Integrand size = 43, antiderivative size = 577 \[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=-\frac {221184 (B c-2 b C) \sqrt {-64+(b+c x)^3}}{91 c^3 \left (4-4 \sqrt {3}-b-c x\right )}+\frac {1152 \left (\frac {91 \left (b B c-A c^2-b^2 C\right ) (b+c x)}{c}-\frac {55 (B c-2 b C) (b+c x)^2}{c}\right ) \sqrt {-64+(b+c x)^3}}{5005 c^2}-\frac {2 \left (\frac {13 \left (b B c-A c^2-b^2 C\right ) (b+c x)}{c}-\frac {11 (B c-2 b C) (b+c x)^2}{c}\right ) \left (-64+(b+c x)^3\right )^{3/2}}{143 c^2}+\frac {2 C \left (-64+(b+c x)^3\right )^{5/2}}{15 c^3}+\frac {221184 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (B c-2 b C) (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} E\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right )|-7+4 \sqrt {3}\right )}{91 c^3 \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}}-\frac {36864\ 3^{3/4} \sqrt {2-\sqrt {3}} \left (220 \left (1+\sqrt {3}\right ) (B c-2 b C)-91 \left (b B c-A c^2-b^2 C\right )\right ) (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right ),-7+4 \sqrt {3}\right )}{5005 c^3 \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}} \] Output:
-221184/91*(B*c-2*C*b)*(-64+(c*x+b)^3)^(1/2)/c^3/(4-4*3^(1/2)-b-c*x)+1152/ 5005*(91*(-A*c^2+B*b*c-C*b^2)*(c*x+b)/c-55*(B*c-2*C*b)*(c*x+b)^2/c)*(-64+( c*x+b)^3)^(1/2)/c^2-2/143*(13*(-A*c^2+B*b*c-C*b^2)*(c*x+b)/c-11*(B*c-2*C*b )*(c*x+b)^2/c)*(-64+(c*x+b)^3)^(3/2)/c^2+2/15*C*(-64+(c*x+b)^3)^(5/2)/c^3+ 221184/91*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*(B*c-2*C*b)*(-c*x-b+4)*((16+4* c*x+4*b+(c*x+b)^2)/(4-4*3^(1/2)-b-c*x)^2)^(1/2)*EllipticE((4+4*3^(1/2)-b-c *x)/(4-4*3^(1/2)-b-c*x),2*I-I*3^(1/2))/c^3/(-(-c*x-b+4)/(4-4*3^(1/2)-b-c*x )^2)^(1/2)/(-64+(c*x+b)^3)^(1/2)-36864/5005*3^(3/4)*(1/2*6^(1/2)-1/2*2^(1/ 2))*(220*(1+3^(1/2))*(B*c-2*C*b)+91*A*c^2-91*B*b*c+91*C*b^2)*(-c*x-b+4)*(( 16+4*c*x+4*b+(c*x+b)^2)/(4-4*3^(1/2)-b-c*x)^2)^(1/2)*EllipticF((4+4*3^(1/2 )-b-c*x)/(4-4*3^(1/2)-b-c*x),2*I-I*3^(1/2))/c^3/(-(-c*x-b+4)/(4-4*3^(1/2)- b-c*x)^2)^(1/2)/(-64+(c*x+b)^3)^(1/2)
Result contains complex when optimal does not.
