ) \sqrt{c+d x}}{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)^{3/2}}{3 d^6}+\frac{2 b (b C d-5 b c D+2 a d D) (c+d x)^{5/2}}{5 d^6}+\frac{2 b^2 D (c+d x)^{7/2}}{7 d^6}\\ \end{align*}

Mathematica [A]  time = 0.668733, size = 287, normalized size = 0.89 \[ \frac{2 \left (105 (c+d x)^2 \left (a^2 d^2 (C d-3 c D)+2 a b d \left (B d^2+6 c^2 D-3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )+35 (c+d x)^3 \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (B d^2+10 c^2 D-4 c C d\right )\right )-105 (c+d x) (b c-a d) \left (b \left (-2 A d^3+3 B c d^2-4 c^2 C d+5 c^3 D\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )+35 (b c-a d)^2 \left (-A d^3+B c d^2-c^2 C d+c^3 D\right )+21 b (c+d x)^4 (2 a d D-5 b c D+b C d)+15 b^2 D (c+d x)^5\right )}{105 d^6 (c+d x)^{3/2}} \]

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]

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

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Maple [A]  time = 0.006, size = 505, normalized size = 1.6 \begin{align*} -{\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}}}} \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)^(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+70*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 = 2.00347, size = 531, normalized size = 1.65 \begin{align*} \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} \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)^(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 [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)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 133.078, size = 377, normalized size = 1.17 \begin{align*} \frac{2 D b^{2} \left (c + d x\right )^{\frac{7}{2}}}{7 d^{6}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (2 C b^{2} d + 4 D a b d - 10 D b^{2} c\right )}{5 d^{6}} + \frac{\left (c + d x\right )^{\frac{3}{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 )}{3 d^{6}} + \frac{\sqrt{c + d x} \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 )}{d^{6}} - \frac{2 \left (a d - b c\right ) \left (2 A b d^{3} + B a d^{3} - 3 B b c d^{2} - 2 C a c d^{2} + 4 C b c^{2} d + 3 D a c^{2} d - 5 D b c^{3}\right )}{d^{6} \sqrt{c + d x}} + \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 )}{3 d^{6} \left (c + d x\right )^{\frac{3}{2}}} \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)**(5/2),x)

[Out]

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

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Giac [B]  time = 2.27425, size = 840, normalized size = 2.61 \begin{align*} -\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}} \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)^(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*s
qrt(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)/d^42