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

Optimal result

Integrand size = 32, antiderivative size = 324 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2 (b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {c+d x}}{d^6}+\frac {2 (b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) (c+d x)^{3/2}}{3 d^6}+\frac {2 \left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^{5/2}}{5 d^6}+\frac {2 \left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{7/2}}{7 d^6}+\frac {2 b (b C d-5 b c D+2 a d D) (c+d x)^{9/2}}{9 d^6}+\frac {2 b^2 D (c+d x)^{11/2}}{11 d^6} \]

[Out]

2/3*(-a*d+b*c)*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(2*A*d^3-3*B*c*d^2+4*C*c^2*d-5*D*c^3))*(d*x+c)^(3/2)/d^6+2/5*(a
^2*d^2*(C*d-3*D*c)-2*a*b*d*(-B*d^2+3*C*c*d-6*D*c^2)+b^2*(A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*(d*x+c)^(5/2)/d^
6+2/7*(a^2*d^2*D+2*a*b*d*(C*d-4*D*c)-b^2*(-B*d^2+4*C*c*d-10*D*c^2))*(d*x+c)^(7/2)/d^6+2/9*b*(C*b*d+2*D*a*d-5*D
*b*c)*(d*x+c)^(9/2)/d^6+2/11*b^2*D*(d*x+c)^(11/2)/d^6+2*(-a*d+b*c)^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(1/
2)/d^6

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {1634} \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2 (c+d x)^{5/2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{5 d^6}+\frac {2 (c+d x)^{7/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{7 d^6}+\frac {2 (c+d x)^{3/2} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{3 d^6}+\frac {2 \sqrt {c+d x} (b c-a d)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6}+\frac {2 b (c+d x)^{9/2} (2 a d D-5 b c D+b C d)}{9 d^6}+\frac {2 b^2 D (c+d x)^{11/2}}{11 d^6} \]

[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 1634

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*} \text {integral}& = \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] (verified)

Time = 0.25 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {c+d x} \left (33 a^2 d^2 \left (-48 c^3 D+8 c^2 d (7 C+3 D x)-2 c d^2 (35 B+x (14 C+9 D x))+d^3 (105 A+x (35 B+3 x (7 C+5 D x)))\right )+22 a b d \left (128 c^4 D-16 c^3 d (9 C+4 D x)+24 c^2 d^2 (7 B+x (3 C+2 D x))+d^4 x (105 A+x (63 B+5 x (9 C+7 D x)))-2 c d^3 (105 A+x (42 B+x (27 C+20 D x)))\right )+b^2 \left (-1280 c^5 D+128 c^4 d (11 C+5 D x)-16 c^3 d^2 \left (99 B+44 C x+30 D x^2\right )+8 c^2 d^3 \left (231 A+x \left (99 B+66 C x+50 D x^2\right )\right )+d^5 x^2 (693 A+5 x (99 B+7 x (11 C+9 D x)))-2 c d^4 x (462 A+x (297 B+5 x (44 C+35 D x)))\right )\right )}{3465 d^6} \]

[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]*(33*a^2*d^2*(-48*c^3*D + 8*c^2*d*(7*C + 3*D*x) - 2*c*d^2*(35*B + x*(14*C + 9*D*x)) + d^3*(105
*A + x*(35*B + 3*x*(7*C + 5*D*x)))) + 22*a*b*d*(128*c^4*D - 16*c^3*d*(9*C + 4*D*x) + 24*c^2*d^2*(7*B + x*(3*C
+ 2*D*x)) + d^4*x*(105*A + x*(63*B + 5*x*(9*C + 7*D*x))) - 2*c*d^3*(105*A + x*(42*B + x*(27*C + 20*D*x)))) + b
^2*(-1280*c^5*D + 128*c^4*d*(11*C + 5*D*x) - 16*c^3*d^2*(99*B + 44*C*x + 30*D*x^2) + 8*c^2*d^3*(231*A + x*(99*
B + 66*C*x + 50*D*x^2)) + d^5*x^2*(693*A + 5*x*(99*B + 7*x*(11*C + 9*D*x))) - 2*c*d^4*x*(462*A + x*(297*B + 5*
x*(44*C + 35*D*x))))))/(3465*d^6)

Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {d x +c}\, \left (\left (\frac {x^{2} \left (\frac {5}{11} D x^{3}+\frac {5}{9} C \,x^{2}+\frac {5}{7} B x +A \right ) b^{2}}{5}+\frac {2 a x \left (\frac {1}{3} D x^{3}+\frac {3}{7} C \,x^{2}+\frac {3}{5} B x +A \right ) b}{3}+a^{2} \left (A +\frac {1}{7} D x^{3}+\frac {1}{5} C \,x^{2}+\frac {1}{3} B x \right )\right ) d^{5}-\frac {4 \left (\frac {x \left (\frac {25}{66} D x^{3}+\frac {10}{21} C \,x^{2}+\frac {9}{14} B x +A \right ) b^{2}}{5}+a \left (\frac {4}{21} D x^{3}+\frac {9}{35} C \,x^{2}+\frac {2}{5} B x +A \right ) b +\frac {a^{2} \left (\frac {9}{35} D x^{2}+\frac {2}{5} C x +B \right )}{2}\right ) c \,d^{4}}{3}+\frac {8 \left (\left (\frac {50}{231} D x^{3}+\frac {2}{7} C \,x^{2}+\frac {3}{7} B x +A \right ) b^{2}+2 a \left (\frac {2}{7} D x^{2}+\frac {3}{7} C x +B \right ) b +a^{2} \left (\frac {3 D x}{7}+C \right )\right ) c^{2} d^{3}}{15}-\frac {16 c^{3} \left (\left (\frac {10}{33} D x^{2}+\frac {4}{9} C x +B \right ) b^{2}+2 a \left (\frac {4 D x}{9}+C \right ) b +D a^{2}\right ) d^{2}}{35}+\frac {128 \left (\left (\frac {5 D x}{11}+C \right ) b +2 D a \right ) b \,c^{4} d}{315}-\frac {256 D b^{2} c^{5}}{693}\right )}{d^{6}}\)	\(285\)
derivativedivides	\(\frac {\frac {2 D b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a d -b c \right ) b D+b^{2} \left (C d -3 D c \right )\right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a d -b c \right )^{2} D+2 \left (a d -b c \right ) b \left (C d -3 D c \right )+b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a d -b c \right )^{2} \left (C d -3 D c \right )+2 \left (a d -b c \right ) b \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a d -b c \right )^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+2 \left (a d -b c \right ) b \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \left (a d -b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \sqrt {d x +c}}{d^{6}}\)	\(319\)
default	\(\frac {\frac {2 D b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a d -b c \right ) b D+b^{2} \left (C d -3 D c \right )\right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a d -b c \right )^{2} D+2 \left (a d -b c \right ) b \left (C d -3 D c \right )+b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a d -b c \right )^{2} \left (C d -3 D c \right )+2 \left (a d -b c \right ) b \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a d -b c \right )^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+2 \left (a d -b c \right ) b \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \left (a d -b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \sqrt {d x +c}}{d^{6}}\)	\(319\)
gosper	\(\frac {2 \sqrt {d x +c}\, \left (315 D b^{2} x^{5} d^{5}+385 C \,b^{2} d^{5} x^{4}+770 D a b \,d^{5} x^{4}-350 D b^{2} c \,d^{4} x^{4}+495 B \,b^{2} d^{5} x^{3}+990 C a b \,d^{5} x^{3}-440 C \,b^{2} c \,d^{4} x^{3}+495 D a^{2} d^{5} x^{3}-880 D a b c \,d^{4} x^{3}+400 D b^{2} c^{2} d^{3} x^{3}+693 A \,b^{2} d^{5} x^{2}+1386 B a b \,d^{5} x^{2}-594 B \,b^{2} c \,d^{4} x^{2}+693 C \,a^{2} d^{5} x^{2}-1188 C a b c \,d^{4} x^{2}+528 C \,b^{2} c^{2} d^{3} x^{2}-594 D a^{2} c \,d^{4} x^{2}+1056 D a b \,c^{2} d^{3} x^{2}-480 D b^{2} c^{3} d^{2} x^{2}+2310 A a b \,d^{5} x -924 A \,b^{2} c \,d^{4} x +1155 B \,a^{2} d^{5} x -1848 B a b c \,d^{4} x +792 B \,b^{2} c^{2} d^{3} x -924 C \,a^{2} c \,d^{4} x +1584 C a b \,c^{2} d^{3} x -704 C \,b^{2} c^{3} d^{2} x +792 D a^{2} c^{2} d^{3} x -1408 D a b \,c^{3} d^{2} x +640 D b^{2} c^{4} d x +3465 a^{2} A \,d^{5}-4620 A a b c \,d^{4}+1848 A \,b^{2} c^{2} d^{3}-2310 B \,a^{2} c \,d^{4}+3696 B a b \,c^{2} d^{3}-1584 B \,b^{2} c^{3} d^{2}+1848 C \,a^{2} c^{2} d^{3}-3168 C a b \,c^{3} d^{2}+1408 C \,b^{2} c^{4} d -1584 D a^{2} c^{3} d^{2}+2816 D a b \,c^{4} d -1280 D b^{2} c^{5}\right )}{3465 d^{6}}\)	\(505\)
trager	\(\frac {2 \sqrt {d x +c}\, \left (315 D b^{2} x^{5} d^{5}+385 C \,b^{2} d^{5} x^{4}+770 D a b \,d^{5} x^{4}-350 D b^{2} c \,d^{4} x^{4}+495 B \,b^{2} d^{5} x^{3}+990 C a b \,d^{5} x^{3}-440 C \,b^{2} c \,d^{4} x^{3}+495 D a^{2} d^{5} x^{3}-880 D a b c \,d^{4} x^{3}+400 D b^{2} c^{2} d^{3} x^{3}+693 A \,b^{2} d^{5} x^{2}+1386 B a b \,d^{5} x^{2}-594 B \,b^{2} c \,d^{4} x^{2}+693 C \,a^{2} d^{5} x^{2}-1188 C a b c \,d^{4} x^{2}+528 C \,b^{2} c^{2} d^{3} x^{2}-594 D a^{2} c \,d^{4} x^{2}+1056 D a b \,c^{2} d^{3} x^{2}-480 D b^{2} c^{3} d^{2} x^{2}+2310 A a b \,d^{5} x -924 A \,b^{2} c \,d^{4} x +1155 B \,a^{2} d^{5} x -1848 B a b c \,d^{4} x +792 B \,b^{2} c^{2} d^{3} x -924 C \,a^{2} c \,d^{4} x +1584 C a b \,c^{2} d^{3} x -704 C \,b^{2} c^{3} d^{2} x +792 D a^{2} c^{2} d^{3} x -1408 D a b \,c^{3} d^{2} x +640 D b^{2} c^{4} d x +3465 a^{2} A \,d^{5}-4620 A a b c \,d^{4}+1848 A \,b^{2} c^{2} d^{3}-2310 B \,a^{2} c \,d^{4}+3696 B a b \,c^{2} d^{3}-1584 B \,b^{2} c^{3} d^{2}+1848 C \,a^{2} c^{2} d^{3}-3168 C a b \,c^{3} d^{2}+1408 C \,b^{2} c^{4} d -1584 D a^{2} c^{3} d^{2}+2816 D a b \,c^{4} d -1280 D b^{2} c^{5}\right )}{3465 d^{6}}\)	\(505\)