Time = 11.46 (sec) , antiderivative size = 504, normalized size of antiderivative = 0.87 \[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\frac {2 \left (-\left (\left (64-b^3-3 b^2 c x-3 b c^2 x^2-c^3 x^3\right ) \left (56 b^6 C-42 b^5 c (5 B+2 C x)+1001 C \left (-64+c^3 x^3\right )^2+105 b^4 c^2 (13 A+x (3 B+C x))+3 c^3 x \left (91 A \left (-896+5 c^3 x^3\right )+55 B x \left (-1024+7 c^3 x^3\right )\right )+4 b^3 \left (105 c^3 x (13 A+8 B x)+17 C \left (-512+35 c^3 x^3\right )\right )+6 b^2 c \left (B \left (12608+1015 c^3 x^3\right )+x \left (7808 C+1365 A c^3 x+805 c^3 C x^3\right )\right )+6 b c^2 \left (182 A \left (-224+5 c^3 x^3\right )+x \left (88 C x \left (-88+7 c^3 x^3\right )+3 B \left (-5184+245 c^3 x^3\right )\right )\right )\right )\right )-165888 \sqrt {2} \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \sqrt {16+b^2+4 c x+c^2 x^2+2 b (2+c x)} \left (-110 \left (3 i+\sqrt {3}\right ) (B c-2 b C) E\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+i \left ((220-91 b) B c+91 A c^2+b (-440+91 b) C\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )\right )}{15015 c^3 \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3}} \] Input:
Integrate[(A + B*x + C*x^2)*(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3 )^(3/2),x]
Output:
(2*(-((64 - b^3 - 3*b^2*c*x - 3*b*c^2*x^2 - c^3*x^3)*(56*b^6*C - 42*b^5*c* (5*B + 2*C*x) + 1001*C*(-64 + c^3*x^3)^2 + 105*b^4*c^2*(13*A + x*(3*B + C* x)) + 3*c^3*x*(91*A*(-896 + 5*c^3*x^3) + 55*B*x*(-1024 + 7*c^3*x^3)) + 4*b ^3*(105*c^3*x*(13*A + 8*B*x) + 17*C*(-512 + 35*c^3*x^3)) + 6*b^2*c*(B*(126 08 + 1015*c^3*x^3) + x*(7808*C + 1365*A*c^3*x + 805*c^3*C*x^3)) + 6*b*c^2* (182*A*(-224 + 5*c^3*x^3) + x*(88*C*x*(-88 + 7*c^3*x^3) + 3*B*(-5184 + 245 *c^3*x^3))))) - 165888*Sqrt[2]*Sqrt[((-I)*(-4 + b + c*x))/(3*I + Sqrt[3])] *Sqrt[16 + b^2 + 4*c*x + c^2*x^2 + 2*b*(2 + c*x)]*(-110*(3*I + Sqrt[3])*(B *c - 2*b*C)*EllipticE[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4 ))], (2*Sqrt[3])/(3*I + Sqrt[3])] + I*((220 - 91*b)*B*c + 91*A*c^2 + b*(-4 40 + 91*b)*C)*EllipticF[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1 /4))], (2*Sqrt[3])/(3*I + Sqrt[3])])))/(15015*c^3*Sqrt[-64 + b^3 + 3*b^2*c *x + 3*b*c^2*x^2 + c^3*x^3])
Time = 1.96 (sec) , antiderivative size = 710, normalized size of antiderivative = 1.23, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2459, 2392, 27, 2392, 27, 2425, 793, 2419, 760, 2418}
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.
