Integrand size = 32, antiderivative size = 434 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 (b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^7 \sqrt {c+d x}}-\frac {2 (b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) \sqrt {c+d x}}{d^7}-\frac {2 (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{3/2}}{3 d^7}+\frac {2 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{5/2}}{5 d^7}+\frac {2 b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{7/2}}{7 d^7}+\frac {2 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{9/2}}{9 d^7}+\frac {2 b^3 D (c+d x)^{11/2}}{11 d^7} \] Output:
2*(-a*d+b*c)^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^7/(d*x+c)^(1/2)-2*(-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)^(1/2)/d^7-2/3*(-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)^(3/2)/ d^7+2/5*(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)^(5/2)/d^7+2/7*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)^(7/2 )/d^7+2/9*b^2*(C*b*d+3*D*a*d-6*D*b*c)*(d*x+c)^(9/2)/d^7+2/11*b^3*D*(d*x+c) ^(11/2)/d^7
Time = 0.47 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \left (231 a^3 d^3 \left (48 c^3 D-8 c^2 d (5 C-3 D x)+2 c d^2 (15 B-x (10 C+3 D x))+d^3 \left (-15 A+x \left (15 B+5 C x+3 D x^2\right )\right )\right )+99 a^2 b d^2 \left (-384 c^4 D+48 c^3 d (7 C-4 D x)-8 c^2 d^2 (35 B-3 x (7 C+2 D x))+2 c d^3 (105 A-x (70 B+3 x (7 C+4 D x)))+d^4 x (105 A+x (35 B+3 x (7 C+5 D x)))\right )+33 a b^2 d \left (1280 c^5 D-128 c^4 d (9 C-5 D x)+16 c^3 d^2 (63 B-2 x (18 C+5 D x))+8 c^2 d^3 (-105 A+x (63 B+2 x (9 C+5 D x)))+d^5 x^2 (105 A+x (63 B+5 x (9 C+7 D x)))-2 c d^4 x (210 A+x (63 B+x (36 C+25 D x)))\right )+b^3 \left (-15360 c^6 D+1280 c^5 d (11 C-6 D x)-128 c^4 d^2 (99 B-5 x (11 C+3 D x))+16 c^3 d^3 (693 A-2 x (198 B+5 x (11 C+6 D x)))+d^6 x^3 (693 A+5 x (99 B+7 x (11 C+9 D x)))+8 c^2 d^4 x (693 A+x (198 B+5 x (22 C+15 D x)))-2 c d^5 x^2 (693 A+x (396 B+5 x (55 C+42 D x)))\right )\right )}{3465 d^7 \sqrt {c+d x}} \] Input:
Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]
Output:
(2*(231*a^3*d^3*(48*c^3*D - 8*c^2*d*(5*C - 3*D*x) + 2*c*d^2*(15*B - x*(10* C + 3*D*x)) + d^3*(-15*A + x*(15*B + 5*C*x + 3*D*x^2))) + 99*a^2*b*d^2*(-3 84*c^4*D + 48*c^3*d*(7*C - 4*D*x) - 8*c^2*d^2*(35*B - 3*x*(7*C + 2*D*x)) + 2*c*d^3*(105*A - x*(70*B + 3*x*(7*C + 4*D*x))) + d^4*x*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)))) + 33*a*b^2*d*(1280*c^5*D - 128*c^4*d*(9*C - 5*D*x) + 16*c^3*d^2*(63*B - 2*x*(18*C + 5*D*x)) + 8*c^2*d^3*(-105*A + x*(63*B + 2* x*(9*C + 5*D*x))) + d^5*x^2*(105*A + x*(63*B + 5*x*(9*C + 7*D*x))) - 2*c*d ^4*x*(210*A + x*(63*B + x*(36*C + 25*D*x)))) + b^3*(-15360*c^6*D + 1280*c^ 5*d*(11*C - 6*D*x) - 128*c^4*d^2*(99*B - 5*x*(11*C + 3*D*x)) + 16*c^3*d^3* (693*A - 2*x*(198*B + 5*x*(11*C + 6*D*x))) + d^6*x^3*(693*A + 5*x*(99*B + 7*x*(11*C + 9*D*x))) + 8*c^2*d^4*x*(693*A + x*(198*B + 5*x*(22*C + 15*D*x) )) - 2*c*d^5*x^2*(693*A + x*(396*B + 5*x*(55*C + 42*D*x))))))/(3465*d^7*Sq rt[c + d*x])
