\(\int \frac {(a+b x)^3 (A+B x+C x^2+D x^3)}{\sqrt {c+d x}} \, dx\) [1]

Optimal result

Integrand size = 32, antiderivative size = 436 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=-\frac {2 (b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {c+d x}}{d^7}-\frac {2 (b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) (c+d x)^{3/2}}{3 d^7}-\frac {2 (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{5/2}}{5 d^7}+\frac {2 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{7/2}}{7 d^7}+\frac {2 b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{9/2}}{9 d^7}+\frac {2 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{11/2}}{11 d^7}+\frac {2 b^3 D (c+d x)^{13/2}}{13 d^7} \]

[Out]

-2/3*(-a*d+b*c)^2*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(3*A*d^3-4*B*c*d^2+5*C*c^2*d-6*D*c^3))*(d*x+c)^(3/2)/d^7-2/5
*(-a*d+b*c)*(a^2*d^2*(C*d-3*D*c)-a*b*d*(-3*B*d^2+8*C*c*d-15*D*c^2)+b^2*(3*A*d^3-6*B*c*d^2+10*C*c^2*d-15*D*c^3)
)*(d*x+c)^(5/2)/d^7+2/7*(a^3*d^3*D+3*a^2*b*d^2*(C*d-4*D*c)-3*a*b^2*d*(-B*d^2+4*C*c*d-10*D*c^2)+b^3*(A*d^3-4*B*
c*d^2+10*C*c^2*d-20*D*c^3))*(d*x+c)^(7/2)/d^7+2/9*b*(3*a^2*d^2*D+3*a*b*d*(C*d-5*D*c)-b^2*(-B*d^2+5*C*c*d-15*D*
c^2))*(d*x+c)^(9/2)/d^7+2/11*b^2*(C*b*d+3*D*a*d-6*D*b*c)*(d*x+c)^(11/2)/d^7+2/13*b^3*D*(d*x+c)^(13/2)/d^7-2*(-
a*d+b*c)^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(1/2)/d^7

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 436, 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} \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=-\frac {2 (c+d x)^{5/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{5 d^7}+\frac {2 b (c+d x)^{9/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{9 d^7}+\frac {2 (c+d x)^{7/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{7 d^7}-\frac {2 (c+d x)^{3/2} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{3 d^7}-\frac {2 \sqrt {c+d x} (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7}+\frac {2 b^2 (c+d x)^{11/2} (3 a d D-6 b c D+b C d)}{11 d^7}+\frac {2 b^3 D (c+d x)^{13/2}}{13 d^7} \]

[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 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)^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] (verified)

Time = 0.45 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {c+d x} \left (429 a^3 d^3 \left (-48 c^3 D+8 c^2 d (7 C+3 D x)-2 c d^2 (35 B+x (14 C+9 D x))+d^3 (105 A+x (35 B+3 x (7 C+5 D x)))\right )+429 a^2 b d^2 \left (128 c^4 D-16 c^3 d (9 C+4 D x)+24 c^2 d^2 (7 B+x (3 C+2 D x))+d^4 x (105 A+x (63 B+5 x (9 C+7 D x)))-2 c d^3 (105 A+x (42 B+x (27 C+20 D x)))\right )+39 a b^2 d \left (-1280 c^5 D+128 c^4 d (11 C+5 D x)-16 c^3 d^2 \left (99 B+44 C x+30 D x^2\right )+8 c^2 d^3 \left (231 A+x \left (99 B+66 C x+50 D x^2\right )\right )+d^5 x^2 (693 A+5 x (99 B+7 x (11 C+9 D x)))-2 c d^4 x (462 A+x (297 B+5 x (44 C+35 D x)))\right )+b^3 \left (15360 c^6 D-1280 c^5 d (13 C+6 D x)-16 c^3 d^3 \left (1287 A+572 B x+390 C x^2+300 D x^3\right )+128 c^4 d^2 (143 B+5 x (13 C+9 D x))+5 d^6 x^3 \left (1287 A+7 x \left (143 B+117 C x+99 D x^2\right )\right )+8 c^2 d^4 x \left (1287 A+x \left (858 B+650 C x+525 D x^2\right )\right )-2 c d^5 x^2 (3861 A+5 x (572 B+7 x (65 C+54 D x)))\right )\right )}{45045 d^7} \]

