Integrand size = 35, antiderivative size = 791 \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=-\frac {4 a^3 e \sqrt {a+b x^3}}{105 b^2}+\frac {54 a^2 (23 b c-8 a f) x \sqrt {a+b x^3}}{21505 b^2}+\frac {54 a^2 (5 b d-2 a g) x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {2 a^2 e x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 f x^4 \sqrt {a+b x^3}}{4301 b}+\frac {54 a^2 g x^5 \sqrt {a+b x^3}}{6175 b}-\frac {216 a^3 (5 b d-2 a g) \sqrt {a+b x^3}}{8645 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac {2 a x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac {108 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} (5 b d-2 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {36\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (1729 \sqrt [3]{b} (23 b c-8 a f)-8602 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (5 b d-2 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{37182145 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
-4/105*a^3*e*(b*x^3+a)^(1/2)/b^2+54/21505*a^2*(-8*a*f+23*b*c)*x*(b*x^3+a)^ (1/2)/b^2+54/8645*a^2*(-2*a*g+5*b*d)*x^2*(b*x^3+a)^(1/2)/b^2+2/105*a^2*e*x ^3*(b*x^3+a)^(1/2)/b+54/4301*a^2*f*x^4*(b*x^3+a)^(1/2)/b+54/6175*a^2*g*x^5 *(b*x^3+a)^(1/2)/b-216/8645*a^3*(-2*a*g+5*b*d)*(b*x^3+a)^(1/2)/b^(8/3)/((1 +3^(1/2))*a^(1/3)+b^(1/3)*x)+2/3900225*x^3*(b*x^3+a)^(3/2)*(156009*g*x^5+1 69575*f*x^4+185725*e*x^3+205275*d*x^2+229425*c*x)+2/185910725*a*x^3*(b*x^3 +a)^(1/2)*(3522519*g*x^5+4279275*f*x^4+5311735*e*x^3+6774075*d*x^2+8947575 *c*x)+108/8645*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a^(10/3)*(-2*a*g+5*b*d)*( a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a ^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticE(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+3 ^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)/b^(8/3)/(a^(1/3)*(a^(1/3)+b^(1/3 )*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)-36/37182145* 3^(3/4)*(1/2*6^(1/2)+1/2*2^(1/2))*a^3*(1729*b^(1/3)*(-8*a*f+23*b*c)-8602*( 1-3^(1/2))*a^(1/3)*(-2*a*g+5*b*d))*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b ^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticF(( (1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2 *I)/b^(8/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2 )^(1/2)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.37 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.23 \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {2 \sqrt {a+b x^3} \left (-\left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} \left (10 a (7429 e+21 x (380 f+391 g x))-b x \left (229425 c+17 x \left (12075 d+19 x \left (575 e+525 f x+483 g x^2\right )\right )\right )\right )+9975 a^2 (-23 b c+8 a f) x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+41055 a^2 (-5 b d+2 a g) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{3900225 b^2 \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[x^3*(a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]
Output:
(2*Sqrt[a + b*x^3]*(-((a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*(10*a*(7429*e + 21 *x*(380*f + 391*g*x)) - b*x*(229425*c + 17*x*(12075*d + 19*x*(575*e + 525* f*x + 483*g*x^2))))) + 9975*a^2*(-23*b*c + 8*a*f)*x*Hypergeometric2F1[-3/2 , 1/3, 4/3, -((b*x^3)/a)] + 41055*a^2*(-5*b*d + 2*a*g)*x^2*Hypergeometric2 F1[-3/2, 2/3, 5/3, -((b*x^3)/a)]))/(3900225*b^2*Sqrt[1 + (b*x^3)/a])
Time = 3.95 (sec) , antiderivative size = 789, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2365, 27, 2365, 27, 2375, 27, 2375, 27, 2427, 27, 2028, 2427, 27, 2028, 2427, 27, 2425, 793, 2417, 759, 2416}
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.
