Integrand size = 35, antiderivative size = 681 \[ \int x^2 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {2 a (5 b c-2 a f) \sqrt {a+b x^3}}{45 b^2}+\frac {6 a (17 b d-8 a g) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a e x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a f x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a g x^4 \sqrt {a+b x^3}}{187 b}-\frac {24 a^2 e \sqrt {a+b x^3}}{91 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {4\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (1547 b d-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-728 a g\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{85085 b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
2/45*a*(-2*a*f+5*b*c)*(b*x^3+a)^(1/2)/b^2+6/935*a*(-8*a*g+17*b*d)*x*(b*x^3 +a)^(1/2)/b^2+6/91*a*e*x^2*(b*x^3+a)^(1/2)/b+2/45*a*f*x^3*(b*x^3+a)^(1/2)/ b+6/187*a*g*x^4*(b*x^3+a)^(1/2)/b+2/109395*x^2*(6435*g*x^5+7293*f*x^4+8415 *e*x^3+9945*d*x^2+12155*c*x)*(b*x^3+a)^(1/2)-24/91*a^2*e*(b*x^3+a)^(1/2)/b ^(5/3)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))+12/91*3^(1/4)*a^(7/3)*e*(a^(1/3)+b^ (1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3 ^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*((a^(2/3)-a^(1/3)*b^(1/3 )*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)/b^(5/3)/(b*x^3+a )^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1 /2)-4/85085*3^(3/4)*a^2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*( 1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1547*b*d-728*a *g-1870*a^(1/3)*b^(2/3)*e*(1-3^(1/2)))*(1/2*6^(1/2)+1/2*2^(1/2))*((a^(2/3) -a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)/b ^(7/3)/(b*x^3+a)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+ 3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.27 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.23 \[ \int x^2 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {2 \sqrt {a+b x^3} \left (-\left (\left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \left (26 a (187 f+180 g x)-b \left (12155 c+9945 d x+33 x^2 (255 e+13 x (17 f+15 g x))\right )\right )\right )+585 a (-17 b d+8 a g) x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )-8415 a b e x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{109395 b^2 \sqrt {1+\frac {b x^3}{a}}} \]
(2*Sqrt[a + b*x^3]*(-((a + b*x^3)*Sqrt[1 + (b*x^3)/a]*(26*a*(187*f + 180*g *x) - b*(12155*c + 9945*d*x + 33*x^2*(255*e + 13*x*(17*f + 15*g*x))))) + 5 85*a*(-17*b*d + 8*a*g)*x*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)] - 8415*a*b*e*x^2*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)]))/(109395* b^2*Sqrt[1 + (b*x^3)/a])
Time = 1.71 (sec) , antiderivative size = 691, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.457, Rules used = {2365, 27, 2375, 27, 2375, 27, 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.
| \(\displaystyle \int x^2 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {3}{2} a \int \frac {2 x^2 \left (6435 g x^4+7293 f x^3+8415 e x^2+9945 d x+12155 c\right )}{109395 \sqrt {b x^3+a}}dx+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {x^2 \left (6435 g x^4+7293 f x^3+8415 e x^2+9945 d x+12155 c\right )}{\sqrt {b x^3+a}}dx}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {a \left (\frac {2 \int \frac {11 x^2 \left (7293 b f x^3+8415 b e x^2+585 (17 b d-8 a g) x+12155 b c\right )}{2 \sqrt {b x^3+a}}dx}{11 b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\int \frac {x^2 \left (7293 b f x^3+8415 b e x^2+585 (17 b d-8 a g) x+12155 b c\right )}{\sqrt {b x^3+a}}dx}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle \frac {a \left (\frac {\frac {2 \int \frac {9 x^2 \left (8415 b^2 e x^2+585 b (17 b d-8 a g) x+2431 b (5 b c-2 a f)\right )}{2 \sqrt {b x^3+a}}dx}{9 