[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]*(429*a^3*d^3*(-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*(10
5*A + x*(35*B + 3*x*(7*C + 5*D*x)))) + 429*a^2*b*d^2*(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))
)) + 39*a*b^2*d*(-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*(2
31*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)))) + b^3*(15360*c^6*D - 1280*c^5*d*(13*C + 6*D*x) - 16*c^3*d^3*(1287*A + 572*B*
x + 390*C*x^2 + 300*D*x^3) + 128*c^4*d^2*(143*B + 5*x*(13*C + 9*D*x)) + 5*d^6*x^3*(1287*A + 7*x*(143*B + 117*C
*x + 99*D*x^2)) + 8*c^2*d^4*x*(1287*A + x*(858*B + 650*C*x + 525*D*x^2)) - 2*c*d^5*x^2*(3861*A + 5*x*(572*B +
7*x*(65*C + 54*D*x))))))/(45045*d^7)

Maple [A] (verified)

Time = 2.30 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(\frac {\frac {2 D b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (a d -b c \right ) b^{2} D+b^{3} \left (C d -3 D c \right )\right ) \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 \left (a d -b c \right )^{2} b D+3 \left (a d -b c \right ) b^{2} \left (C d -3 D c \right )+b^{3} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a d -b c \right )^{3} D+3 \left (a d -b c \right )^{2} b \left (C d -3 D c \right )+3 \left (a d -b c \right ) b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{3} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a d -b c \right )^{3} \left (C d -3 D c \right )+3 \left (a d -b c \right )^{2} b \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+3 \left (a d -b c \right ) b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a d -b c \right )^{3} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+3 \left (a d -b c \right )^{2} b \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \left (a d -b c \right )^{3} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \sqrt {d x +c}}{d^{7}}\)	\(440\)
default	\(\frac {\frac {2 D b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (a d -b c \right ) b^{2} D+b^{3} \left (C d -3 D c \right )\right ) \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 \left (a d -b c \right )^{2} b D+3 \left (a d -b c \right ) b^{2} \left (C d -3 D c \right )+b^{3} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a d -b c \right )^{3} D+3 \left (a d -b c \right )^{2} b \left (C d -3 D c \right )+3 \left (a d -b c \right ) b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{3} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a d -b c \right )^{3} \left (C d -3 D c \right )+3 \left (a d -b c \right )^{2} b \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+3 \left (a d -b c \right ) b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a d -b c \right )^{3} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+3 \left (a d -b c \right )^{2} b \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \left (a d -b c \right )^{3} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \sqrt {d x +c}}{d^{7}}\)	\(440\)