[In]

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

[Out]

2*(d*x+c)^(1/2)*((1/5*x^2*(5/11*D*x^3+5/9*C*x^2+5/7*B*x+A)*b^2+2/3*a*x*(1/3*D*x^3+3/7*C*x^2+3/5*B*x+A)*b+a^2*(
A+1/7*D*x^3+1/5*C*x^2+1/3*B*x))*d^5-4/3*(1/5*x*(25/66*D*x^3+10/21*C*x^2+9/14*B*x+A)*b^2+a*(4/21*D*x^3+9/35*C*x
^2+2/5*B*x+A)*b+1/2*a^2*(9/35*D*x^2+2/5*C*x+B))*c*d^4+8/15*((50/231*D*x^3+2/7*C*x^2+3/7*B*x+A)*b^2+2*a*(2/7*D*
x^2+3/7*C*x+B)*b+a^2*(3/7*D*x+C))*c^2*d^3-16/35*c^3*((10/33*D*x^2+4/9*C*x+B)*b^2+2*a*(4/9*D*x+C)*b+D*a^2)*d^2+
128/315*((5/11*D*x+C)*b+2*D*a)*b*c^4*d-256/693*D*b^2*c^5)/d^6

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (315 \, D b^{2} d^{5} x^{5} - 1280 \, D b^{2} c^{5} + 3465 \, A a^{2} d^{5} + 1848 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} - 2310 \, {\left (B a^{2} + 2 \, A a b\right )} c d^{4} - 35 \, {\left (10 \, D b^{2} c d^{4} - 11 \, {\left (2 \, D a b + C b^{2}\right )} d^{5}\right )} x^{4} + 5 \, {\left (80 \, D b^{2} c^{2} d^{3} + 99 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{5} - 88 \, {\left (2 \, D a b c + C b^{2} c\right )} d^{4}\right )} x^{3} - 1584 \, {\left (D a^{2} c^{3} + {\left (2 \, C a b + B b^{2}\right )} c^{3}\right )} d^{2} - 3 \, {\left (160 \, D b^{2} c^{3} d^{2} - 231 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{5} + 198 \, {\left (D a^{2} c + {\left (2 \, C a b + B b^{2}\right )} c\right )} d^{4} - 176 \, {\left (2 \, D a b c^{2} + C b^{2} c^{2}\right )} d^{3}\right )} x^{2} + 1408 \, {\left (2 \, D a b c^{4} + C b^{2} c^{4}\right )} d + {\left (640 \, D b^{2} c^{4} d - 924 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{4} + 1155 \, {\left (B a^{2} + 2 \, A a b\right )} d^{5} + 792 \, {\left (D a^{2} c^{2} + {\left (2 \, C a b + B b^{2}\right )} c^{2}\right )} d^{3} - 704 \, {\left (2 \, D a b c^{3} + C b^{2} c^{3}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{3465 \, d^{6}} \]

[In]

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

[Out]

2/3465*(315*D*b^2*d^5*x^5 - 1280*D*b^2*c^5 + 3465*A*a^2*d^5 + 1848*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 - 2310*(B
*a^2 + 2*A*a*b)*c*d^4 - 35*(10*D*b^2*c*d^4 - 11*(2*D*a*b + C*b^2)*d^5)*x^4 + 5*(80*D*b^2*c^2*d^3 + 99*(D*a^2 +
2*C*a*b + B*b^2)*d^5 - 88*(2*D*a*b*c + C*b^2*c)*d^4)*x^3 - 1584*(D*a^2*c^3 + (2*C*a*b + B*b^2)*c^3)*d^2 - 3*(
160*D*b^2*c^3*d^2 - 231*(C*a^2 + 2*B*a*b + A*b^2)*d^5 + 198*(D*a^2*c + (2*C*a*b + B*b^2)*c)*d^4 - 176*(2*D*a*b
*c^2 + C*b^2*c^2)*d^3)*x^2 + 1408*(2*D*a*b*c^4 + C*b^2*c^4)*d + (640*D*b^2*c^4*d - 924*(C*a^2 + 2*B*a*b + A*b^
2)*c*d^4 + 1155*(B*a^2 + 2*A*a*b)*d^5 + 792*(D*a^2*c^2 + (2*C*a*b + B*b^2)*c^2)*d^3 - 704*(2*D*a*b*c^3 + C*b^2
*c^3)*d^2)*x)*sqrt(d*x + c)/d^6

Sympy [A] (verification not implemented)

