Integrand size = 32, antiderivative size = 434 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 (b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^7 \sqrt {c+d x}}-\frac {2 (b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) \sqrt {c+d x}}{d^7}-\frac {2 (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{3/2}}{3 d^7}+\frac {2 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{5/2}}{5 d^7}+\frac {2 b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{7/2}}{7 d^7}+\frac {2 b^2 (b C d-6 b c D+3 a d D) (c+d x)^{9/2}}{9 d^7}+\frac {2 b^3 D (c+d x)^{11/2}}{11 d^7} \]
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Time = 0.25 (sec) , antiderivative size = 434, 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} \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=-\frac {2 (c+d x)^{3/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{3 d^7}+\frac {2 b (c+d x)^{7/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{7 d^7}+\frac {2 (c+d x)^{5/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{5 d^7}-\frac {2 \sqrt {c+d x} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7}+\frac {2 (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7 \sqrt {c+d x}}+\frac {2 b^2 (c+d x)^{9/2} (3 a d D-6 b c D+b C d)}{9 d^7}+\frac {2 b^3 D (c+d x)^{11/2}}{11 d^7} \]
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Rule 1634
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 (c+d x)^{3/2}}+\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 )}{d^6 \sqrt {c+d x}}+\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 ) \sqrt {c+d x}}{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)^{3/2}}{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)^{5/2}}{d^6}+\frac {b^2 (b C d-6 b c D+3 a d D) (c+d x)^{7/2}}{d^6}+\frac {b^3 D (c+d x)^{9/2}}{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 )}{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} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \left (231 a^3 d^3 \left (48 c^3 D-8 c^2 d (5 C-3 D x)+2 c d^2 (15 B-x (10 C+3 D x))+d^3 \left (-15 A+x \left (15 B+5 C x+3 D x^2\right )\right )\right )+99 a^2 b d^2 \left (-384 c^4 D+48 c^3 d (7 C-4 D x)-8 c^2 d^2 (35 B-3 x (7 C+2 D x))+2 c d^3 (105 A-x (70 B+3 x (7 C+4 D x)))+d^4 x (105 A+x (35 B+3 x (7 C+5 D x)))\right )+33 a b^2 d \left (1280 c^5 D-128 c^4 d (9 C-5 D x)+16 c^3 d^2 (63 B-2 x (18 C+5 D x))+8 c^2 d^3 (-105 A+x (63 B+2 x (9 C+5 D x)))+d^5 x^2 (105 A+x (63 B+5 x (9 C+7 D x)))-2 c d^4 x (210 A+x (63 B+x (36 C+25 D x)))\right )+b^3 \left (-15360 c^6 D+1280 c^5 d (11 C-6 D x)-128 c^4 d^2 (99 B-5 x (11 C+3 D x))+16 c^3 d^3 (693 A-2 x (198 B+5 x (11 C+6 D x)))+d^6 x^3 (693 A+5 x (99 B+7 x (11 C+9 D x)))+8 c^2 d^4 x (693 A+x (198 B+5 x (22 C+15 D x)))-2 c d^5 x^2 (693 A+x (396 B+5 x (55 C+42 D x)))\right )\right )}{3465 d^7 \sqrt {c+d x}} \]