| \(\displaystyle \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {9}{2} a \int \frac {2 x^3 \sqrt {b x^3+a} \left (156009 g x^4+169575 f x^3+185725 e x^2+205275 d x+229425 c\right )}{3900225}dx+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \int x^3 \sqrt {b x^3+a} \left (156009 g x^4+169575 f x^3+185725 e x^2+205275 d x+229425 c\right )dx}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {3 a \left (\frac {3}{2} a \int \frac {2 x^3 \left (3522519 g x^4+4279275 f x^3+5311735 e x^2+6774075 d x+8947575 c\right )}{429 \sqrt {b x^3+a}}dx+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \int \frac {x^3 \left (3522519 g x^4+4279275 f x^3+5311735 e x^2+6774075 d x+8947575 c\right )}{\sqrt {b x^3+a}}dx+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {2 \int \frac {65 x^3 \left (855855 b f x^3+1062347 b e x^2+270963 (5 b d-2 a g) x+1789515 b c\right )}{2 \sqrt {b x^3+a}}dx}{13 b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \int \frac {x^3 \left (855855 b f x^3+1062347 b e x^2+270963 (5 b d-2 a g) x+1789515 b c\right )}{\sqrt {b x^3+a}}dx}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {2 \int \frac {11 x^3 \left (1062347 b^2 e x^2+270963 b (5 b d-2 a g) x+77805 b (23 b c-8 a f)\right )}{2 \sqrt {b x^3+a}}dx}{11 b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\int \frac {x^3 \left (1062347 b^2 e x^2+270963 b (5 b d-2 a g) x+77805 b (23 b c-8 a f)\right )}{\sqrt {b x^3+a}}dx}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2 \int -\frac {3 \left (-812889 b^2 (5 b d-2 a g) x^4-233415 b^2 (23 b c-8 a f) x^3+2124694 a b^2 e x^2\right )}{2 \sqrt {b x^3+a}}dx}{9 b}+\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\int \frac {-812889 b^2 (5 b d-2 a g) x^4-233415 b^2 (23 b c-8 a f) x^3+2124694 a b^2 e x^2}{\sqrt {b x^3+a}}dx}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\int \frac {x^2 \left (-812889 (5 b d-2 a g) x^2 b^2+2124694 a e b^2-233415 (23 b c-8 a f) x b^2\right )}{\sqrt {b x^3+a}}dx}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\frac {2 \int \frac {7 \left (-233415 (23 b c-8 a f) x^3 b^3+2124694 a e x^2 b^3+464508 a (5 b d-2 a g) x b^2\right )}{2 \sqrt {b x^3+a}}dx}{7 b}-232254 b x^2 \sqrt {a+b x^3} (5 b d-2 a g)}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\frac {\int \frac {-233415 (23 b c-8 a f) x^3 b^3+2124694 a e x^2 b^3+464508 a (5 b d-2 a g) x b^2}{\sqrt {b x^3+a}}dx}{b}-232254 b x^2 \sqrt {a+b x^3} (5 b d-2 a g)}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\frac {\int \frac {x \left (-233415 (23 b c-8 a f) x^2 b^3+2124694 a e x b^3+464508 a (5 b d-2 a g) b^2\right )}{\sqrt {b x^3+a}}dx}{b}-232254 b x^2 \sqrt {a+b x^3} (5 b d-2 a g)}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\frac {\frac {2 \int \frac {5 \left (1062347 a e x^2 b^4+46683 a (23 b c-8 a f) b^3+232254 a (5 b d-2 a g) x b^3\right )}{\sqrt {b x^3+a}}dx}{5 b}-93366 b^2 x \sqrt {a+b x^3} (23 b c-8 a f)}{b}-232254 b x^2 \sqrt {a+b x^3} (5 b d-2 a g)}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\frac {\frac {2 \int \frac {1062347 a e x^2 b^4+46683 a (23 b c-8 a f) b^3+232254 a (5 b d-2 a g) x b^3}{\sqrt {b x^3+a}}dx}{b}-93366 b^2 x \sqrt {a+b x^3} (23 b c-8 a f)}{b}-232254 b x^2 \sqrt {a+b x^3} (5 b d-2 a g)}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2425 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\frac {\frac {2 \left (1062347 a b^4 e \int \frac {x^2}{\sqrt {b