pseudoelliptic	\(\frac {2 \sqrt {d x +c}\, \left (\left (\frac {x^{3} \left (\frac {7}{13} D x^{3}+\frac {7}{11} C \,x^{2}+\frac {7}{9} B x +A \right ) b^{3}}{7}+\frac {3 a \,x^{2} \left (\frac {5}{11} D x^{3}+\frac {5}{9} C \,x^{2}+\frac {5}{7} B x +A \right ) b^{2}}{5}+a^{2} x \left (\frac {1}{3} D x^{3}+\frac {3}{7} C \,x^{2}+\frac {3}{5} B x +A \right ) b +a^{3} \left (A +\frac {1}{7} D x^{3}+\frac {1}{5} C \,x^{2}+\frac {1}{3} B x \right )\right ) d^{6}-2 c \left (\left (\frac {6}{143} D x^{5}+\frac {5}{99} C \,x^{4}+\frac {4}{63} x^{3} B +\frac {3}{35} A \,x^{2}\right ) b^{3}+\frac {2 a x \left (\frac {25}{66} D x^{3}+\frac {10}{21} C \,x^{2}+\frac {9}{14} B x +A \right ) b^{2}}{5}+a^{2} \left (\frac {4}{21} D x^{3}+\frac {9}{35} C \,x^{2}+\frac {2}{5} B x +A \right ) b +\frac {a^{3} \left (\frac {9}{35} D x^{2}+\frac {2}{5} C x +B \right )}{3}\right ) d^{5}+\frac {8 \left (\frac {\left (\frac {175}{429} D x^{3}+\frac {50}{99} C \,x^{2}+\frac {2}{3} B x +A \right ) x \,b^{3}}{7}+a \left (\frac {50}{231} D x^{3}+\frac {2}{7} C \,x^{2}+\frac {3}{7} B x +A \right ) b^{2}+a^{2} \left (\frac {2}{7} D x^{2}+\frac {3}{7} C x +B \right ) b +\frac {a^{3} \left (\frac {3 D x}{7}+C \right )}{3}\right ) c^{2} d^{4}}{5}-\frac {16 c^{3} \left (\left (\frac {100}{429} D x^{3}+\frac {10}{33} C \,x^{2}+\frac {4}{9} B x +A \right ) b^{3}+3 a \left (\frac {10}{33} D x^{2}+\frac {4}{9} C x +B \right ) b^{2}+3 a^{2} \left (\frac {4 D x}{9}+C \right ) b +D a^{3}\right ) d^{3}}{35}+\frac {128 b \,c^{4} \left (\left (\frac {45}{143} D x^{2}+\frac {5}{11} C x +B \right ) b^{2}+3 \left (\frac {5 D x}{11}+C \right ) a b +3 D a^{2}\right ) d^{2}}{315}-\frac {256 b^{2} c^{5} \left (\left (\frac {6 D x}{13}+C \right ) b +3 D a \right ) d}{693}+\frac {1024 D b^{3} c^{6}}{3003}\right )}{d^{7}}\)	\(442\)
gosper	\(\frac {2 \sqrt {d x +c}\, \left (3465 D b^{3} x^{6} d^{6}+4095 C \,b^{3} d^{6} x^{5}+12285 D a \,b^{2} d^{6} x^{5}-3780 D b^{3} c \,d^{5} x^{5}+5005 B \,b^{3} d^{6} x^{4}+15015 C a \,b^{2} d^{6} x^{4}-4550 C \,b^{3} c \,d^{5} x^{4}+15015 D a^{2} b \,d^{6} x^{4}-13650 D a \,b^{2} c \,d^{5} x^{4}+4200 D b^{3} c^{2} d^{4} x^{4}+6435 A \,b^{3} d^{6} x^{3}+19305 B a \,b^{2} d^{6} x^{3}-5720 B \,b^{3} c \,d^{5} x^{3}+19305 C \,a^{2} b \,d^{6} x^{3}-17160 C a \,b^{2} c \,d^{5} x^{3}+5200 C \,b^{3} c^{2} d^{4} x^{3}+6435 D a^{3} d^{6} x^{3}-17160 D a^{2} b c \,d^{5} x^{3}+15600 D a \,b^{2} c^{2} d^{4} x^{3}-4800 D b^{3} c^{3} d^{3} x^{3}+27027 A a \,b^{2} d^{6} x^{2}-7722 A \,b^{3} c \,d^{5} x^{2}+27027 B \,a^{2} b \,d^{6} x^{2}-23166 B a \,b^{2} c \,d^{5} x^{2}+6864 B \,b^{3} c^{2} d^{4} x^{2}+9009 C \,a^{3} d^{6} x^{2}-23166 C \,a^{2} b c \,d^{5} x^{2}+20592 C a \,b^{2} c^{2} d^{4} x^{2}-6240 C \,b^{3} c^{3} d^{3} x^{2}-7722 D a^{3} c \,d^{5} x^{2}+20592 D a^{2} b \,c^{2} d^{4} x^{2}-18720 D a \,b^{2} c^{3} d^{3} x^{2}+5760 D b^{3} c^{4} d^{2} x^{2}+45045 A \,a^{2} b \,d^{6} x -36036 A a \,b^{2} c \,d^{5} x +10296 A \,b^{3} c^{2} d^{4} x +15015 B \,a^{3} d^{6} x -36036 B \,a^{2} b c \,d^{5} x +30888 B a \,b^{2} c^{2} d^{4} x -9152 B \,b^{3} c^{3} d^{3} x -12012 C \,a^{3} c \,d^{5} x +30888 C \,a^{2} b \,c^{2} d^{4} x -27456 C a \,b^{2} c^{3} d^{3} x +8320 C \,b^{3} c^{4} d^{2} x +10296 D a^{3} c^{2} d^{4} x -27456 D a^{2} b \,c^{3} d^{3} x +24960 D a \,b^{2} c^{4} d^{2} x -7680 D b^{3} c^{5} d x +45045 a^{3} A \,d^{6}-90090 A \,a^{2} b c \,d^{5}+72072 A a \,b^{2} c^{2} d^{4}-20592 A \,b^{3} c^{3} d^{3}-30030 B \,a^{3} c \,d^{5}+72072 B \,a^{2} b \,c^{2} d^{4}-61776 B a \,b^{2} c^{3} d^{3}+18304 B \,b^{3} c^{4} d^{2}+24024 C \,a^{3} c^{2} d^{4}-61776 C \,a^{2} b \,c^{3} d^{3}+54912 C a \,b^{2} c^{4} d^{2}-16640 C \,b^{3} c^{5} d -20592 D a^{3} c^{3} d^{3}+54912 D a^{2} b \,c^{4} d^{2}-49920 D a \,b^{2} c^{5} d +15360 D b^{3} c^{6}\right )}{45045 d^{7}}\)	\(841\)
trager	\(\frac {2 \sqrt {d x +c}\, \left (3465 D b^{3} x^{6} d^{6}+4095 C \,b^{3} d^{6} x^{5}+12285 D a \,b^{2} d^{6} x^{5}-3780 D b^{3} c \,d^{5} x^{5}+5005 B \,b^{3} d^{6} x^{4}+15015 C a \,b^{2} d^{6} x^{4}-4550 C \,b^{3} c \,d^{5} x^{4}+15015 D a^{2} b \,d^{6} x^{4}-13650 D a \,b^{2} c \,d^{5} x^{4}+4200 D b^{3} c^{2} d^{4} x^{4}+6435 A \,b^{3} d^{6} x^{3}+19305 B a \,b^{2} d^{6} x^{3}-5720 B \,b^{3} c \,d^{5} x^{3}+19305 C \,a^{2} b \,d^{6} x^{3}-17160 C a \,b^{2} c \,d^{5} x^{3}+5200 C \,b^{3} c^{2} d^{4} x^{3}+6435 D a^{3} d^{6} x^{3}-17160 D a^{2} b c \,d^{5} x^{3}+15600 D a \,b^{2} c^{2} d^{4} x^{3}-4800 D b^{3} c^{3} d^{3} x^{3}+27027 A a \,b^{2} d^{6} x^{2}-7722 A \,b^{3} c \,d^{5} x^{2}+27027 B \,a^{2} b \,d^{6} x^{2}-23166 B a \,b^{2} c \,d^{5} x^{2}+6864 B \,b^{3} c^{2} d^{4} x^{2}+9009 C \,a^{3} d^{6} x^{2}-23166 C \,a^{2} b c \,d^{5} x^{2}+20592 C a \,b^{2} c^{2} d^{4} x^{2}-6240 C \,b^{3} c^{3} d^{3} x^{2}-7722 D a^{3} c \,d^{5} x^{2}+20592 D a^{2} b \,c^{2} d^{4} x^{2}-18720 D a \,b^{2} c^{3} d^{3} x^{2}+5760 D b^{3} c^{4} d^{2} x^{2}+45045 A \,a^{2} b \,d^{6} x -36036 A a \,b^{2} c \,d^{5} x +10296 A \,b^{3} c^{2} d^{4} x +15015 B \,a^{3} d^{6} x -36036 B \,a^{2} b c \,d^{5} x +30888 B a \,b^{2} c^{2} d^{4} x -9152 B \,b^{3} c^{3} d^{3} x -12012 C \,a^{3} c \,d^{5} x +30888 C \,a^{2} b \,c^{2} d^{4} x -27456 C a \,b^{2} c^{3} d^{3} x +8320 C \,b^{3} c^{4} d^{2} x +10296 D a^{3} c^{2} d^{4} x -27456 D a^{2} b \,c^{3} d^{3} x +24960 D a \,b^{2} c^{4} d^{2} x -7680 D b^{3} c^{5} d x +45045 a^{3} A \,d^{6}-90090 A \,a^{2} b c \,d^{5}+72072 A a \,b^{2} c^{2} d^{4}-20592 A \,b^{3} c^{3} d^{3}-30030 B \,a^{3} c \,d^{5}+72072 B \,a^{2} b \,c^{2} d^{4}-61776 B a \,b^{2} c^{3} d^{3}+18304 B \,b^{3} c^{4} d^{2}+24024 C \,a^{3} c^{2} d^{4}-61776 C \,a^{2} b \,c^{3} d^{3}+54912 C a \,b^{2} c^{4} d^{2}-16640 C \,b^{3} c^{5} d -20592 D a^{3} c^{3} d^{3}+54912 D a^{2} b \,c^{4} d^{2}-49920 D a \,b^{2} c^{5} d +15360 D b^{3} c^{6}\right )}{45045 d^{7}}\)	\(841\)