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Time = 1.77 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (-\frac {x^{3} \left (\frac {5}{11} D x^{3}+\frac {5}{9} C \,x^{2}+\frac {5}{7} B x +A \right ) b^{3}}{5}-a \,x^{2} \left (\frac {1}{3} D x^{3}+\frac {3}{7} C \,x^{2}+\frac {3}{5} B x +A \right ) b^{2}-3 a^{2} \left (A +\frac {1}{7} D x^{3}+\frac {1}{5} C \,x^{2}+\frac {1}{3} B x \right ) x b +a^{3} \left (-\frac {1}{5} D x^{3}-\frac {1}{3} C \,x^{2}-B x +A \right )\right ) d^{6}-6 c \left (-\frac {x^{2} \left (\frac {10}{33} D x^{3}+\frac {25}{63} C \,x^{2}+\frac {4}{7} B x +A \right ) b^{3}}{15}-\frac {2 a \left (\frac {5}{42} D x^{3}+\frac {6}{35} C \,x^{2}+\frac {3}{10} B x +A \right ) x \,b^{2}}{3}+a^{2} \left (-\frac {4}{35} D x^{3}-\frac {1}{5} C \,x^{2}-\frac {2}{3} B x +A \right ) b +\frac {a^{3} \left (-\frac {1}{5} D x^{2}-\frac {2}{3} C x +B \right )}{3}\right ) d^{5}+8 \left (-\frac {\left (\frac {25}{231} D x^{3}+\frac {10}{63} C \,x^{2}+\frac {2}{7} B x +A \right ) x \,b^{3}}{5}+a \left (-\frac {2}{21} D x^{3}-\frac {6}{35} C \,x^{2}-\frac {3}{5} B x +A \right ) b^{2}+a^{2} \left (-\frac {6}{35} D x^{2}-\frac {3}{5} C x +B \right ) b +\frac {a^{3} \left (-\frac {3 D x}{5}+C \right )}{3}\right ) c^{2} d^{4}-\frac {16 c^{3} \left (\left (-\frac {20}{231} D x^{3}-\frac {10}{63} C \,x^{2}-\frac {4}{7} B x +A \right ) b^{3}+3 a \left (-\frac {10}{63} D x^{2}-\frac {4}{7} C x +B \right ) b^{2}+3 a^{2} \left (-\frac {4 D x}{7}+C \right ) b +D a^{3}\right ) d^{3}}{5}+\frac {128 b \,c^{4} \left (\left (-\frac {5}{33} D x^{2}-\frac {5}{9} C x +B \right ) b^{2}+3 a \left (-\frac {5 D x}{9}+C \right ) b +3 D a^{2}\right ) d^{2}}{35}-\frac {256 \left (\left (-\frac {6 D x}{11}+C \right ) b +3 D a \right ) b^{2} c^{5} d}{63}+\frac {1024 D b^{3} c^{6}}{231}\right )}{\sqrt {d x +c}\, d^{7}}\) | \(440\) |
gosper | \(-\frac {2 \left (-315 D b^{3} x^{6} d^{6}-385 C \,b^{3} d^{6} x^{5}-1155 D a \,b^{2} d^{6} x^{5}+420 D b^{3} c \,d^{5} x^{5}-495 B \,b^{3} d^{6} x^{4}-1485 C a \,b^{2} d^{6} x^{4}+550 C \,b^{3} c \,d^{5} x^{4}-1485 D a^{2} b \,d^{6} x^{4}+1650 D a \,b^{2} c \,d^{5} x^{4}-600 D b^{3} c^{2} d^{4} x^{4}-693 A \,b^{3} d^{6} x^{3}-2079 B a \,b^{2} d^{6} x^{3}+792 B \,b^{3} c \,d^{5} x^{3}-2079 C \,a^{2} b \,d^{6} x^{3}+2376 C a \,b^{2} c \,d^{5} x^{3}-880 C \,b^{3} c^{2} d^{4} x^{3}-693 D a^{3} d^{6} x^{3}+2376 D a^{2} b c \,d^{5} x^{3}-2640 D a \,b^{2} c^{2} d^{4} x^{3}+960 D b^{3} c^{3} d^{3} x^{3}-3465 A a \,b^{2} d^{6} x^{2}+1386 A \,b^{3} c \,d^{5} x^{2}-3465 B \,a^{2} b \,d^{6} x^{2}+4158 B a \,b^{2} c \,d^{5} x^{2}-1584 B \,b^{3} c^{2} d^{4} x^{2}-1155 C \,a^{3} d^{6} x^{2}+4158 C \,a^{2} b c \,d^{5} x^{2}-4752 C a \,b^{2} c^{2} d^{4} x^{2}+1760 