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

Optimal. Leaf size=324 \[ \frac{2 (c+d x)^{5/2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )}{5 d^6}+\frac{2 (c+d x)^{7/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{7 d^6}+\frac{2 (c+d x)^{3/2} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right )\right )}{3 d^6}+\frac{2 \sqrt{c+d x} (b c-a d)^2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^6}+\frac{2 b (c+d x)^{9/2} (2 a d D-5 b c D+b C d)}{9 d^6}+\frac{2 b^2 D (c+d x)^{11/2}}{11 d^6} \]

[Out]

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

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Rubi [A]  time = 0.27208, antiderivative size = 324, 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} \[ \frac{2 (c+d x)^{5/2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )}{5 d^6}+\frac{2 (c+d x)^{7/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{7 d^6}+\frac{2 (c+d x)^{3/2} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right )\right )}{3 d^6}+\frac{2 \sqrt{c+d x} (b c-a d)^2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^6}+\frac{2 b (c+d x)^{9/2} (2 a d D-5 b c D+b C d)}{9 d^6}+\frac{2 b^2 D (c+d x)^{11/2}}{11 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^6 + (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))*(c + d*x)^(3/2))/(3*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)^(5/2))/(
5*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)^(7/2))/(7*d^6) +
(2*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(9/2))/(9*d^6) + (2*b^2*D*(c + d*x)^(11/2))/(11*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 )}{\sqrt{c+d x}} \, 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 \sqrt{c+d x}}+\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 ) \sqrt{c+d x}}{d^5}+\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 ) (c+d x)^{3/2}}{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)^{5/2}}{d^5}+\frac{b (b C d-5 b c D+2 a d D) (c+d x)^{7/2}}{d^5}+\frac{b^2 D (c+d x)^{9/2}}{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 ) \sqrt{c+d x}}{d^6}+\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 ) (c+d x)^{3/2}}{3 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)^{5/2}}{5 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)^{7/2}}{7 d^6}+\frac{2 b (b C d-5 b c D+2 a d D) (c+d x)^{9/2}}{9 d^6}+\frac{2 b^2 D (c+d x)^{11/2}}{11 d^6}\\ \end{align*}

Mathematica [A]  time = 0.642523, size = 287, normalized size = 0.89 \[ \frac{2 \sqrt{c+d x} \left (693 (c+d x)^2 \left (a^2 d^2 (C d-3 c D)+2 a b d \left (B d^2+6 c^2 D-3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )+495 (c+d x)^3 \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (B d^2+10 c^2 D-4 c C d\right )\right )+1155 (c+d x) (b c-a d) \left (b \left (-2 A d^3+3 B c d^2-4 c^2 C d+5 c^3 D\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )+3465 (b c-a d)^2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )+385 b (c+d x)^4 (2 a d D-5 b c D+b C d)+315 b^2 D (c+d x)^5\right )}{3465 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(3465*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) + 1155*(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) + 693*(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 + 495*(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 + 385*b*(b*C*d - 5*b*c*D + 2*a*
d*D)*(c + d*x)^4 + 315*b^2*D*(c + d*x)^5))/(3465*d^6)