x^3+a}}dx+\int \frac {46683 a (23 b c-8 a f) b^3+232254 a (5 b d-2 a g) x b^3}{\sqrt {b x^3+a}}dx\right )}{b}-93366 b^2 x \sqrt {a+b x^3} (23 b c-8 a f)}{b}-232254 b x^2 \sqrt {a+b x^3} (5 b d-2 a g)}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 793 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\frac {\frac {2 \left (\int \frac {46683 a (23 b c-8 a f) b^3+232254 a (5 b d-2 a g) x b^3}{\sqrt {b x^3+a}}dx+\frac {2124694}{3} a b^3 e \sqrt {a+b x^3}\right )}{b}-93366 b^2 x \sqrt {a+b x^3} (23 b c-8 a f)}{b}-232254 b x^2 \sqrt {a+b x^3} (5 b d-2 a g)}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\frac {\frac {2 \left (27 a b^{8/3} \left (1729 \sqrt [3]{b} (23 b c-8 a f)-8602 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (5 b d-2 a g)\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+232254 a b^{8/3} (5 b d-2 a g) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {2124694}{3} a b^3 e \sqrt {a+b x^3}\right )}{b}-93366 b^2 x \sqrt {a+b x^3} (23 b c-8 a f)}{b}-232254 b x^2 \sqrt {a+b x^3} (5 b d-2 a g)}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\frac {\frac {2 \left (232254 a b^{8/3} (5 b d-2 a g) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a b^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (1729 \sqrt [3]{b} (23 b c-8 a f)-8602 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (5 b d-2 a g)\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2124694}{3} a b^3 e \sqrt {a+b x^3}\right )}{b}-93366 b^2 x \sqrt {a+b x^3} (23 b c-8 a f)}{b}-232254 b x^2 \sqrt {a+b x^3} (5 b d-2 a g)}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {3 a \left (\frac {1}{143} a \left (\frac {5 \left (\frac {\frac {2124694}{9} b e x^3 \sqrt {a+b x^3}-\frac {\frac {\frac {2 \left (\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a b^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (1729 \sqrt [3]{b} (23 b c-8 a f)-8602 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (5 b d-2 a g)\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+232254 a b^{8/3} (5 b d-2 a g) \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )+\frac {2124694}{3} a b^3 e \sqrt {a+b x^3}\right )}{b}-93366 b^2 x \sqrt {a+b x^3} (23 b c-8 a f)}{b}-232254 b x^2 \sqrt {a+b x^3} (5 b d-2 a g)}{3 b}}{b}+155610 f x^4 \sqrt {a+b x^3}\right )}{b}+\frac {541926 g x^5 \sqrt {a+b x^3}}{b}\right )+\frac {2}{429} x^3 \sqrt {a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )\right )}{1300075}+\frac {2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}\) |
Input:
Int[x^3*(a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]
Output:
(2*x^3*(a + b*x^3)^(3/2)*(229425*c*x + 205275*d*x^2 + 185725*e*x^3 + 16957 5*f*x^4 + 156009*g*x^5))/3900225 + (3*a*((2*x^3*Sqrt[a + b*x^3]*(8947575*c *x + 6774075*d*x^2 + 5311735*e*x^3 + 4279275*f*x^4 + 3522519*g*x^5))/429 + (a*((541926*g*x^5*Sqrt[a + b*x^3])/b + (5*(155610*f*x^4*Sqrt[a + b*x^3] + ((2124694*b*e*x^3*Sqrt[a + b*x^3])/9 - (-232254*b*(5*b*d - 2*a*g)*x^2*Sqr t[a + b*x^3] + (-93366*b^2*(23*b*c - 8*a*f)*x*Sqrt[a + b*x^3] + (2*((21246 94*a*b^3*e*Sqrt[a + b*x^3])/3 + 232254*a*b^(8/3)*(5*b*d - 2*a*g)*((2*Sqrt[ a + b*x^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^(1/4)*Sqrt[ 2 - Sqrt[3]]*a^(1/3)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3) *x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[ArcSin[ ((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(b^(1/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[ 3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])) + (18*3^(3/4)*Sqrt[2 + Sqrt[ 3]]*a*b^(7/3)*(1729*b^(1/3)*(23*b*c - 8*a*f) - 8602*(1 - Sqrt[3])*a^(1/3)* (5*b*d - 2*a*g))*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])))/b)/b)/(3*b))/b))/b))/143))/1300075