[In]

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

[Out]

2/d^7*(1/13*D*b^3*(d*x+c)^(13/2)+1/11*(3*(a*d-b*c)*b^2*D+b^3*(C*d-3*D*c))*(d*x+c)^(11/2)+1/9*(3*(a*d-b*c)^2*b*
D+3*(a*d-b*c)*b^2*(C*d-3*D*c)+b^3*(B*d^2-2*C*c*d+3*D*c^2))*(d*x+c)^(9/2)+1/7*((a*d-b*c)^3*D+3*(a*d-b*c)^2*b*(C
*d-3*D*c)+3*(a*d-b*c)*b^2*(B*d^2-2*C*c*d+3*D*c^2)+b^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3))*(d*x+c)^(7/2)+1/5*((a*d-b
*c)^3*(C*d-3*D*c)+3*(a*d-b*c)^2*b*(B*d^2-2*C*c*d+3*D*c^2)+3*(a*d-b*c)*b^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3))*(d*x+
c)^(5/2)+1/3*((a*d-b*c)^3*(B*d^2-2*C*c*d+3*D*c^2)+3*(a*d-b*c)^2*b*(A*d^3-B*c*d^2+C*c^2*d-D*c^3))*(d*x+c)^(3/2)
+(a*d-b*c)^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (3465 \, D b^{3} d^{6} x^{6} + 15360 \, D b^{3} c^{6} + 45045 \, A a^{3} d^{6} + 24024 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - 30030 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} - 315 \, {\left (12 \, D b^{3} c d^{5} - 13 \, {\left (3 \, D a b^{2} + C b^{3}\right )} d^{6}\right )} x^{5} + 35 \, {\left (120 \, D b^{3} c^{2} d^{4} + 143 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{6} - 130 \, {\left (3 \, D a b^{2} c + C b^{3} c\right )} d^{5}\right )} x^{4} - 20592 \, {\left (D a^{3} c^{3} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3}\right )} d^{3} - 5 \, {\left (960 \, D b^{3} c^{3} d^{3} - 1287 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{6} + 1144 \, {\left (3 \, D a^{2} b c + {\left (3 \, C a b^{2} + B b^{3}\right )} c\right )} d^{5} - 1040 \, {\left (3 \, D a b^{2} c^{2} + C b^{3} c^{2}\right )} d^{4}\right )} x^{3} + 18304 \, {\left (3 \, D a^{2} b c^{4} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4}\right )} d^{2} + 3 \, {\left (1920 \, D b^{3} c^{4} d^{2} + 3003 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6} - 2574 \, {\left (D a^{3} c + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c\right )} d^{5} + 2288 \, {\left (3 \, D a^{2} b c^{2} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{2}\right )} d^{4} - 2080 \, {\left (3 \, D a b^{2} c^{3} + C b^{3} c^{3}\right )} d^{3}\right )} x^{2} - 16640 \, {\left (3 \, D a b^{2} c^{5} + C b^{3} c^{5}\right )} d - {\left (7680 \, D b^{3} c^{5} d + 12012 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} - 15015 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6} - 10296 \, {\left (D a^{3} c^{2} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2}\right )} d^{4} + 9152 \, {\left (3 \, D a^{2} b c^{3} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{3}\right )} d^{3} - 8320 \, {\left (3 \, D a b^{2} c^{4} + C b^{3} c^{4}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{7}} \]

[In]

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

[Out]

2/45045*(3465*D*b^3*d^6*x^6 + 15360*D*b^3*c^6 + 45045*A*a^3*d^6 + 24024*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^
4 - 30030*(B*a^3 + 3*A*a^2*b)*c*d^5 - 315*(12*D*b^3*c*d^5 - 13*(3*D*a*b^2 + C*b^3)*d^6)*x^5 + 35*(120*D*b^3*c^
2*d^4 + 143*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^6 - 130*(3*D*a*b^2*c + C*b^3*c)*d^5)*x^4 - 20592*(D*a^3*c^3 + (3
*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3)*d^3 - 5*(960*D*b^3*c^3*d^3 - 1287*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d
^6 + 1144*(3*D*a^2*b*c + (3*C*a*b^2 + B*b^3)*c)*d^5 - 1040*(3*D*a*b^2*c^2 + C*b^3*c^2)*d^4)*x^3 + 18304*(3*D*a
^2*b*c^4 + (3*C*a*b^2 + B*b^3)*c^4)*d^2 + 3*(1920*D*b^3*c^4*d^2 + 3003*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^6 - 2
574*(D*a^3*c + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c)*d^5 + 2288*(3*D*a^2*b*c^2 + (3*C*a*b^2 + B*b^3)*c^2)*d^4 - 2
080*(3*D*a*b^2*c^3 + C*b^3*c^3)*d^3)*x^2 - 16640*(3*D*a*b^2*c^5 + C*b^3*c^5)*d - (7680*D*b^3*c^5*d + 12012*(C*
a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^5 - 15015*(B*a^3 + 3*A*a^2*b)*d^6 - 10296*(D*a^3*c^2 + (3*C*a^2*b + 3*B*a*b^2
+ A*b^3)*c^2)*d^4 + 9152*(3*D*a^2*b*c^3 + (3*C*a*b^2 + B*b^3)*c^3)*d^3 - 8320*(3*D*a*b^2*c^4 + C*b^3*c^4)*d^2
)*x)*sqrt(d*x + c)/d^7