Time = 0.77 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2123, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \int \left (\frac {\sqrt {c+d x} (b c-a d) \left (-a^2 d^2 (C d-3 c D)+a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )-\left (b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{d^6}+\frac {b (c+d x)^{5/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{d^6}+\frac {(c+d x)^{3/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{d^6}+\frac {(b c-a d)^2 \left (b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )-a d \left (-B d^2-3 c^2 D+2 c C d\right )\right )}{d^6 \sqrt {c+d x}}+\frac {(a d-b c)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6 (c+d x)^{3/2}}+\frac {b^2 (c+d x)^{7/2} (3 a d D-6 b c D+b C d)}{d^6}+\frac {b^3 D (c+d x)^{9/2}}{d^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 (c+d x)^{3/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{3 d^7}+\frac {2 b (c+d x)^{7/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{7 d^7}+\frac {2 (c+d x)^{5/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{5 d^7}-\frac {2 \sqrt {c+d x} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7}+\frac {2 (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7 \sqrt {c+d x}}+\frac {2 b^2 (c+d x)^{9/2} (3 a d D-6 b c D+b C d)}{9 d^7}+\frac {2 b^3 D (c+d x)^{11/2}}{11 d^7}\) |
Input:
Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]
Output:
(2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^7*Sqrt[c + d*x]) - (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))*Sqrt[c + d*x])/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)^(3/2))/(3*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*(1 0*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(5/2))/(5*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)^(7/2))/(7*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^(9 /2))/(9*d^7) + (2*b^3*D*(c + d*x)^(11/2))/(11*d^7)
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])
Time = 0.62 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (-\frac {x^{3} \left (\frac {5}{11} D x^{3}+\frac {5}{9} C \,x^{2}+\frac {5}{7} B x +A \right ) b^{3}}{5}-x^{2} \left (\frac {1}{3} D x^{3}+\frac {3}{7} C \,x^{2}+\frac {3}{5} B x +A \right ) a \,b^{2}-3 x \left (\frac {1}{5} C \,x^{2}+\frac {1}{3} B x +\frac {1}{7} D x^{3}+A \right ) a^{2} b +a^{3} \left (-\frac {1}{5} D x^{3}-\frac {1}{3} C \,x^{2}-B x +A \right )\right ) d^{6}-6 \left (-\frac {x^{2} \left (\frac {10}{33} D x^{3}+\frac {25}{63} C \,x^{2}+\frac {4}{7} B x +A \right ) b^{3}}{15}-\frac {2 x \left (\frac {5}{42} D x^{3}+\frac {6}{35} C \,x^{2}+\frac {3}{10} B x +A \right ) a \,b^{2}}{3}+a^{2} \left (-\frac {4}{35} D x^{3}-\frac {1}{5} C \,x^{2}-\frac {2}{3} B x +A \right ) b +\frac {\left (-\frac {1}{5} D x^{2}-\frac {2}{3} C x +B \right ) a^{3}}{3}\right ) c \,d^{5}+8 \left (-\frac {x \left (\frac {25}{231} D x^{3}+\frac {10}{63} C \,x^{2}+\frac {2}{7} B x +A \right ) b^{3}}{5}+a \left (-\frac {2}{21} D x^{3}-\frac {6}{35} C \,x^{2}-\frac {3}{5} B x +A \right ) b^{2}+a^{2} \left (-\frac {6}{35} D x^{2}-\frac {3}{5} C x +B \right ) b +\frac {\left (-\frac {3 D x}{5}+C \right ) a^{3}}{3}\right ) c^{2} d^{4}-\frac {16 \left (\left (-\frac {20}{231} D x^{3}-\frac {10}{63} C \,x^{2}-\frac {4}{7} B x +A \right ) b^{3}+3 \left (-\frac {10}{63} D x^{2}-\frac {4}{7} C x +B \right ) a \,b^{2}+3 \left (-\frac {4 D x}{7}+C \right ) a^{2} b +a^{3} D\right ) c^{3} d^{3}}{5}+\frac {128 \left (\left (-\frac {5}{33} D x^{2}-\frac {5}{9} C x +B \right ) b^{2}+3 \left (-\frac {5 D x}{9}+C \right ) a b +3 D a^{2}\right ) c^{4} b \,d^{2}}{35}-\frac {256 c^{5} b^{2} \left (\left (-\frac {6 D x}{11}+C \right ) b +3 D a \right ) d}{63}+\frac {1024 D b^{3} c^{6}}{231}\right )}{\sqrt {x d +c}\, d^{7}}\) | \(440\) |