b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\frac {\int \frac {x^2 \left (8415 b^2 e x^2+585 b (17 b d-8 a g) x+2431 b (5 b c-2 a f)\right )}{\sqrt {b x^3+a}}dx}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {2 \int -\frac {-4095 b^2 (17 b d-8 a g) x^3-17017 b^2 (5 b c-2 a f) x^2+33660 a b^2 e x}{2 \sqrt {b x^3+a}}dx}{7 b}+\frac {16830}{7} b e x^2 \sqrt {a+b x^3}}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {16830}{7} b e x^2 \sqrt {a+b x^3}-\frac {\int \frac {-4095 b^2 (17 b d-8 a g) x^3-17017 b^2 (5 b c-2 a f) x^2+33660 a b^2 e x}{\sqrt {b x^3+a}}dx}{7 b}}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {16830}{7} b e x^2 \sqrt {a+b x^3}-\frac {\int \frac {x \left (-4095 (17 b d-8 a g) x^2 b^2+33660 a e b^2-17017 (5 b c-2 a f) x b^2\right )}{\sqrt {b x^3+a}}dx}{7 b}}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {16830}{7} b e x^2 \sqrt {a+b x^3}-\frac {\frac {2 \int \frac {5 \left (-17017 (5 b c-2 a f) x^2 b^3+33660 a e x b^3+1638 a (17 b d-8 a g) b^2\right )}{2 \sqrt {b x^3+a}}dx}{5 b}-1638 b x \sqrt {a+b x^3} (17 b d-8 a g)}{7 b}}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {16830}{7} b e x^2 \sqrt {a+b x^3}-\frac {\frac {\int \frac {-17017 (5 b c-2 a f) x^2 b^3+33660 a e x b^3+1638 a (17 b d-8 a g) b^2}{\sqrt {b x^3+a}}dx}{b}-1638 b x \sqrt {a+b x^3} (17 b d-8 a g)}{7 b}}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 2425 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {16830}{7} b e x^2 \sqrt {a+b x^3}-\frac {\frac {\int \frac {33660 a e x b^3+1638 a (17 b d-8 a g) b^2}{\sqrt {b x^3+a}}dx-17017 b^3 (5 b c-2 a f) \int \frac {x^2}{\sqrt {b x^3+a}}dx}{b}-1638 b x \sqrt {a+b x^3} (17 b d-8 a g)}{7 b}}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 793 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {16830}{7} b e x^2 \sqrt {a+b x^3}-\frac {\frac {\int \frac {33660 a e x b^3+1638 a (17 b d-8 a g) b^2}{\sqrt {b x^3+a}}dx-\frac {34034}{3} b^2 \sqrt {a+b x^3} (5 b c-2 a f)}{b}-1638 b x \sqrt {a+b x^3} (17 b d-8 a g)}{7 b}}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {16830}{7} b e x^2 \sqrt {a+b x^3}-\frac {\frac {33660 a b^{8/3} e \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+18 a b^2 \left (-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-728 a g+1547 b d\right ) \int \frac {1}{\sqrt {b x^3+a}}dx-\frac {34034}{3} b^2 \sqrt {a+b x^3} (5 b c-2 a f)}{b}-1638 b x \sqrt {a+b x^3} (17 b d-8 a g)}{7 b}}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {16830}{7} b e x^2 \sqrt {a+b x^3}-\frac {\frac {33660 a b^{8/3} e \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {12\ 3^{3/4} \sqrt {2+\sqrt {3}} a b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-728 a g+1547 b d\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {34034}{3} b^2 \sqrt {a+b x^3} (5 b c-2 a f)}{b}-1638 b x \sqrt {a+b x^3} (17 b d-8 a g)}{7 b}}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {a \left (\frac {\frac {\frac {16830}{7} b e x^2 \sqrt {a+b x^3}-\frac {\frac {\frac {12\ 3^{3/4} \sqrt {2+\sqrt {3}} a b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-728 a g+1547 b d\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+33660 a b^{8/3} e \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )-\frac {34034}{3} b^2 \sqrt {a+b x^3} (5 b c-2 a f)}{b}-1638 b x \sqrt {a+b x^3} (17 b d-8 a g)}{7 b}}{b}+\frac {4862}{3} f x^3 \sqrt {a+b x^3}}{b}+\frac {1170 g x^4 \sqrt {a+b x^3}}{b}\right )}{36465}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}\) |