| \(\displaystyle \int \left (b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3-64\right )^{3/2} \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2} \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )+C \left (\frac {b}{c}+x\right )^2\right )d\left (\frac {b}{c}+x\right )\) |
| \(\Big \downarrow \) 2392 |
| \(\displaystyle \frac {2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2} \left (195 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+165 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+143 C \left (\frac {b}{c}+x\right )^3\right )}{2145}-288 \int \frac {2 \left (143 C \left (\frac {b}{c}+x\right )^2+165 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )+195 \left (A-\frac {b (B c-b C)}{c^2}\right )\right ) \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{2145}d\left (\frac {b}{c}+x\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2} \left (195 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+165 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+143 C \left (\frac {b}{c}+x\right )^3\right )}{2145}-\frac {192}{715} \int \left (143 C \left (\frac {b}{c}+x\right )^2+165 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )+195 \left (A-\frac {b (B c-b C)}{c^2}\right )\right ) \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}d\left (\frac {b}{c}+x\right )\) |
| \(\Big \downarrow \) 2392 |
| \(\displaystyle \frac {2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2} \left (195 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+165 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+143 C \left (\frac {b}{c}+x\right )^3\right )}{2145}-\frac {192}{715} \left (\frac {2}{63} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (2457 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+1485 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+1001 C \left (\frac {b}{c}+x\right )^3\right )-96 \int \frac {2 \left (1001 C \left (\frac {b}{c}+x\right )^2+1485 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )+2457 \left (A-\frac {b (B c-b C)}{c^2}\right )\right )}{63 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2} \left (195 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+165 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+143 C \left (\frac {b}{c}+x\right )^3\right )}{2145}-\frac {192}{715} \left (\frac {2}{63} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (2457 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+1485 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+1001 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{21} \int \frac {1001 C \left (\frac {b}{c}+x\right )^2+1485 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )+2457 \left (A-\frac {b (B c-b C)}{c^2}\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )\right )\) |
| \(\Big \downarrow \) 2425 |
| \(\displaystyle \frac {2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2} \left (195 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+165 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+143 C \left (\frac {b}{c}+x\right )^3\right )}{2145}-\frac {192}{715} \left (\frac {2}{63} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (2457 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+1485 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+1001 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{21} \left (\int \frac {2457 \left (A-\frac {b (B c-b C)}{c^2}\right )+1485 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )+1001 C \int \frac {\left (\frac {b}{c}+x\right )^2}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )\right )\right )\) |
| \(\Big \downarrow \) 793 |
| \(\displaystyle \frac {2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2} \left (195 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+165 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+143 C \left (\frac {b}{c}+x\right )^3\right )}{2145}-\frac {192}{715} \left (\frac {2}{63} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (2457 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+1485 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+1001 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{21} \left (\int \frac {2457 \left (A-\frac {b (B c-b C)}{c^2}\right )+1485 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )+\frac {2002 C \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{3 c^3}\right )\right )\) |
| \(\Big \downarrow \) 2419 |