gosper | \(-\frac {2 \left (-315 D x^{6} b^{3} d^{6}-385 C \,x^{5} b^{3} d^{6}-1155 D x^{5} a \,b^{2} d^{6}+420 D x^{5} b^{3} c \,d^{5}-495 B \,x^{4} b^{3} d^{6}-1485 C \,x^{4} a \,b^{2} d^{6}+550 C \,x^{4} b^{3} c \,d^{5}-1485 D x^{4} a^{2} b \,d^{6}+1650 D x^{4} a \,b^{2} c \,d^{5}-600 D x^{4} b^{3} c^{2} d^{4}-693 A \,x^{3} b^{3} d^{6}-2079 B \,x^{3} a \,b^{2} d^{6}+792 B \,x^{3} b^{3} c \,d^{5}-2079 C \,x^{3} a^{2} b \,d^{6}+2376 C \,x^{3} a \,b^{2} c \,d^{5}-880 C \,x^{3} b^{3} c^{2} d^{4}-693 D x^{3} a^{3} d^{6}+2376 D x^{3} a^{2} b c \,d^{5}-2640 D x^{3} a \,b^{2} c^{2} d^{4}+960 D x^{3} b^{3} c^{3} d^{3}-3465 A \,x^{2} a \,b^{2} d^{6}+1386 A \,x^{2} b^{3} c \,d^{5}-3465 B \,x^{2} a^{2} b \,d^{6}+4158 B \,x^{2} a \,b^{2} c \,d^{5}-1584 B \,x^{2} b^{3} c^{2} d^{4}-1155 C \,x^{2} a^{3} d^{6}+4158 C \,x^{2} a^{2} b c \,d^{5}-4752 C \,x^{2} a \,b^{2} c^{2} d^{4}+1760 C \,x^{2} b^{3} c^{3} d^{3}+1386 D x^{2} a^{3} c \,d^{5}-4752 D x^{2} a^{2} b \,c^{2} d^{4}+5280 D x^{2} a \,b^{2} c^{3} d^{3}-1920 D x^{2} b^{3} c^{4} d^{2}-10395 A x \,a^{2} b \,d^{6}+13860 A x a \,b^{2} c \,d^{5}-5544 A x \,b^{3} c^{2} d^{4}-3465 B x \,a^{3} d^{6}+13860 B x \,a^{2} b c \,d^{5}-16632 B x a \,b^{2} c^{2} d^{4}+6336 B x \,b^{3} c^{3} d^{3}+4620 C x \,a^{3} c \,d^{5}-16632 C x \,a^{2} b \,c^{2} d^{4}+19008 C x a \,b^{2} c^{3} d^{3}-7040 C x \,b^{3} c^{4} d^{2}-5544 D x \,a^{3} c^{2} d^{4}+19008 D x \,a^{2} b \,c^{3} d^{3}-21120 D x a \,b^{2} c^{4} d^{2}+7680 D x \,b^{3} c^{5} d +3465 A \,a^{3} d^{6}-20790 A \,a^{2} b c \,d^{5}+27720 A a \,b^{2} c^{2} d^{4}-11088 A \,b^{3} c^{3} d^{3}-6930 B \,a^{3} c \,d^{5}+27720 B \,a^{2} b \,c^{2} d^{4}-33264 B a \,b^{2} c^{3} d^{3}+12672 B \,b^{3} c^{4} d^{2}+9240 C \,a^{3} c^{2} d^{4}-33264 C \,a^{2} b \,c^{3} d^{3}+38016 C a \,b^{2} c^{4} d^{2}-14080 C \,b^{3} c^{5} d -11088 D a^{3} c^{3} d^{3}+38016 D a^{2} b \,c^{4} d^{2}-42240 D a \,b^{2} c^{5} d +15360 D b^{3} c^{6}\right )}{3465 \sqrt {x d +c}\, d^{7}}\) | \(841\) |
trager | \(-\frac {2 \left (-315 D x^{6} b^{3} d^{6}-385 C \,x^{5} b^{3} d^{6}-1155 D x^{5} a \,b^{2} d^{6}+420 D x^{5} b^{3} c \,d^{5}-495 B \,x^{4} b^{3} d^{6}-1485 C \,x^{4} a \,b^{2} d^{6}+550 C \,x^{4} b^{3} c \,d^{5}-1485 D x^{4} a^{2} b \,d^{6}+1650 D x^{4} a \,b^{2} c \,d^{5}-600 D x^{4} b^{3} c^{2} d^{4}-693 A \,x^{3} b^{3} d^{6}-2079 B \,x^{3} a \,b^{2} d^{6}+792 B \,x^{3} b^{3} c \,d^{5}-2079 C \,x^{3} a^{2} b \,d^{6}+2376 C \,x^{3} a \,b^{2} c \,d^{5}-880 C \,x^{3} b^{3} c^{2} d^{4}-693 D x^{3} a^{3} d^{6}+2376 D x^{3} a^{2} b c \,d^{5}-2640 D x^{3} a \,b^{2} c^{2} d^{4}+960 D x^{3} b^{3} c^{3} d^{3}-3465 A \,x^{2} a \,b^{2} d^{6}+1386 A \,x^{2} b^{3} c \,d^{5}-3465 B \,x^{2} a^{2} b \,d^{6}+4158 B \,x^{2} a \,b^{2} c \,d^{5}-1584 B \,x^{2} b^{3} c^{2} d^{4}-1155 C \,x^{2} a^{3} d^{6}+4158 C \,x^{2} a^{2} b c \,d^{5}-4752 C \,x^{2} a \,b^{2} c^{2} d^{4}+1760 C \,x^{2} b^{3} c^{3} d^{3}+1386 D x^{2} a^{3} c \,d^{5}-4752 D x^{2} a^{2} b \,c^{2} d^{4}+5280 D x^{2} a \,b^{2} c^{3} d^{3}-1920 D