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (454) = 908\).

Time = 1.80 (sec) , antiderivative size = 1027, normalized size of antiderivative = 2.36 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (\frac {D b^{3} \left (c + d x\right )^{\frac {13}{2}}}{13 d^{6}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \left (C b^{3} d + 3 D a b^{2} d - 6 D b^{3} c\right )}{11 d^{6}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (B b^{3} d^{2} + 3 C a b^{2} d^{2} - 5 C b^{3} c d + 3 D a^{2} b d^{2} - 15 D a b^{2} c d + 15 D b^{3} c^{2}\right )}{9 d^{6}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (A b^{3} d^{3} + 3 B a b^{2} d^{3} - 4 B b^{3} c d^{2} + 3 C a^{2} b d^{3} - 12 C a b^{2} c d^{2} + 10 C b^{3} c^{2} d + D a^{3} d^{3} - 12 D a^{2} b c d^{2} + 30 D a b^{2} c^{2} d - 20 D b^{3} c^{3}\right )}{7 d^{6}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (3 A a b^{2} d^{4} - 3 A b^{3} c d^{3} + 3 B a^{2} b d^{4} - 9 B a b^{2} c d^{3} + 6 B b^{3} c^{2} d^{2} + C a^{3} d^{4} - 9 C a^{2} b c d^{3} + 18 C a b^{2} c^{2} d^{2} - 10 C b^{3} c^{3} d - 3 D a^{3} c d^{3} + 18 D a^{2} b c^{2} d^{2} - 30 D a b^{2} c^{3} d + 15 D b^{3} c^{4}\right )}{5 d^{6}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (3 A a^{2} b d^{5} - 6 A a b^{2} c d^{4} + 3 A b^{3} c^{2} d^{3} + B a^{3} d^{5} - 6 B a^{2} b c d^{4} + 9 B a b^{2} c^{2} d^{3} - 4 B b^{3} c^{3} d^{2} - 2 C a^{3} c d^{4} + 9 C a^{2} b c^{2} d^{3} - 12 C a b^{2} c^{3} d^{2} + 5 C b^{3} c^{4} d + 3 D a^{3} c^{2} d^{3} - 12 D a^{2} b c^{3} d^{2} + 15 D a b^{2} c^{4} d - 6 D b^{3} c^{5}\right )}{3 d^{6}} + \frac {\sqrt {c + d x} \left (A a^{3} d^{6} - 3 A a^{2} b c d^{5} + 3 A a b^{2} c^{2} d^{4} - A b^{3} c^{3} d^{3} - B a^{3} c d^{5} + 3 B a^{2} b c^{2} d^{4} - 3 B a b^{2} c^{3} d^{3} + B b^{3} c^{4} d^{2} + C a^{3} c^{2} d^{4} - 3 C a^{2} b c^{3} d^{3} + 3 C a b^{2} c^{4} d^{2} - C b^{3} c^{5} d - D a^{3} c^{3} d^{3} + 3 D a^{2} b c^{4} d^{2} - 3 D a b^{2} c^{5} d + D b^{3} c^{6}\right )}{d^{6}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {A a^{3} x + \frac {D b^{3} x^{7}}{7} + \frac {x^{6} \left (C b^{3} + 3 D a b^{2}\right )}{6} + \frac {x^{5} \left (B b^{3} + 