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] & & PosQ[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[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> M odule[{q = Expon[Pq, x], i}, Simp[(c*x)^m*(a + b*x^n)^p*Sum[Coeff[Pq, x, i] *(x^(i + 1)/(m + n*p + i + 1)), {i, 0, q}], x] + Simp[a*n*p Int[(c*x)^m*( a + b*x^n)^(p - 1)*Sum[Coeff[Pq, x, i]*(x^i/(m + n*p + i + 1)), {i, 0, q}], x], x]] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wi th[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)/(b*c^(q - n + 1)*(m + q + n*p + 1))), x] + Si mp[1/(b*(m + q + n*p + 1)) Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(P q - Pqq*x^q) - a*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] / ; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + ( q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 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 + Sqrt [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] && PosQ[a] && Eq Q[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] && PosQ[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[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x ]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), x] + Simp[1/(b*(q + n*p + 1)) Int[ExpandToSum[b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ [p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Time = 1.26 (sec) , antiderivative size = 1161, normalized size of antiderivative = 1.47
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1161\) |
| risch | \(\text {Expression too large to display}\) | \(1198\) |
| default | \(\text {Expression too large to display}\) | \(1764\) |
Input:
int(x^3*(b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c),x,method=_RETURNVERBOSE)
Output:
2/25*b*g*x^11*(b*x^3+a)^(1/2)+2/23*f*b*x^10*(b*x^3+a)^(1/2)+2/21*b*e*x^9*( b*x^3+a)^(1/2)+2/19*(28/25*a*b*g+b^2*d)/b*x^8*(b*x^3+a)^(1/2)+2/17*(26/23* a*b*f+b^2*c)/b*x^7*(b*x^3+a)^(1/2)+16/105*a*e*x^6*(b*x^3+a)^(1/2)+2/13*(a^ 2*g+2*d*a*b-16/19*a/b*(28/25*a*b*g+b^2*d))/b*x^5*(b*x^3+a)^(1/2)+2/11*(f*a ^2+2*a*b*c-14/17*a/b*(26/23*a*b*f+b^2*c))/b*x^4*(b*x^3+a)^(1/2)+2/105*a^2* e*x^3*(b*x^3+a)^(1/2)/b+2/7*(a^2*d-10/13*a/b*(a^2*g+2*d*a*b-16/19*a/b*(28/ 25*a*b*g+b^2*d)))/b*x^2*(b*x^3+a)^(1/2)+2/5*(a^2*c-8/11*a/b*(f*a^2+2*a*b*c -14/17*a/b*(26/23*a*b*f+b^2*c)))/b*x*(b*x^3+a)^(1/2)-4/105*a^3*e*(b*x^3+a) ^(1/2)/b^2+4/15*I*a/b^2*(a^2*c-8/11*a/b*(f*a^2+2*a*b*c-14/17*a/b*(26/23*a* b*f+b^2*c)))*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/ 2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3 ))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/ 2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3 ))^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)- 1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2) /b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^ (1/2))+8/21*I*a/b^2*(a^2*d-10/13*a/b*(a^2*g+2*d*a*b-16/19*a/b*(28/25*a*b*g +b^2*d)))*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/ b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/ (-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/...