x^{2} b^{3} c^{4} d^{2}-10395 A x \,a^{2} b \,d^{6}+13860 A x a \,b^{2} c \,d^{5}-5544 A x \,b^{3} c^{2} d^{4}-3465 B x \,a^{3} d^{6}+13860 B x \,a^{2} b c \,d^{5}-16632 B x a \,b^{2} c^{2} d^{4}+6336 B x \,b^{3} c^{3} d^{3}+4620 C x \,a^{3} c \,d^{5}-16632 C x \,a^{2} b \,c^{2} d^{4}+19008 C x a \,b^{2} c^{3} d^{3}-7040 C x \,b^{3} c^{4} d^{2}-5544 D x \,a^{3} c^{2} d^{4}+19008 D x \,a^{2} b \,c^{3} d^{3}-21120 D x a \,b^{2} c^{4} d^{2}+7680 D x \,b^{3} c^{5} d +3465 A \,a^{3} d^{6}-20790 A \,a^{2} b c \,d^{5}+27720 A a \,b^{2} c^{2} d^{4}-11088 A \,b^{3} c^{3} d^{3}-6930 B \,a^{3} c \,d^{5}+27720 B \,a^{2} b \,c^{2} d^{4}-33264 B a \,b^{2} c^{3} d^{3}+12672 B \,b^{3} c^{4} d^{2}+9240 C \,a^{3} c^{2} d^{4}-33264 C \,a^{2} b \,c^{3} d^{3}+38016 C a \,b^{2} c^{4} d^{2}-14080 C \,b^{3} c^{5} d -11088 D a^{3} c^{3} d^{3}+38016 D a^{2} b \,c^{4} d^{2}-42240 D a \,b^{2} c^{5} d +15360 D b^{3} c^{6}\right )}{3465 \sqrt {x d +c}\, d^{7}}\) | \(841\) |
orering | \(-\frac {2 \left (-315 D x^{6} b^{3} d^{6}-385 C \,x^{5} b^{3} d^{6}-1155 D x^{5} a \,b^{2} d^{6}+420 D x^{5} b^{3} c \,d^{5}-495 B \,x^{4} b^{3} d^{6}-1485 C \,x^{4} a \,b^{2} d^{6}+550 C \,x^{4} b^{3} c \,d^{5}-1485 D x^{4} a^{2} b \,d^{6}+1650 D x^{4} a \,b^{2} c \,d^{5}-600 D x^{4} b^{3} c^{2} d^{4}-693 A \,x^{3} b^{3} d^{6}-2079 B \,x^{3} a \,b^{2} d^{6}+792 B \,x^{3} b^{3} c \,d^{5}-2079 C \,x^{3} a^{2} b \,d^{6}+2376 C \,x^{3} a \,b^{2} c \,d^{5}-880 C \,x^{3} b^{3} c^{2} d^{4}-693 D x^{3} a^{3} d^{6}+2376 D x^{3} a^{2} b c \,d^{5}-2640 D x^{3} a \,b^{2} c^{2} d^{4}+960 D x^{3} b^{3} c^{3} d^{3}-3465 A \,x^{2} a \,b^{2} d^{6}+1386 A \,x^{2} b^{3} c \,d^{5}-3465 B \,x^{2} a^{2} b \,d^{6}+4158 B \,x^{2} a \,b^{2} c \,d^{5}-1584 B \,x^{2} b^{3} c^{2} d^{4}-1155 C \,x^{2} a^{3} d^{6}+4158 C \,x^{2} a^{2} b c \,d^{5}-4752 C \,x^{2} a \,b^{2} c^{2} d^{4}+1760 C \,x^{2} b^{3} c^{3} d^{3}+1386 D x^{2} a^{3} c \,d^{5}-4752 D x^{2} a^{2} b \,c^{2} d^{4}+5280 D x^{2} a \,b^{2} c^{3} d^{3}-1920 D x^{2} b^{3} c^{4} d^{2}-10395 A x \,a^{2} b \,d^{6}+13860 A x a \,b^{2} c \,d^{5}-5544 A x \,b^{3} c^{2} d^{4}-3465 B x \,a^{3} d^{6}+13860 B x \,a^{2} b c \,d^{5}-16632 B x a \,b^{2} c^{2} d^{4}+6336 B x \,b^{3} c^{3} d^{3}+4620 C x \,a^{3} c \,d^{5}-16632 C x \,a^{2} b \,c^{2} d^{4}+19008 C x a \,b^{2} c^{3} d^{3}-7040 C x \,b^{3} c^{4} d^{2}-5544 D x \,a^{3} c^{2} d^{4}+19008 D x \,a^{2} b \,c^{3} d^{3}-21120 D x a \,b^{2} c^{4} d^{2}+7680 D x \,b^{3} c^{5} d +3465 A \,a^{3} d^{6}-20790 A \,a^{2} b c \,d^{5}+27720 A a \,b^{2} c^{2} d^{4}-11088 A \,b^{3} c^{3} d^{3}-6930 B \,a^{3} c \,d^{5}+27720 B \,a^{2} b \,c^{2} d^{4}-33264 B a \,b^{2} c^{3} d^{3}+12672 B \,b^{3} c^{4} d^{2}+9240 C \,a^{3} c^{2} d^{4}-33264 C \,a^{2} b \,c^{3} d^{3}+38016 C a \,b^{2} c^{4} d^{2}-14080 C \,b^{3} c^{5} d -11088 D a^{3} c^{3} d^{3}+38016 D a^{2} b \,c^{4} d^{2}-42240 D a \,b^{2} c^{5} d +15360 D b^{3} c^{6}\right )}{3465 \sqrt {x d +c}\, d^{7}}\) | \(841\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1023\) |
default | \(\text {Expression too large to display}\) | \(1023\) |
Input:
int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*((-1/5*x^3*(5/11*D*x^3+5/9*C*x^2+5/7*B*x+A)*b^3-x^2*(1/3*D*x^3+3/7*C*x^ 2+3/5*B*x+A)*a*b^2-3*x*(1/5*C*x^2+1/3*B*x+1/7*D*x^3+A)*a^2*b+a^3*(-1/5*D*x ^3-1/3*C*x^2-B*x+A))*d^6-6*(-1/15*x^2*(10/33*D*x^3+25/63*C*x^2+4/7*B*x+A)* b^3-2/3*x*(5/42*D*x^3+6/35*C*x^2+3/10*B*x+A)*a*b^2+a^2*(-4/35*D*x^3-1/5*C* x^2-2/3*B*x+A)*b+1/3*(-1/5*D*x^2-2/3*C*x+B)*a^3)*c*d^5+8*(-1/5*x*(25/231*D *x^3+10/63*C*x^2+2/7*B*x+A)*b^3+a*(-2/21*D*x^3-6/35*C*x^2-3/5*B*x+A)*b^2+a ^2*(-6/35*D*x^2-3/5*C*x+B)*b+1/3*(-3/5*D*x+C)*a^3)*c^2*d^4-16/5*((-20/231* D*x^3-10/63*C*x^2-4/7*B*x+A)*b^3+3*(-10/63*D*x^2-4/7*C*x+B)*a*b^2+3*(-4/7* D*x+C)*a^2*b+a^3*D)*c^3*d^3+128/35*((-5/33*D*x^2-5/9*C*x+B)*b^2+3*(-5/9*D* x+C)*a*b+3*D*a^2)*c^4*b*d^2-256/63*c^5*b^2*((-6/11*D*x+C)*b+3*D*a)*d+1024/ 231*D*b^3*c^6)/(d*x+c)^(1/2)/d^7
Time = 0.08 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (315 \, D b^{3} d^{6} x^{6} - 15360 \, D b^{3} c^{6} - 3465 \, A a^{3} d^{6} + 14080 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d - 12672 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} + 11088 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} - 9240 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} + 6930 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} - 35 \, {\left (12 \, D b^{3} c d^{5} - 11 \, {\left (3 \, D a b^{2} + C b^{3}\right )} d^{6}\right )} x^{5} + 5 \, {\left (120 \, D b^{3} c^{2} d^{4} - 110 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c d^{5} + 99 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{6}\right )} x^{4} - {\left (960 \, D b^{3} c^{3} d^{3} - 880 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d^{4} + 792 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{5} - 693 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{6}\right )} x^{3} + {\left (1920 \, D b^{3} c^{4} d^{2} - 1760 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d^{3} + 1584 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{4} - 1386 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{5} + 1155 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6}\right )} x^{2} - {\left (7680 \, D b^{3} c^{5} d - 7040 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d^{2} + 6336 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{3} - 5544 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{4} + 4620 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} - 3465 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6}\right )} x\right )} \sqrt {d x + c}}{3465 \, {\left (d^{8} x + c d^{7}\right )}} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm="fricas ")
Output:
2/3465*(315*D*b^3*d^6*x^6 - 15360*D*b^3*c^6 - 3465*A*a^3*d^6 + 14080*(3*D* a*b^2 + C*b^3)*c^5*d - 12672*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 + 110 88*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 - 9240*(C*a^3 + 3*B*a^2 *b + 3*A*a*b^2)*c^2*d^4 + 6930*(B*a^3 + 3*A*a^2*b)*c*d^5 - 35*(12*D*b^3*c* d^5 - 11*(3*D*a*b^2 + C*b^3)*d^6)*x^5 + 5*(120*D*b^3*c^2*d^4 - 110*(3*D*a* b^2 + C*b^3)*c*d^5 + 99*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^6)*x^4 - (960*D* b^3*c^3*d^3 - 880*(3*D*a*b^2 + C*b^3)*c^2*d^4 + 792*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^5 - 693*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^6)*x^3 + ( 1920*D*b^3*c^4*d^2 - 1760*(3*D*a*b^2 + C*b^3)*c^3*d^3 + 1584*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^2*d^4 - 1386*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)* c*d^5 + 1155*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^6)*x^2 - (7680*D*b^3*c^5*d - 7040*(3*D*a*b^2 + C*b^3)*c^4*d^2 + 6336*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)* c^3*d^3 - 5544*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^4 + 4620*(C*a ^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^5 - 3465*(B*a^3 + 3*A*a^2*b)*d^6)*x)*sqrt( d*x + c)/(d^8*x + c*d^7)