3 C a b^{2} + 3 D a^{2} b\right )}{5} + \frac {x^{4} \left (A b^{3} + 3 B a b^{2} + 3 C a^{2} b + D a^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 A a b^{2} + 3 B a^{2} b + C a^{3}\right )}{3} + \frac {x^{2} \cdot \left (3 A a^{2} b + B a^{3}\right )}{2}}{\sqrt {c}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(D*b**3*(c + d*x)**(13/2)/(13*d**6) + (c + d*x)**(11/2)*(C*b**3*d + 3*D*a*b**2*d - 6*D*b**3*c)/(1
1*d**6) + (c + d*x)**(9/2)*(B*b**3*d**2 + 3*C*a*b**2*d**2 - 5*C*b**3*c*d + 3*D*a**2*b*d**2 - 15*D*a*b**2*c*d +
15*D*b**3*c**2)/(9*d**6) + (c + d*x)**(7/2)*(A*b**3*d**3 + 3*B*a*b**2*d**3 - 4*B*b**3*c*d**2 + 3*C*a**2*b*d**
3 - 12*C*a*b**2*c*d**2 + 10*C*b**3*c**2*d + D*a**3*d**3 - 12*D*a**2*b*c*d**2 + 30*D*a*b**2*c**2*d - 20*D*b**3*
c**3)/(7*d**6) + (c + d*x)**(5/2)*(3*A*a*b**2*d**4 - 3*A*b**3*c*d**3 + 3*B*a**2*b*d**4 - 9*B*a*b**2*c*d**3 + 6
*B*b**3*c**2*d**2 + C*a**3*d**4 - 9*C*a**2*b*c*d**3 + 18*C*a*b**2*c**2*d**2 - 10*C*b**3*c**3*d - 3*D*a**3*c*d*
*3 + 18*D*a**2*b*c**2*d**2 - 30*D*a*b**2*c**3*d + 15*D*b**3*c**4)/(5*d**6) + (c + d*x)**(3/2)*(3*A*a**2*b*d**5
- 6*A*a*b**2*c*d**4 + 3*A*b**3*c**2*d**3 + B*a**3*d**5 - 6*B*a**2*b*c*d**4 + 9*B*a*b**2*c**2*d**3 - 4*B*b**3*
c**3*d**2 - 2*C*a**3*c*d**4 + 9*C*a**2*b*c**2*d**3 - 12*C*a*b**2*c**3*d**2 + 5*C*b**3*c**4*d + 3*D*a**3*c**2*d
**3 - 12*D*a**2*b*c**3*d**2 + 15*D*a*b**2*c**4*d - 6*D*b**3*c**5)/(3*d**6) + sqrt(c + d*x)*(A*a**3*d**6 - 3*A*
a**2*b*c*d**5 + 3*A*a*b**2*c**2*d**4 - A*b**3*c**3*d**3 - B*a**3*c*d**5 + 3*B*a**2*b*c**2*d**4 - 3*B*a*b**2*c*
*3*d**3 + B*b**3*c**4*d**2 + C*a**3*c**2*d**4 - 3*C*a**2*b*c**3*d**3 + 3*C*a*b**2*c**4*d**2 - C*b**3*c**5*d -
D*a**3*c**3*d**3 + 3*D*a**2*b*c**4*d**2 - 3*D*a*b**2*c**5*d + D*b**3*c**6)/d**6)/d, Ne(d, 0)), ((A*a**3*x + D*
b**3*x**7/7 + x**6*(C*b**3 + 3*D*a*b**2)/6 + x**5*(B*b**3 + 3*C*a*b**2 + 3*D*a**2*b)/5 + x**4*(A*b**3 + 3*B*a*
b**2 + 3*C*a**2*b + D*a**3)/4 + x**3*(3*A*a*b**2 + 3*B*a**2*b + C*a**3)/3 + x**2*(3*A*a**2*b + B*a**3)/2)/sqrt
(c), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\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}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 854 vs. \(2 (412) = 824\).