Time = 0.11 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.33 \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=-\frac {2 \, {\left (1400490 \, {\left (23 \, a^{3} b c - 8 \, a^{4} f\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 6967620 \, {\left (5 \, a^{3} b d - 2 \, a^{4} g\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (22309287 \, b^{4} g x^{11} + 24249225 \, b^{4} f x^{10} + 26558675 \, b^{4} e x^{9} + 42493880 \, a b^{3} e x^{6} + 1174173 \, {\left (25 \, b^{4} d + 28 \, a b^{3} g\right )} x^{8} + 5311735 \, a^{2} b^{2} e x^{3} + 1426425 \, {\left (23 \, b^{4} c + 26 \, a b^{3} f\right )} x^{7} + 90321 \, {\left (550 \, a b^{3} d + 27 \, a^{2} b^{2} g\right )} x^{5} - 10623470 \, a^{3} b e + 129675 \, {\left (460 \, a b^{3} c + 27 \, a^{2} b^{2} f\right )} x^{4} + 1741905 \, {\left (5 \, a^{2} b^{2} d - 2 \, a^{3} b g\right )} x^{2} + 700245 \, {\left (23 \, a^{2} b^{2} c - 8 \, a^{3} b f\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{557732175 \, b^{3}} \] Input:
integrate(x^3*(b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="fric as")
Output:
-2/557732175*(1400490*(23*a^3*b*c - 8*a^4*f)*sqrt(b)*weierstrassPInverse(0 , -4*a/b, x) - 6967620*(5*a^3*b*d - 2*a^4*g)*sqrt(b)*weierstrassZeta(0, -4 *a/b, weierstrassPInverse(0, -4*a/b, x)) - (22309287*b^4*g*x^11 + 24249225 *b^4*f*x^10 + 26558675*b^4*e*x^9 + 42493880*a*b^3*e*x^6 + 1174173*(25*b^4* d + 28*a*b^3*g)*x^8 + 5311735*a^2*b^2*e*x^3 + 1426425*(23*b^4*c + 26*a*b^3 *f)*x^7 + 90321*(550*a*b^3*d + 27*a^2*b^2*g)*x^5 - 10623470*a^3*b*e + 1296 75*(460*a*b^3*c + 27*a^2*b^2*f)*x^4 + 1741905*(5*a^2*b^2*d - 2*a^3*b*g)*x^ 2 + 700245*(23*a^2*b^2*c - 8*a^3*b*f)*x)*sqrt(b*x^3 + a))/b^3
Time = 4.05 (sec) , antiderivative size = 512, normalized size of antiderivative = 0.65 \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx =\text {Too large to display} \] Input:
integrate(x**3*(b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c),x)
Output:
a**(3/2)*c*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*p i)/a)/(3*gamma(7/3)) + a**(3/2)*d*x**5*gamma(5/3)*hyper((-1/2, 5/3), (8/3, ), b*x**3*exp_polar(I*pi)/a)/(3*gamma(8/3)) + a**(3/2)*f*x**7*gamma(7/3)*h yper((-1/2, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + a** (3/2)*g*x**8*gamma(8/3)*hyper((-1/2, 8/3), (11/3,), b*x**3*exp_polar(I*pi) /a)/(3*gamma(11/3)) + sqrt(a)*b*c*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3 ,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + sqrt(a)*b*d*x**8*gamma(8/3 )*hyper((-1/2, 8/3), (11/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(11/3)) + sqrt(a)*b*f*x**10*gamma(10/3)*hyper((-1/2, 10/3), (13/3,), b*x**3*exp_pola r(I*pi)/a)/(3*gamma(13/3)) + sqrt(a)*b*g*x**11*gamma(11/3)*hyper((-1/2, 11 /3), (14/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(14/3)) + a*e*Piecewise((- 4*a**2*sqrt(a + b*x**3)/(45*b**2) + 2*a*x**3*sqrt(a + b*x**3)/(45*b) + 2*x **6*sqrt(a + b*x**3)/15, Ne(b, 0)), (sqrt(a)*x**6/6, True)) + b*e*Piecewis e((16*a**3*sqrt(a + b*x**3)/(315*b**3) - 8*a**2*x**3*sqrt(a + b*x**3)/(315 *b**2) + 2*a*x**6*sqrt(a + b*x**3)/(105*b) + 2*x**9*sqrt(a + b*x**3)/21, N e(b, 0)), (sqrt(a)*x**9/9, True))