Time = 84.52 (sec) , antiderivative size = 853, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {D b^{3} \left (c + d x\right )^{\frac {11}{2}}}{11 d^{6}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (C b^{3} d + 3 D a b^{2} d - 6 D b^{3} c\right )}{9 d^{6}} + \frac {\left (c + d x\right )^{\frac {7}{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 )}{7 d^{6}} + \frac {\left (c + d x\right )^{\frac {5}{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 )}{5 d^{6}} + \frac {\left (c + d x\right )^{\frac {3}{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 )}{3 d^{6}} + \frac {\sqrt {c + d x} \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 )}{d^{6}} + \frac {\left (a d - b c\right )^{3} \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{6} \sqrt {c + d x}}\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}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(3/2),x)
Output:
Piecewise((2*(D*b**3*(c + d*x)**(11/2)/(11*d**6) + (c + d*x)**(9/2)*(C*b** 3*d + 3*D*a*b**2*d - 6*D*b**3*c)/(9*d**6) + (c + d*x)**(7/2)*(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)/(7*d**6) + (c + d*x)**(5/2)*(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)/ (5*d**6) + (c + d*x)**(3/2)*(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)/(3*d**6) + sqrt( c + d*x)*(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)/d**6 + (a*d - b*c)**3*(-A*d**3 + B*c*d**2 - C*c**2*d + D*c* *3)/(d**6*sqrt(c + d*x)))/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)/c**(3/2), True))
Time = 0.04 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {315 \, {\left (d x + c\right )}^{\frac {11}{2}} D b^{3} - 385 \, {\left (6 \, D b^{3} c - {\left (3 \, D a b^{2} + C b^{3}\right )} d\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 495 \, {\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 {7}{2}} - 693 \, {\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 {5}{2}} + 1155 \, {\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 {3}{2}} - 3465 \, {\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 )} \sqrt {d x + c}}{d^{6}} - \frac {3465 \, {\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} d^{6}}\right )}}{3465 \, d} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm="maxima ")
Output:
2/3465*((315*(d*x + c)^(11/2)*D*b^3 - 385*(6*D*b^3*c - (3*D*a*b^2 + C*b^3) *d)*(d*x + c)^(9/2) + 495*(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)^(7/2) - 693*(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)^(5/2) + 1155*(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)^(3/2) - 3465*(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)*sqrt(d*x + c))/d^6 - 3465*(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^6))/d
Leaf count of result is larger than twice the leaf count of optimal. 1067 vs. \(2 (412) = 824\).