| \(\displaystyle \frac {2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2} \left (195 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+165 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+143 C \left (\frac {b}{c}+x\right )^3\right )}{2145}-\frac {192}{715} \left (\frac {2}{63} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (2457 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+1485 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+1001 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{21} \left (\frac {27 \left (91 c \left (A-\frac {b (B c-b C)}{c^2}\right )+220 \left (1+\sqrt {3}\right ) \left (B-\frac {2 b C}{c}\right )\right ) \int \frac {1}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{c}-\frac {1485 (B c-2 b C) \int \frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{c^2}+\frac {2002 C \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{3 c^3}\right )\right )\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2} \left (195 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+165 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+143 C \left (\frac {b}{c}+x\right )^3\right )}{2145}-\frac {192}{715} \left (\frac {2}{63} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (2457 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+1485 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+1001 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{21} \left (-\frac {1485 (B c-2 b C) \int \frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{c^2}-\frac {9\ 3^{3/4} \sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \left (91 c \left (A-\frac {b (B c-b C)}{c^2}\right )+220 \left (1+\sqrt {3}\right ) \left (B-\frac {2 b C}{c}\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{c^2 \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}+\frac {2002 C \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{3 c^3}\right )\right )\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2} \left (195 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+165 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+143 C \left (\frac {b}{c}+x\right )^3\right )}{2145}-\frac {192}{715} \left (\frac {2}{63} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (2457 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+1485 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+1001 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{21} \left (-\frac {9\ 3^{3/4} \sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \left (91 c \left (A-\frac {b (B c-b C)}{c^2}\right )+220 \left (1+\sqrt {3}\right ) \left (B-\frac {2 b C}{c}\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{c^2 \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}-\frac {1485 (B c-2 b C) \left (\frac {2 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{c \left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )}-\frac {2 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} E\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right )|-7+4 \sqrt {3}\right )}{c \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\right )}{c^2}+\frac {2002 C \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{3 c^3}\right )\right )\) |
Input:
Int[(A + B*x + C*x^2)*(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^(3/2 ),x]
Output:
(2*(-64 + c^3*(b/c + x)^3)^(3/2)*(195*(A - (b*(B*c - b*C))/c^2)*(b/c + x) + 165*(B - (2*b*C)/c)*(b/c + x)^2 + 143*C*(b/c + x)^3))/2145 - (192*((2*Sq rt[-64 + c^3*(b/c + x)^3]*(2457*(A - (b*(B*c - b*C))/c^2)*(b/c + x) + 1485 *(B - (2*b*C)/c)*(b/c + x)^2 + 1001*C*(b/c + x)^3))/63 - (64*((2002*C*Sqrt [-64 + c^3*(b/c + x)^3])/(3*c^3) - (1485*(B*c - 2*b*C)*((2*Sqrt[-64 + c^3* (b/c + x)^3])/(c*(4*(1 - Sqrt[3]) - c*(b/c + x))) - (2*3^(1/4)*Sqrt[2 + Sq rt[3]]*(4 - c*(b/c + x))*Sqrt[(16 + 4*c*(b/c + x) + c^2*(b/c + x)^2)/(4*(1 - Sqrt[3]) - c*(b/c + x))^2]*EllipticE[ArcSin[(4*(1 + Sqrt[3]) - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))], -7 + 4*Sqrt[3]])/(c*Sqrt[-((4 - c*( b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))^2)]*Sqrt[-64 + c^3*(b/c + x)^3]) ))/c^2 - (9*3^(3/4)*Sqrt[2 - Sqrt[3]]*(220*(1 + Sqrt[3])*(B - (2*b*C)/c) + 91*c*(A - (b*(B*c - b*C))/c^2))*(4 - c*(b/c + x))*Sqrt[(16 + 4*c*(b/c + x ) + c^2*(b/c + x)^2)/(4*(1 - Sqrt[3]) - c*(b/c + x))^2]*EllipticF[ArcSin[( 4*(1 + Sqrt[3]) - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))], -7 + 4*Sq rt[3]])/(c^2*Sqrt[-((4 - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))^2)]* Sqrt[-64 + c^3*(b/c + x)^3])))/21))/715
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) *s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) ^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{q = Expon[Pq , x], i}, Simp[(a + b*x^n)^p*Sum[Coeff[Pq, x, i]*(x^(i + 1)/(n*p + i + 1)), {i, 0, q}], x] + Simp[a*n*p Int[(a + b*x^n)^(p - 1)*Sum[Coeff[Pq, x, i]* (x^i/(n*p + i + 1)), {i, 0, q}], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x ] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 + Sqrt[3])*d*s)/r Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r Int[((1 + Sqrt[3])*s + r*x)/Sq rt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && NeQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Pq, x, n - 1] Int[x^(n - 1)*(a + b*x^n)^p, x], x] + Int[ExpandToSum[Pq - Coeff[Pq, x, n - 1]*x^(n - 1), x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, p}, x] && PolyQ[P q, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1575 vs. \(2 (515 ) = 1030\).