\[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int { {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{3} \,d x } \] Input:
integrate(x^3*(b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="maxi ma")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)*x^3, x)
\[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int { {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{3} \,d x } \] Input:
integrate(x^3*(b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="giac ")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)*x^3, x)
Timed out. \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int x^3\,{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \] Input:
int(x^3*(a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4),x)
Output:
int(x^3*(a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4), x)
\[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {-\frac {4 \sqrt {b \,x^{3}+a}\, a^{3} e}{105}-\frac {432 \sqrt {b \,x^{3}+a}\, a^{3} f x}{21505}-\frac {108 \sqrt {b \,x^{3}+a}\, a^{3} g \,x^{2}}{8645}+\frac {54 \sqrt {b \,x^{3}+a}\, a^{2} b c x}{935}+\frac {54 \sqrt {b \,x^{3}+a}\, a^{2} b d \,x^{2}}{1729}+\frac {2 \sqrt {b \,x^{3}+a}\, a^{2} b e \,x^{3}}{105}+\frac {54 \sqrt {b \,x^{3}+a}\, a^{2} b f \,x^{4}}{4301}+\frac {54 \sqrt {b \,x^{3}+a}\, a^{2} b g \,x^{5}}{6175}+\frac {40 \sqrt {b \,x^{3}+a}\, a \,b^{2} c \,x^{4}}{187}+\frac {44 \sqrt {b \,x^{3}+a}\, a \,b^{2} d \,x^{5}}{247}+\frac {16 \sqrt {b \,x^{3}+a}\, a \,b^{2} e \,x^{6}}{105}+\frac {52 \sqrt {b \,x^{3}+a}\, a \,b^{2} f \,x^{7}}{391}+\frac {56 \sqrt {b \,x^{3}+a}\, a \,b^{2} g \,x^{8}}{475}+\frac {2 \sqrt {b \,x^{3}+a}\, b^{3} c \,x^{7}}{17}+\frac {2 \sqrt {b \,x^{3}+a}\, b^{3} d \,x^{8}}{19}+\frac {2 \sqrt {b \,x^{3}+a}\, b^{3} e \,x^{9}}{21}+\frac {2 \sqrt {b \,x^{3}+a}\, b^{3} f \,x^{10}}{23}+\frac {2 \sqrt {b \,x^{3}+a}\, b^{3} g \,x^{11}}{25}+\frac {432 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a^{4} f}{21505}-\frac {54 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a^{3} b c}{935}+\frac {216 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) a^{4} g}{8645}-\frac {108 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) a^{3} b d}{1729}}{b^{2}} \] Input:
int(x^3*(b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c),x)
Output:
(2*( - 10623470*sqrt(a + b*x**3)*a**3*e - 5601960*sqrt(a + b*x**3)*a**3*f* x - 3483810*sqrt(a + b*x**3)*a**3*g*x**2 + 16105635*sqrt(a + b*x**3)*a**2* b*c*x + 8709525*sqrt(a + b*x**3)*a**2*b*d*x**2 + 5311735*sqrt(a + b*x**3)* a**2*b*e*x**3 + 3501225*sqrt(a + b*x**3)*a**2*b*f*x**4 + 2438667*sqrt(a + b*x**3)*a**2*b*g*x**5 + 59650500*sqrt(a + b*x**3)*a*b**2*c*x**4 + 49676550 *sqrt(a + b*x**3)*a*b**2*d*x**5 + 42493880*sqrt(a + b*x**3)*a*b**2*e*x**6 + 37087050*sqrt(a + b*x**3)*a*b**2*f*x**7 + 32876844*sqrt(a + b*x**3)*a*b* *2*g*x**8 + 32807775*sqrt(a + b*x**3)*b**3*c*x**7 + 29354325*sqrt(a + b*x* *3)*b**3*d*x**8 + 26558675*sqrt(a + b*x**3)*b**3*e*x**9 + 24249225*sqrt(a + b*x**3)*b**3*f*x**10 + 22309287*sqrt(a + b*x**3)*b**3*g*x**11 + 5601960* int(sqrt(a + b*x**3)/(a + b*x**3),x)*a**4*f - 16105635*int(sqrt(a + b*x**3 )/(a + b*x**3),x)*a**3*b*c + 6967620*int((sqrt(a + b*x**3)*x)/(a + b*x**3) ,x)*a**4*g - 17419050*int((sqrt(a + b*x**3)*x)/(a + b*x**3),x)*a**3*b*d))/ (557732175*b**2)