Time = 0.14 (sec) , antiderivative size = 1067, normalized size of antiderivative = 2.46 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm="giac")
Output:
-2*(D*b^3*c^6 - 3*D*a*b^2*c^5*d - C*b^3*c^5*d + 3*D*a^2*b*c^4*d^2 + 3*C*a* b^2*c^4*d^2 + B*b^3*c^4*d^2 - D*a^3*c^3*d^3 - 3*C*a^2*b*c^3*d^3 - 3*B*a*b^ 2*c^3*d^3 - A*b^3*c^3*d^3 + C*a^3*c^2*d^4 + 3*B*a^2*b*c^2*d^4 + 3*A*a*b^2* c^2*d^4 - B*a^3*c*d^5 - 3*A*a^2*b*c*d^5 + A*a^3*d^6)/(sqrt(d*x + c)*d^7) + 2/3465*(315*(d*x + c)^(11/2)*D*b^3*d^70 - 2310*(d*x + c)^(9/2)*D*b^3*c*d^ 70 + 7425*(d*x + c)^(7/2)*D*b^3*c^2*d^70 - 13860*(d*x + c)^(5/2)*D*b^3*c^3 *d^70 + 17325*(d*x + c)^(3/2)*D*b^3*c^4*d^70 - 20790*sqrt(d*x + c)*D*b^3*c ^5*d^70 + 1155*(d*x + c)^(9/2)*D*a*b^2*d^71 + 385*(d*x + c)^(9/2)*C*b^3*d^ 71 - 7425*(d*x + c)^(7/2)*D*a*b^2*c*d^71 - 2475*(d*x + c)^(7/2)*C*b^3*c*d^ 71 + 20790*(d*x + c)^(5/2)*D*a*b^2*c^2*d^71 + 6930*(d*x + c)^(5/2)*C*b^3*c ^2*d^71 - 34650*(d*x + c)^(3/2)*D*a*b^2*c^3*d^71 - 11550*(d*x + c)^(3/2)*C *b^3*c^3*d^71 + 51975*sqrt(d*x + c)*D*a*b^2*c^4*d^71 + 17325*sqrt(d*x + c) *C*b^3*c^4*d^71 + 1485*(d*x + c)^(7/2)*D*a^2*b*d^72 + 1485*(d*x + c)^(7/2) *C*a*b^2*d^72 + 495*(d*x + c)^(7/2)*B*b^3*d^72 - 8316*(d*x + c)^(5/2)*D*a^ 2*b*c*d^72 - 8316*(d*x + c)^(5/2)*C*a*b^2*c*d^72 - 2772*(d*x + c)^(5/2)*B* b^3*c*d^72 + 20790*(d*x + c)^(3/2)*D*a^2*b*c^2*d^72 + 20790*(d*x + c)^(3/2 )*C*a*b^2*c^2*d^72 + 6930*(d*x + c)^(3/2)*B*b^3*c^2*d^72 - 41580*sqrt(d*x + c)*D*a^2*b*c^3*d^72 - 41580*sqrt(d*x + c)*C*a*b^2*c^3*d^72 - 13860*sqrt( d*x + c)*B*b^3*c^3*d^72 + 693*(d*x + c)^(5/2)*D*a^3*d^73 + 2079*(d*x + c)^ (5/2)*C*a^2*b*d^73 + 2079*(d*x + c)^(5/2)*B*a*b^2*d^73 + 693*(d*x + c)^...