C \,b^{3} c^{3} d^{3} x^{2}+1386 D a^{3} c \,d^{5} x^{2}-4752 D a^{2} b \,c^{2} d^{4} x^{2}+5280 D a \,b^{2} c^{3} d^{3} x^{2}-1920 D b^{3} c^{4} d^{2} x^{2}-10395 A \,a^{2} b \,d^{6} x +13860 A a \,b^{2} c \,d^{5} x -5544 A \,b^{3} c^{2} d^{4} x -3465 B \,a^{3} d^{6} x +13860 B \,a^{2} b c \,d^{5} x -16632 B a \,b^{2} c^{2} d^{4} x +6336 B \,b^{3} c^{3} d^{3} x +4620 C \,a^{3} c \,d^{5} x -16632 C \,a^{2} b \,c^{2} d^{4} x +19008 C a \,b^{2} c^{3} d^{3} x -7040 C \,b^{3} c^{4} d^{2} x -5544 D a^{3} c^{2} d^{4} x +19008 D a^{2} b \,c^{3} d^{3} x -21120 D a \,b^{2} c^{4} d^{2} x +7680 D b^{3} c^{5} d x +3465 a^{3} A \,d^{6}-20790 A \,a^{2} b c \,d^{5}+27720 A a \,b^{2} c^{2} d^{4}-11088 A \,b^{3} c^{3} d^{3}-6930 B \,a^{3} c \,d^{5}+27720 B \,a^{2} b \,c^{2} d^{4}-33264 B a \,b^{2} c^{3} d^{3}+12672 B \,b^{3} c^{4} d^{2}+9240 C \,a^{3} c^{2} d^{4}-33264 C \,a^{2} b \,c^{3} d^{3}+38016 C a \,b^{2} c^{4} d^{2}-14080 C \,b^{3} c^{5} d -11088 D a^{3} c^{3} d^{3}+38016 D a^{2} b \,c^{4} d^{2}-42240 D a \,b^{2} c^{5} d +15360 D b^{3} c^{6}\right )}{3465 \sqrt {d x +c}\, d^{7}}\) | \(841\) |
trager | \(-\frac {2 \left (-315 D b^{3} x^{6} d^{6}-385 C \,b^{3} d^{6} x^{5}-1155 D a \,b^{2} d^{6} x^{5}+420 D b^{3} c \,d^{5} x^{5}-495 B \,b^{3} d^{6} x^{4}-1485 C a \,b^{2} d^{6} x^{4}+550 C \,b^{3} c \,d^{5} x^{4}-1485 D a^{2} b \,d^{6} x^{4}+1650 D a \,b^{2} c \,d^{5} x^{4}-600 D b^{3} c^{2} d^{4} x^{4}-693 A \,b^{3} d^{6} x^{3}-2079 B a \,b^{2} d^{6} x^{3}+792 B \,b^{3} c \,d^{5} x^{3}-2079 C \,a^{2} b \,d^{6} x^{3}+2376 C a \,b^{2} c \,d^{5} x^{3}-880 C \,b^{3} c^{2} d^{4} x^{3}-693 D a^{3} d^{6} x^{3}+2376 D a^{2} b c \,d^{5} x^{3}-2640 D a \,b^{2} c^{2} d^{4} x^{3}+960 D b^{3} c^{3} d^{3} x^{3}-3465 A a \,b^{2} d^{6} x^{2}+1386 A \,b^{3} c \,d^{5} x^{2}-3465 B \,a^{2} b \,d^{6} x^{2}+4158 B a \,b^{2} c \,d^{5} x^{2}-1584 B \,b^{3} c^{2} d^{4} x^{2}-1155 C \,a^{3} d^{6} x^{2}+4158 C \,a^{2} b c \,d^{5} x^{2}-4752 C a \,b^{2} c^{2} d^{4} x^{2}+1760 C \,b^{3} c^{3} d^{3} x^{2}+1386 D a^{3} c \,d^{5} x^{2}-4752 D a^{2} b \,c^{2} d^{4} x^{2}+5280 D a \,b^{2} c^{3} d^{3} x^{2}-1920 D b^{3} c^{4} d^{2} x^{2}-10395 A \,a^{2} b \,d^{6} x +13860 A a \,b^{2} c \,d^{5} x -5544 A \,b^{3} c^{2} d^{4} x -3465 B \,a^{3} d^{6} x +13860 B \,a^{2} b c \,d^{5} x -16632 B a \,b^{2} c^{2} d^{4} x +6336 B \,b^{3} c^{3} d^{3} x +4620 C \,a^{3} c \,d^{5} x -16632 C \,a^{2} b \,c^{2} d^{4} x +19008 C a \,b^{2} c^{3} d^{3} x -7040 C \,b^{3} c^{4} d^{2} x -5544 D a^{3} c^{2} d^{4} x +19008 D a^{2} b \,c^{3} d^{3} x -21120 D a \,b^{2} c^{4} d^{2} x +7680 D b^{3} c^{5} d x +3465 a^{3} A \,d^{6}-20790 A \,a^{2} b c \,d^{5}+27720 A a \,b^{2} c^{2} d^{4}-11088 A \,b^{3} c^{3} d^{3}-6930 B \,a^{3} c \,d^{5}+27720 B \,a^{2} b \,c^{2} d^{4}-33264 B a \,b^{2} c^{3} d^{3}+12672 B \,b^{3} c^{4} d^{2}+9240 C \,a^{3} c^{2} d^{4}-33264 C \,a^{2} b \,c^{3} d^{3}+38016 C a \,b^{2} c^{4} d^{2}-14080 C \,b^{3} c^{5} d -11088 D a^{3} c^{3} d^{3}+38016 D a^{2} b \,c^{4} d^{2}-42240 D a \,b^{2} c^{5} d +15360 D b^{3} c^{6}\right )}{3465 \sqrt {d x +c}\, d^{7}}\) | \(841\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1023\) |