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Maple [A]  time = 0.007, size = 505, normalized size = 1.6 \begin{align*}{\frac{630\,{b}^{2}D{x}^{5}{d}^{5}+770\,C{b}^{2}{d}^{5}{x}^{4}+1540\,Dab{d}^{5}{x}^{4}-700\,D{b}^{2}c{d}^{4}{x}^{4}+990\,B{b}^{2}{d}^{5}{x}^{3}+1980\,Cab{d}^{5}{x}^{3}-880\,C{b}^{2}c{d}^{4}{x}^{3}+990\,D{a}^{2}{d}^{5}{x}^{3}-1760\,Dabc{d}^{4}{x}^{3}+800\,D{b}^{2}{c}^{2}{d}^{3}{x}^{3}+1386\,A{b}^{2}{d}^{5}{x}^{2}+2772\,Bab{d}^{5}{x}^{2}-1188\,B{b}^{2}c{d}^{4}{x}^{2}+1386\,C{a}^{2}{d}^{5}{x}^{2}-2376\,Cabc{d}^{4}{x}^{2}+1056\,C{b}^{2}{c}^{2}{d}^{3}{x}^{2}-1188\,D{a}^{2}c{d}^{4}{x}^{2}+2112\,Dab{c}^{2}{d}^{3}{x}^{2}-960\,D{b}^{2}{c}^{3}{d}^{2}{x}^{2}+4620\,Aab{d}^{5}x-1848\,A{b}^{2}c{d}^{4}x+2310\,B{a}^{2}{d}^{5}x-3696\,Babc{d}^{4}x+1584\,B{b}^{2}{c}^{2}{d}^{3}x-1848\,C{a}^{2}c{d}^{4}x+3168\,Cab{c}^{2}{d}^{3}x-1408\,C{b}^{2}{c}^{3}{d}^{2}x+1584\,D{a}^{2}{c}^{2}{d}^{3}x-2816\,Dab{c}^{3}{d}^{2}x+1280\,D{b}^{2}{c}^{4}dx+6930\,{a}^{2}A{d}^{5}-9240\,Aabc{d}^{4}+3696\,A{b}^{2}{c}^{2}{d}^{3}-4620\,B{a}^{2}c{d}^{4}+7392\,Bab{c}^{2}{d}^{3}-3168\,B{b}^{2}{c}^{3}{d}^{2}+3696\,C{a}^{2}{c}^{2}{d}^{3}-6336\,Cab{c}^{3}{d}^{2}+2816\,C{b}^{2}{c}^{4}d-3168\,D{a}^{2}{c}^{3}{d}^{2}+5632\,Dab{c}^{4}d-2560\,D{b}^{2}{c}^{5}}{3465\,{d}^{6}}\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)^(1/2),x)

[Out]

2/3465*(d*x+c)^(1/2)*(315*D*b^2*d^5*x^5+385*C*b^2*d^5*x^4+770*D*a*b*d^5*x^4-350*D*b^2*c*d^4*x^4+495*B*b^2*d^5*
x^3+990*C*a*b*d^5*x^3-440*C*b^2*c*d^4*x^3+495*D*a^2*d^5*x^3-880*D*a*b*c*d^4*x^3+400*D*b^2*c^2*d^3*x^3+693*A*b^
2*d^5*x^2+1386*B*a*b*d^5*x^2-594*B*b^2*c*d^4*x^2+693*C*a^2*d^5*x^2-1188*C*a*b*c*d^4*x^2+528*C*b^2*c^2*d^3*x^2-
594*D*a^2*c*d^4*x^2+1056*D*a*b*c^2*d^3*x^2-480*D*b^2*c^3*d^2*x^2+2310*A*a*b*d^5*x-924*A*b^2*c*d^4*x+1155*B*a^2
*d^5*x-1848*B*a*b*c*d^4*x+792*B*b^2*c^2*d^3*x-924*C*a^2*c*d^4*x+1584*C*a*b*c^2*d^3*x-704*C*b^2*c^3*d^2*x+792*D
*a^2*c^2*d^3*x-1408*D*a*b*c^3*d^2*x+640*D*b^2*c^4*d*x+3465*A*a^2*d^5-4620*A*a*b*c*d^4+1848*A*b^2*c^2*d^3-2310*
B*a^2*c*d^4+3696*B*a*b*c^2*d^3-1584*B*b^2*c^3*d^2+1848*C*a^2*c^2*d^3-3168*C*a*b*c^3*d^2+1408*C*b^2*c^4*d-1584*
D*a^2*c^3*d^2+2816*D*a*b*c^4*d-1280*D*b^2*c^5)/d^6

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Maxima [A]  time = 1.57989, size = 522, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (315 \,{\left (d x + c\right )}^{\frac{11}{2}} D b^{2} - 385 \,{\left (5 \, D b^{2} c -{\left (2 \, D a b + C b^{2}\right )} d\right )}{\left (d x + c\right )}^{\frac{9}{2}} + 495 \,{\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{7}{2}} - 693 \,{\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{5}{2}} + 1155 \,{\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 )}^{\frac{3}{2}} - 3465 \,{\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}\right )}}{3465 \, d^{6}} \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)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(d*x + c)^(11/2)*D*b^2 - 385*(5*D*b^2*c - (2*D*a*b + C*b^2)*d)*(d*x + c)^(9/2) + 495*(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)^(7/2) - 693*(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)^(5/2) + 1155*(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)^(3/2) - 3465*(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^6

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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)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 124.994, size = 1510, normalized size = 4.66 \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)**2*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2),x)

[Out]