Timed out. \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^3\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(3/2),x)
Output:
int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(3/2), x)
Time = 0.14 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {\frac {64}{5} a \,b^{3} c^{2} d^{3} x -\frac {96}{35} a^{2} b \,c^{4} d^{2}-2 a^{4} d^{5}-\frac {512}{693} b^{3} c^{6}+16 a^{3} b c \,d^{4}+8 a^{3} b \,d^{5} x +\frac {8}{15} a^{3} c^{2} d^{4} x -\frac {2}{15} a^{3} c \,d^{5} x^{2}-32 a^{2} b^{2} c^{2} d^{3}+4 a^{2} b^{2} d^{5} x^{2}+\frac {6}{7} a^{2} b \,d^{6} x^{4}+\frac {128}{5} a \,b^{3} c^{3} d^{2}+\frac {8}{5} a \,b^{3} d^{5} x^{3}+\frac {256}{105} a \,b^{2} c^{5} d +\frac {2}{3} a \,b^{2} d^{6} x^{5}-\frac {128}{35} b^{4} c^{3} d^{2} x +\frac {32}{35} b^{4} c^{2} d^{3} x^{2}-\frac {16}{35} b^{4} c \,d^{4} x^{3}-\frac {256}{693} b^{3} c^{5} d x +\frac {64}{693} b^{3} c^{4} d^{2} x^{2}-\frac {32}{693} b^{3} c^{3} d^{3} x^{3}+\frac {20}{693} b^{3} c^{2} d^{4} x^{4}-\frac {2}{99} b^{3} c \,d^{5} x^{5}+\frac {16}{15} a^{3} c^{3} d^{3}+\frac {2}{5} a^{3} d^{6} x^{3}-\frac {256}{35} b^{4} c^{4} d +\frac {2}{7} b^{4} d^{5} x^{4}+\frac {2}{11} b^{3} d^{6} x^{6}-16 a^{2} b^{2} c \,d^{4} x -\frac {48}{35} a^{2} b \,c^{3} d^{3} x +\frac {12}{35} a^{2} b \,c^{2} d^{4} x^{2}-\frac {6}{35} a^{2} b c \,d^{5} x^{3}-\frac {16}{5} a \,b^{3} c \,d^{4} x^{2}+\frac {128}{105} a \,b^{2} c^{4} d^{2} x -\frac {32}{105} a \,b^{2} c^{3} d^{3} x^{2}+\frac {16}{105} a \,b^{2} c^{2} d^{4} x^{3}-\frac {2}{21} a \,b^{2} c \,d^{5} x^{4}}{\sqrt {d x +c}\, d^{6}} \] Input:
int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x)
Output:
(2*( - 3465*a**4*d**5 + 27720*a**3*b*c*d**4 + 13860*a**3*b*d**5*x + 1848*a **3*c**3*d**3 + 924*a**3*c**2*d**4*x - 231*a**3*c*d**5*x**2 + 693*a**3*d** 6*x**3 - 55440*a**2*b**2*c**2*d**3 - 27720*a**2*b**2*c*d**4*x + 6930*a**2* b**2*d**5*x**2 - 4752*a**2*b*c**4*d**2 - 2376*a**2*b*c**3*d**3*x + 594*a** 2*b*c**2*d**4*x**2 - 297*a**2*b*c*d**5*x**3 + 1485*a**2*b*d**6*x**4 + 4435 2*a*b**3*c**3*d**2 + 22176*a*b**3*c**2*d**3*x - 5544*a*b**3*c*d**4*x**2 + 2772*a*b**3*d**5*x**3 + 4224*a*b**2*c**5*d + 2112*a*b**2*c**4*d**2*x - 528 *a*b**2*c**3*d**3*x**2 + 264*a*b**2*c**2*d**4*x**3 - 165*a*b**2*c*d**5*x** 4 + 1155*a*b**2*d**6*x**5 - 12672*b**4*c**4*d - 6336*b**4*c**3*d**2*x + 15 84*b**4*c**2*d**3*x**2 - 792*b**4*c*d**4*x**3 + 495*b**4*d**5*x**4 - 1280* b**3*c**6 - 640*b**3*c**5*d*x + 160*b**3*c**4*d**2*x**2 - 80*b**3*c**3*d** 3*x**3 + 50*b**3*c**2*d**4*x**4 - 35*b**3*c*d**5*x**5 + 315*b**3*d**6*x**6 ))/(3465*sqrt(c + d*x)*d**6)