default | \(\text {Expression too large to display}\) | \(1023\) |
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Time = 0.28 (sec) , antiderivative size = 677, normalized size of antiderivative = 1.56 \[ \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} - 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} + 99 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{6} - 110 \, {\left (3 \, D a b^{2} c + C b^{3} c\right )} d^{5}\right )} x^{4} + 11088 \, {\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} - {\left (960 \, D b^{3} c^{3} d^{3} - 693 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{6} + 792 \, {\left (3 \, D a^{2} b c + {\left (3 \, C a b^{2} + B b^{3}\right )} c\right )} d^{5} - 880 \, {\left (3 \, D a b^{2} c^{2} + C b^{3} c^{2}\right )} d^{4}\right )} x^{3} - 12672 \, {\left (3 \, D a^{2} b c^{4} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4}\right )} d^{2} + {\left (1920 \, D b^{3} c^{4} d^{2} + 1155 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6} - 1386 \, {\left (D a^{3} c + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c\right )} d^{5} + 1584 \, {\left (3 \, D a^{2} b c^{2} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{2}\right )} d^{4} - 1760 \, {\left (3 \, D a b^{2} c^{3} + C b^{3} c^{3}\right )} d^{3}\right )} x^{2} + 14080 \, {\left (3 \, D a b^{2} c^{5} + C b^{3} c^{5}\right )} d - {\left (7680 \, D b^{3} c^{5} d + 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} - 5544 \, {\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} + 6336 \, {\left (3 \, D a^{2} b c^{3} + {\left (3 \, C a b^{2} + B b^{3}\right )} c^{3}\right )} d^{3} - 7040 \, {\left (3 \, D a b^{2} c^{4} + C b^{3} c^{4}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{3465 \, {\left (d^{8} x + c d^{7}\right )}} \]
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Time = 70.97 (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} \]
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Time = 0.20 (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} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1067 vs. \(2 (412) = 824\).
Time = 0.31 (sec) , antiderivative size = 1067, normalized size of antiderivative = 2.46 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^3\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{3/2}} \,d x \]
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