Time = 0.31 (sec) , antiderivative size = 854, normalized size of antiderivative = 1.96 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\text {Too large to display} \]

[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

Mupad [B] (verification not implemented)

Time = 4.91 (sec) , antiderivative size = 713, normalized size of antiderivative = 1.64 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {5544\,b^3\,c^6\,\sqrt {c+d\,x}\,D-504\,b^3\,c\,{\left (c+d\,x\right )}^{11/2}\,D-9240\,b^3\,c^5\,{\left (c+d\,x\right )}^{3/2}\,D+11088\,b^3\,c^4\,{\left (c+d\,x\right )}^{5/2}\,D-7920\,b^3\,c^3\,{\left (c+d\,x\right )}^{7/2}\,D+3080\,b^3\,c^2\,{\left (c+d\,x\right )}^{9/2}\,D+462\,b^3\,d^6\,x^6\,\sqrt {c+d\,x}\,D}{3003\,d^7}+\frac {2\,C\,{\left (c+d\,x\right )}^{5/2}\,\left (a^3\,d^3-9\,a^2\,b\,c\,d^2+18\,a\,b^2\,c^2\,d-10\,b^3\,c^3\right )}{5\,d^6}+\frac {2\,A\,b^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}+\frac {2\,B\,b^3\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5}+\frac {2\,C\,b^3\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {2\,A\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^4}+\frac {2\,A\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{d^4}+\frac {6\,A\,b^2\,\left (a\,d-b\,c\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4}+\frac {2\,B\,b^2\,\left (3\,a\,d-4\,b\,c\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}-\frac {2\,B\,c\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^5}+\frac {2\,C\,b^2\,\left (3\,a\,d-5\,b\,c\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {6\,B\,b\,{\left (c+d\,x\right )}^{5/2}\,\left (a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{5\,d^5}+\frac {2\,C\,b\,{\left (c+d\,x\right )}^{7/2}\,\left (3\,a^2\,d^2-12\,a\,b\,c\,d+10\,b^2\,c^2\right )}{7\,d^6}+\frac {2\,B\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d-4\,b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5}+\frac {2\,C\,c^2\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^6}-\frac {2\,a^3\,\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\,a\,b^2\,\sqrt {c+d\,x}\,D\,\left (70\,c\,{\left (c+d\,x\right )}^4-840\,c^4\,\left (c+d\,x\right )-360\,c^2\,{\left (c+d\,x\right )}^3+756\,c^3\,{\left (c+d\,x\right )}^2+630\,c^5-63\,d^5\,x^5\right )}{231\,d^6}+\frac {2\,a^2\,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 )}{105\,d^5}-\frac {2\,C\,c\,{\left (a\,d-b\,c\right )}^2\,\left (2\,a\,d-5\,b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6} \]

[In]

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

[Out]

(5544*b^3*c^6*(c + d*x)^(1/2)*D - 504*b^3*c*(c + d*x)^(11/2)*D - 9240*b^3*c^5*(c + d*x)^(3/2)*D + 11088*b^3*c^
4*(c + d*x)^(5/2)*D - 7920*b^3*c^3*(c + d*x)^(7/2)*D + 3080*b^3*c^2*(c + d*x)^(9/2)*D + 462*b^3*d^6*x^6*(c + d
*x)^(1/2)*D)/(3003*d^7) + (2*C*(c + d*x)^(5/2)*(a^3*d^3 - 10*b^3*c^3 + 18*a*b^2*c^2*d - 9*a^2*b*c*d^2))/(5*d^6
) + (2*A*b^3*(c + d*x)^(7/2))/(7*d^4) + (2*B*b^3*(c + d*x)^(9/2))/(9*d^5) + (2*C*b^3*(c + d*x)^(11/2))/(11*d^6
) + (2*A*(a*d - b*c)^3*(c + d*x)^(1/2))/d^4 + (2*A*b*(a*d - b*c)^2*(c + d*x)^(3/2))/d^4 + (6*A*b^2*(a*d - b*c)
*(c + d*x)^(5/2))/(5*d^4) + (2*B*b^2*(3*a*d - 4*b*c)*(c + d*x)^(7/2))/(7*d^5) - (2*B*c*(a*d - b*c)^3*(c + d*x)
^(1/2))/d^5 + (2*C*b^2*(3*a*d - 5*b*c)*(c + d*x)^(9/2))/(9*d^6) + (6*B*b*(c + d*x)^(5/2)*(a^2*d^2 + 2*b^2*c^2
- 3*a*b*c*d))/(5*d^5) + (2*C*b*(c + d*x)^(7/2)*(3*a^2*d^2 + 10*b^2*c^2 - 12*a*b*c*d))/(7*d^6) + (2*B*(a*d - b*
c)^2*(a*d - 4*b*c)*(c + d*x)^(3/2))/(3*d^5) + (2*C*c^2*(a*d - b*c)^3*(c + d*x)^(1/2))/d^6 - (2*a^3*(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*a*b^2*(c + d*x)^(1/2)*D*(70*c*
(c + d*x)^4 - 840*c^4*(c + d*x) - 360*c^2*(c + d*x)^3 + 756*c^3*(c + d*x)^2 + 630*c^5 - 63*d^5*x^5))/(231*d^6)
+ (2*a^2*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))/(105*d^5) - (2*C*c*(a*d - b*c)^2*(2*a*d - 5*b*c)*(c + d*x)^(3/2))/(3*d^6)