Piecewise((-(2*A*a**2*c/sqrt(c + d*x) + 2*A*a**2*(-c/sqrt(c + d*x) - sqrt(c + d*x)) + 4*A*a*b*c*(-c/sqrt(c + d
*x) - sqrt(c + d*x))/d + 4*A*a*b*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d + 2*A*b**2*c*
(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 + 2*A*b**2*(-c**3/sqrt(c + d*x) - 3*c**2*sq
rt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**2 + 2*B*a**2*c*(-c/sqrt(c + d*x) - sqrt(c + d*x))/d
+ 2*B*a**2*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d + 4*B*a*b*c*(c**2/sqrt(c + d*x) + 2
*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 + 4*B*a*b*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x
)**(3/2) - (c + d*x)**(5/2)/5)/d**2 + 2*B*b**2*c*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(3
/2) - (c + d*x)**(5/2)/5)/d**3 + 2*B*b**2*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2)
+ 4*c*(c + d*x)**(5/2)/5 - (c + d*x)**(7/2)/7)/d**3 + 2*C*a**2*c*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c
+ d*x)**(3/2)/3)/d**2 + 2*C*a**2*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)
**(5/2)/5)/d**2 + 4*C*a*b*c*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2
)/5)/d**3 + 4*C*a*b*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2) + 4*c*(c + d*x)**(5/2
)/5 - (c + d*x)**(7/2)/7)/d**3 + 2*C*b**2*c*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/
2) + 4*c*(c + d*x)**(5/2)/5 - (c + d*x)**(7/2)/7)/d**4 + 2*C*b**2*(-c**5/sqrt(c + d*x) - 5*c**4*sqrt(c + d*x)
+ 10*c**3*(c + d*x)**(3/2)/3 - 2*c**2*(c + d*x)**(5/2) + 5*c*(c + d*x)**(7/2)/7 - (c + d*x)**(9/2)/9)/d**4 + 2
*D*a**2*c*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**3 + 2*D*a*
*2*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2) + 4*c*(c + d*x)**(5/2)/5 - (c + d*x)**
(7/2)/7)/d**3 + 4*D*a*b*c*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2) + 4*c*(c + d*x)
**(5/2)/5 - (c + d*x)**(7/2)/7)/d**4 + 4*D*a*b*(-c**5/sqrt(c + d*x) - 5*c**4*sqrt(c + d*x) + 10*c**3*(c + d*x)
**(3/2)/3 - 2*c**2*(c + d*x)**(5/2) + 5*c*(c + d*x)**(7/2)/7 - (c + d*x)**(9/2)/9)/d**4 + 2*D*b**2*c*(-c**5/sq
rt(c + d*x) - 5*c**4*sqrt(c + d*x) + 10*c**3*(c + d*x)**(3/2)/3 - 2*c**2*(c + d*x)**(5/2) + 5*c*(c + d*x)**(7/
2)/7 - (c + d*x)**(9/2)/9)/d**5 + 2*D*b**2*(c**6/sqrt(c + d*x) + 6*c**5*sqrt(c + d*x) - 5*c**4*(c + d*x)**(3/2
) + 4*c**3*(c + d*x)**(5/2) - 15*c**2*(c + d*x)**(7/2)/7 + 2*c*(c + d*x)**(9/2)/3 - (c + d*x)**(11/2)/11)/d**5
)/d, Ne(d, 0)), ((A*a**2*x + D*b**2*x**6/6 + x**5*(C*b**2 + 2*D*a*b)/5 + x**4*(B*b**2 + 2*C*a*b + D*a**2)/4 +
x**3*(A*b**2 + 2*B*a*b + C*a**2)/3 + x**2*(2*A*a*b + B*a**2)/2)/sqrt(c), True))

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Giac [A]  time = 2.29546, size = 753, normalized size = 2.32 \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)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/3465*(3465*sqrt(d*x + c)*A*a^2 + 1155*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B*a^2/d + 2310*((d*x + c)^(3/2)
- 3*sqrt(d*x + c)*c)*A*a*b/d + 231*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*C*a^2/d^2
+ 462*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*B*a*b/d^2 + 231*(3*(d*x + c)^(5/2) -
10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*A*b^2/d^2 + 99*(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^2/d^3 + 198*(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*b/d^3 + 99*(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*b^2/d^3 + 22*(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*b/d^4 + 11*(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*b^2/d^4 + 5*(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*b^2/d^5)/d