Time = 1.50 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (\frac {D b^{2} \left (c + d x\right )^{\frac {11}{2}}}{11 d^{5}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (C b^{2} d + 2 D a b d - 5 D b^{2} c\right )}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (B b^{2} d^{2} + 2 C a b d^{2} - 4 C b^{2} c d + D a^{2} d^{2} - 8 D a b c d + 10 D b^{2} c^{2}\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (A b^{2} d^{3} + 2 B a b d^{3} - 3 B b^{2} c d^{2} + C a^{2} d^{3} - 6 C a b c d^{2} + 6 C b^{2} c^{2} d - 3 D a^{2} c d^{2} + 12 D a b c^{2} d - 10 D b^{2} c^{3}\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (2 A a b d^{4} - 2 A b^{2} c d^{3} + B a^{2} d^{4} - 4 B a b c d^{3} + 3 B b^{2} c^{2} d^{2} - 2 C a^{2} c d^{3} + 6 C a b c^{2} d^{2} - 4 C b^{2} c^{3} d + 3 D a^{2} c^{2} d^{2} - 8 D a b c^{3} d + 5 D b^{2} c^{4}\right )}{3 d^{5}} + \frac {\sqrt {c + d x} \left (A a^{2} d^{5} - 2 A a b c d^{4} + A b^{2} c^{2} d^{3} - B a^{2} c d^{4} + 2 B a b c^{2} d^{3} - B b^{2} c^{3} d^{2} + C a^{2} c^{2} d^{3} - 2 C a b c^{3} d^{2} + C b^{2} c^{4} d - D a^{2} c^{3} d^{2} + 2 D a b c^{4} d - D b^{2} c^{5}\right )}{d^{5}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {A a^{2} x + \frac {D b^{2} x^{6}}{6} + \frac {x^{5} \left (C b^{2} + 2 D a b\right )}{5} + \frac {x^{4} \left (B b^{2} + 2 C a b + D a^{2}\right )}{4} + \frac {x^{3} \left (A b^{2} + 2 B a b + C a^{2}\right )}{3} + \frac {x^{2} \cdot \left (2 A a b + B a^{2}\right )}{2}}{\sqrt {c}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(D*b**2*(c + d*x)**(11/2)/(11*d**5) + (c + d*x)**(9/2)*(C*b**2*d + 2*D*a*b*d - 5*D*b**2*c)/(9*d**
5) + (c + d*x)**(7/2)*(B*b**2*d**2 + 2*C*a*b*d**2 - 4*C*b**2*c*d + D*a**2*d**2 - 8*D*a*b*c*d + 10*D*b**2*c**2)
/(7*d**5) + (c + d*x)**(5/2)*(A*b**2*d**3 + 2*B*a*b*d**3 - 3*B*b**2*c*d**2 + C*a**2*d**3 - 6*C*a*b*c*d**2 + 6*
C*b**2*c**2*d - 3*D*a**2*c*d**2 + 12*D*a*b*c**2*d - 10*D*b**2*c**3)/(5*d**5) + (c + d*x)**(3/2)*(2*A*a*b*d**4
- 2*A*b**2*c*d**3 + B*a**2*d**4 - 4*B*a*b*c*d**3 + 3*B*b**2*c**2*d**2 - 2*C*a**2*c*d**3 + 6*C*a*b*c**2*d**2 -
4*C*b**2*c**3*d + 3*D*a**2*c**2*d**2 - 8*D*a*b*c**3*d + 5*D*b**2*c**4)/(3*d**5) + sqrt(c + d*x)*(A*a**2*d**5 -
2*A*a*b*c*d**4 + A*b**2*c**2*d**3 - B*a**2*c*d**4 + 2*B*a*b*c**2*d**3 - B*b**2*c**3*d**2 + C*a**2*c**2*d**3 -
2*C*a*b*c**3*d**2 + C*b**2*c**4*d - D*a**2*c**3*d**2 + 2*D*a*b*c**4*d - D*b**2*c**5)/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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} D b^{2} - 385 \, {\left (5 \, D b^{2} c - {\left (2 \, D a b + C b^{2}\right )} d\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 495 \, {\left (10 \, D b^{2} c^{2} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 693 \, {\left (10 \, D b^{2} c^{3} - 6 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, D b^{2} c^{4} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{2} - 2 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{3} + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 3465 \, {\left (D b^{2} c^{5} - A a^{2} d^{5} - {\left (2 \, D