Time = 2.72 (sec) , antiderivative size = 1576, normalized size of antiderivative = 2.73
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1576\) |
| default | \(\text {Expression too large to display}\) | \(5082\) |
| elliptic | \(\text {Expression too large to display}\) | \(5698\) |
Input:
int((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(3/2),x,method=_R ETURNVERBOSE)
Output:
2/15015/c^3*(1001*C*c^6*x^6+1155*B*c^6*x^5+3696*C*b*c^5*x^5+1365*A*c^6*x^4 +4410*B*b*c^5*x^4+4830*C*b^2*c^4*x^4+5460*A*b*c^5*x^3+6090*B*b^2*c^4*x^3+2 380*C*b^3*c^3*x^3+8190*A*b^2*c^4*x^2+3360*B*b^3*c^3*x^2+105*C*b^4*c^2*x^2+ 5460*A*b^3*c^3*x+315*B*b^4*c^2*x-84*C*b^5*c*x+1365*A*b^4*c^2-210*B*b^5*c+5 6*C*b^6-128128*C*c^3*x^3-168960*B*c^3*x^2-46464*C*b*c^2*x^2-244608*A*c^3*x -93312*B*b*c^2*x+46848*C*b^2*c*x-244608*A*b*c^2+75648*B*b^2*c-34816*C*b^3+ 4100096*C)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)+110592/5005/c^2*(1 10*c*(B*c-2*C*b)*((-b-2-2*I*3^(1/2))/c+(b-4)/c)*((x+(b-4)/c)/((-b-2-2*I*3^ (1/2))/c+(b-4)/c))^(1/2)*((x-(-b-2+2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^( 1/2))/c))^(1/2)*((x-(-b-2-2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2-2*I*3^(1/2))/c)) ^(1/2)/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)*((-(b-4)/c-(-b-2+2*I*3 ^(1/2))/c)*EllipticE(((x+(b-4)/c)/((-b-2-2*I*3^(1/2))/c+(b-4)/c))^(1/2),(( -(b-4)/c-(-b-2-2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^(1/2))/c))^(1/2))+(-b -2+2*I*3^(1/2))/c*EllipticF(((x+(b-4)/c)/((-b-2-2*I*3^(1/2))/c+(b-4)/c))^( 1/2),((-(b-4)/c-(-b-2-2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^(1/2))/c))^(1/ 2)))+182*A*c^2*((-b-2-2*I*3^(1/2))/c+(b-4)/c)*((x+(b-4)/c)/((-b-2-2*I*3^(1 /2))/c+(b-4)/c))^(1/2)*((x-(-b-2+2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^(1/ 2))/c))^(1/2)*((x-(-b-2-2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2-2*I*3^(1/2))/c))^( 1/2)/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)*EllipticF(((x+(b-4)/c)/( (-b-2-2*I*3^(1/2))/c+(b-4)/c))^(1/2),((-(b-4)/c-(-b-2-2*I*3^(1/2))/c)/(...
Time = 0.08 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.63 \[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\frac {2 \, {\left (30191616 \, {\left (C b^{2} - B b c + A c^{2}\right )} \sqrt {c^{3}} {\rm weierstrassPInverse}\left (0, \frac {256}{c^{3}}, \frac {c x + b}{c}\right ) + 18247680 \, {\left (2 \, C b c - B c^{2}\right )} \sqrt {c^{3}} {\rm weierstrassZeta}\left (0, \frac {256}{c^{3}}, {\rm weierstrassPInverse}\left (0, \frac {256}{c^{3}}, \frac {c x + b}{c}\right )\right ) + {\left (1001 \, C c^{8} x^{6} + 231 \, {\left (16 \, C b c^{7} + 5 \, B c^{8}\right )} x^{5} + 273 \, {\left (5 \, A b^{4} - 896 \, A b\right )} c^{4} + 105 \, {\left (46 \, C b^{2} c^{6} + 42 \, B b c^{7} + 13 \, A c^{8}\right )} x^{4} - 6 \, {\left (35 \, B b^{5} - 12608 \, B b^{2}\right )} c^{3} + 14 \, {\left (435 \, B b^{2} c^{6} + 390 \, A b c^{7} + 2 \, {\left (85 \, C b^{3} - 4576 \, C\right )} c^{5}\right )} x^{3} + 8 \, {\left (7 \, C b^{6} - 4352 \, C b^{3} + 512512 \, C\right )} c^{2} + 3 \, {\left (2730 \, A b^{2} c^{6} + 160 \, {\left (7 \, B b^{3} - 352 \, B\right )} c^{5} + {\left (35 \, C b^{4} - 15488 \, C b\right )} c^{4}\right )} x^{2} + 3 \, {\left (364 \, {\left (5 \, A b^{3} - 224 \, A\right )} c^{5} + 3 \, {\left (35 \, B b^{4} - 10368 \, B b\right )} c^{4} - 4 \, {\left (7 \, C b^{5} - 3904 \, C b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64}\right )}}{15015 \, c^{5}} \] Input:
integrate((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(3/2),x, al gorithm="fricas")
Output:
2/15015*(30191616*(C*b^2 - B*b*c + A*c^2)*sqrt(c^3)*weierstrassPInverse(0, 256/c^3, (c*x + b)/c) + 18247680*(2*C*b*c - B*c^2)*sqrt(c^3)*weierstrassZ eta(0, 256/c^3, weierstrassPInverse(0, 256/c^3, (c*x + b)/c)) + (1001*C*c^ 8*x^6 + 231*(16*C*b*c^7 + 5*B*c^8)*x^5 + 273*(5*A*b^4 - 896*A*b)*c^4 + 105 *(46*C*b^2*c^6 + 42*B*b*c^7 + 13*A*c^8)*x^4 - 6*(35*B*b^5 - 12608*B*b^2)*c ^3 + 14*(435*B*b^2*c^6 + 390*A*b*c^7 + 2*(85*C*b^3 - 4576*C)*c^5)*x^3 + 8* (7*C*b^6 - 4352*C*b^3 + 512512*C)*c^2 + 3*(2730*A*b^2*c^6 + 160*(7*B*b^3 - 352*B)*c^5 + (35*C*b^4 - 15488*C*b)*c^4)*x^2 + 3*(364*(5*A*b^3 - 224*A)*c ^5 + 3*(35*B*b^4 - 10368*B*b)*c^4 - 4*(7*C*b^5 - 3904*C*b^2)*c^3)*x)*sqrt( c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64))/c^5
\[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\int \left (\left (b + c x - 4\right ) \left (b^{2} + 2 b c x + 4 b + c^{2} x^{2} + 4 c x + 16\right )\right )^{\frac {3}{2}} \left (A + B x + C x^{2}\right )\, dx \] Input:
integrate((C*x**2+B*x+A)*(c**3*x**3+3*b*c**2*x**2+3*b**2*c*x+b**3-64)**(3/ 2),x)
Output:
Integral(((b + c*x - 4)*(b**2 + 2*b*c*x + 4*b + c**2*x**2 + 4*c*x + 16))** (3/2)*(A + B*x + C*x**2), x)
\[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\int { {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {3}{2}} {\left (C x^{2} + B x + A\right )} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(3/2),x, al gorithm="maxima")
Output:
integrate((c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)^(3/2)*(C*x^2 + B* x + A), x)
\[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\int { {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {3}{2}} {\left (C x^{2} + B x + A\right )} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(3/2),x, al gorithm="giac")
Output:
integrate((c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)^(3/2)*(C*x^2 + B* x + A), x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\int \left (C\,x^2+B\,x+A\right )\,{\left (b^3+3\,b^2\,c\,x+3\,b\,c^2\,x^2+c^3\,x^3-64\right )}^{3/2} \,d x \] Input:
int((A + B*x + C*x^2)*(b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(3/2) ,x)
Output:
int((A + B*x + C*x^2)*(b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(3/2) , x)
\[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx =\text {Too large to display} \] Input:
int((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(3/2),x)
Output:
(2*(1365*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*a*b**4*c + 5460*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*a*b**3*c* *2*x + 8190*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*a*b** 2*c**3*x**2 + 5460*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64 )*a*b*c**4*x**3 - 244608*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x** 3 - 64)*a*b*c + 1365*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*a*c**5*x**4 - 244608*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x** 3 - 64)*a*c**2*x - 154*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**6 + 231*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)* b**5*c*x + 3465*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b **4*c**2*x**2 + 8470*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**3*c**3*x**3 + 40832*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x **3 - 64)*b**3 + 9240*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**2*c**4*x**4 - 46464*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3* x**3 - 64)*b**2*c*x + 4851*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x **3 - 64)*b*c**5*x**5 - 215424*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c* *3*x**3 - 64)*b*c**2*x**2 + 1001*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*c**6*x**6 - 128128*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*c**3*x**3 + 1058816*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x* *2 + c**3*x**3 - 64) + 15095808*int(sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x...