a b + C b^{2}\right )} c^{4} d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} + {\left (B a^{2} + 2 \, A a b\right )} c d^{4}\right )} \sqrt {d x + c}\right )}}{3465 \, d^{6}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.72 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (3465 \, \sqrt {d x + c} A a^{2} + \frac {1155 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} B a^{2}}{d} + \frac {2310 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} A a b}{d} + \frac {231 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} C a^{2}}{d^{2}} + \frac {462 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} B a b}{d^{2}} + \frac {231 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} A b^{2}}{d^{2}} + \frac {99 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} D a^{2}}{d^{3}} + \frac {198 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} C a b}{d^{3}} + \frac {99 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} B b^{2}}{d^{3}} + \frac {22 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} D a b}{d^{4}} + \frac {11 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} C b^{2}}{d^{4}} + \frac {5 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} D b^{2}}{d^{5}}\right )}}{3465 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 4.09 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2\,A\,\sqrt {c+d\,x}\,\left (3\,b^2\,{\left (c+d\,x\right )}^2+15\,a^2\,d^2+15\,b^2\,c^2-10\,b^2\,c\,\left (c+d\,x\right )+10\,a\,b\,d\,\left (c+d\,x\right )-30\,a\,b\,c\,d\right )}{15\,d^3}+\frac {2\,B\,b^2\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}+\frac {2\,C\,b^2\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5}+\frac {2\,B\,{\left (c+d\,x\right )}^{3/2}\,\left (a^2\,d^2-4\,a\,b\,c\,d+3\,b^2\,c^2\right )}{3\,d^4}+\frac {2\,C\,{\left (c+d\,x\right )}^{5/2}\,\left (a^2\,d^2-6\,a\,b\,c\,d+6\,b^2\,c^2\right )}{5\,d^5}-\frac {2\,B\,c\,{\left (a\,d-b\,c\right )}^2\,\sqrt {c+d\,x}}{d^4}-\frac {4\,C\,c\,{\left (c+d\,x\right )}^{3/2}\,\left (a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{3\,d^5}-\frac {10\,b^2\,c\,D\,\left (\frac {2\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5}+\frac {2\,c^4\,\sqrt {c+d\,x}}{d^5}-\frac {8\,c^3\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5}+\frac {12\,c^2\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5}-\frac {8\,c\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}\right )}{11\,d}+\frac {2\,C\,c^2\,{\left (a\,d-b\,c\right )}^2\,\sqrt {c+d\,x}}{d^5}-\frac {2\,a^2\,\sqrt {c+d\,x}\,D\,\left (6\,c\,{\left (c+d\,x\right )}^2-20\,c^2\,\left (c+d\,x\right )+30\,c^3-5\,d^3\,x^3\right )}{35\,d^4}+\frac {2\,b^2\,x^5\,\sqrt {c+d\,x}\,D}{11\,d}+\frac {2\,B\,b\,\left (2\,a\,d-3\,b\,c\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4}+\frac {4\,C\,b\,\left (a\,d-2\,b\,c\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}+\frac {4\,a\,b\,\sqrt {c+d\,x}\,D\,\left (168\,c^2\,{\left (c+d\,x\right )}^2-280\,c^3\,\left (c+d\,x\right )-40\,c\,{\left (c+d\,x\right )}^3+280\,c^4+35\,d^4\,x^4\right )}{315\,d^5} \]

[In]

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

[Out]

(2*A*(c + d*x)^(1/2)*(3*b^2*(c + d*x)^2 + 15*a^2*d^2 + 15*b^2*c^2 - 10*b^2*c*(c + d*x) + 10*a*b*d*(c + d*x) -
30*a*b*c*d))/(15*d^3) + (2*B*b^2*(c + d*x)^(7/2))/(7*d^4) + (2*C*b^2*(c + d*x)^(9/2))/(9*d^5) + (2*B*(c + d*x)
^(3/2)*(a^2*d^2 + 3*b^2*c^2 - 4*a*b*c*d))/(3*d^4) + (2*C*(c + d*x)^(5/2)*(a^2*d^2 + 6*b^2*c^2 - 6*a*b*c*d))/(5
*d^5) - (2*B*c*(a*d - b*c)^2*(c + d*x)^(1/2))/d^4 - (4*C*c*(c + d*x)^(3/2)*(a^2*d^2 + 2*b^2*c^2 - 3*a*b*c*d))/
(3*d^5) - (10*b^2*c*D*((2*(c + d*x)^(9/2))/(9*d^5) + (2*c^4*(c + d*x)^(1/2))/d^5 - (8*c^3*(c + d*x)^(3/2))/(3*
d^5) + (12*c^2*(c + d*x)^(5/2))/(5*d^5) - (8*c*(c + d*x)^(7/2))/(7*d^5)))/(11*d) + (2*C*c^2*(a*d - b*c)^2*(c +
d*x)^(1/2))/d^5 - (2*a^2*(c + d*x)^(1/2)*D*(6*c*(c + d*x)^2 - 20*c^2*(c + d*x) + 30*c^3 - 5*d^3*x^3))/(35*d^4
) + (2*b^2*x^5*(c + d*x)^(1/2)*D)/(11*d) + (2*B*b*(2*a*d - 3*b*c)*(c + d*x)^(5/2))/(5*d^4) + (4*C*b*(a*d - 2*b
*c)*(c + d*x)^(7/2))/(7*d^5) + (4*a*b*(c + d*x)^(1/2)*D*(168*c^2*(c + d*x)^2 - 280*c^3*(c + d*x) - 40*c*(c + d
*x)^3 + 280*c^4 + 35*d^4*x^4))/(315*d^5)