(2*x^2*Sqrt[a + b*x^3]*(12155*c*x + 9945*d*x^2 + 8415*e*x^3 + 7293*f*x^4 + 6435*g*x^5))/109395 + (a*((1170*g*x^4*Sqrt[a + b*x^3])/b + ((4862*f*x^3*S qrt[a + b*x^3])/3 + ((16830*b*e*x^2*Sqrt[a + b*x^3])/7 - (-1638*b*(17*b*d - 8*a*g)*x*Sqrt[a + b*x^3] + ((-34034*b^2*(5*b*c - 2*a*f)*Sqrt[a + b*x^3]) /3 + 33660*a*b^(8/3)*e*((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 ])) + (12*3^(3/4)*Sqrt[2 + Sqrt[3]]*a*b^(5/3)*(1547*b*d - 1870*(1 - Sqrt[3 ])*a^(1/3)*b^(2/3)*e - 728*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]*Ellipt icF[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 + Sq rt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3]))/b)/(7*b))/b)/b))/36465
3.5.46.3.1 Defintions of rubi rules used
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 = 1.74 (sec) , antiderivative size = 920, normalized size of antiderivative = 1.35
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(920\) |
| risch | \(\text {Expression too large to display}\) | \(1115\) |
| default | \(\text {Expression too large to display}\) | \(1197\) |
2/17*g*x^7*(b*x^3+a)^(1/2)+2/15*f*x^6*(b*x^3+a)^(1/2)+2/13*e*x^5*(b*x^3+a) ^(1/2)+2/11*(3/17*a*g+b*d)/b*x^4*(b*x^3+a)^(1/2)+2/9*(1/5*a*f+b*c)/b*x^3*( b*x^3+a)^(1/2)+6/91*a*e*x^2*(b*x^3+a)^(1/2)/b+2/5*(a*d-8/11*a/b*(3/17*a*g+ b*d))/b*x*(b*x^3+a)^(1/2)+2/3*(a*c-2/3*a/b*(1/5*a*f+b*c))/b*(b*x^3+a)^(1/2 )+4/15*I*a/b^2*(a*d-8/11*a/b*(3/17*a*g+b*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+1/2/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))+8/91*I/b^2*a^2*e*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)*((-3/2 /b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(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))+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.26 \[ \int x^2 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {2 \, {\left (100980 \, a^{2} b^{\frac {3}{2}} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 4914 \, {\left (17 \, a^{2} b d - 8 \, a^{3} g\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (45045 \, b^{3} g x^{7} + 51051 \, b^{3} f x^{6} + 58905 \, b^{3} e x^{5} + 25245 \, a b^{2} e x^{2} + 4095 \, {\left (17 \, b^{3} d + 3 \, a b^{2} g\right )} x^{4} + 85085 \, a b^{2} c - 34034 \, a^{2} b f + 17017 \, {\left (5 \, b^{3} c + a b^{2} f\right )} x^{3} + 2457 \, {\left (17 \, a b^{2} d - 8 \, a^{2} b g\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{765765 \, b^{3}} \]
2/765765*(100980*a^2*b^(3/2)*e*weierstrassZeta(0, -4*a/b, weierstrassPInve rse(0, -4*a/b, x)) - 4914*(17*a^2*b*d - 8*a^3*g)*sqrt(b)*weierstrassPInver se(0, -4*a/b, x) + (45045*b^3*g*x^7 + 51051*b^3*f*x^6 + 58905*b^3*e*x^5 + 25245*a*b^2*e*x^2 + 4095*(17*b^3*d + 3*a*b^2*g)*x^4 + 85085*a*b^2*c - 3403 4*a^2*b*f + 17017*(5*b^3*c + a*b^2*f)*x^3 + 2457*(17*a*b^2*d - 8*a^2*b*g)* x)*sqrt(b*x^3 + a))/b^3
Time = 2.12 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.33 \[ \int x^2 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {\sqrt {a} d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt {a} e x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} g x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + c \left (\begin {cases} \frac {\sqrt {a} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{9 b} & \text {otherwise} \end {cases}\right ) + f \left (\begin {cases} - \frac {4 a^{2} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 a x^{3} \sqrt {a + b x^{3}}}{45 b} + \frac {2 x^{6} \sqrt {a + b x^{3}}}{15} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) \]
sqrt(a)*d*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi )/a)/(3*gamma(7/3)) + sqrt(a)*e*x**5*gamma(5/3)*hyper((-1/2, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(8/3)) + sqrt(a)*g*x**7*gamma(7/3)*hype r((-1/2, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + c*Piec ewise((sqrt(a)*x**3/3, Eq(b, 0)), (2*(a + b*x**3)**(3/2)/(9*b), True)) + f *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))
\[ \int x^2 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int { {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a} x^{2} \,d x } \]
\[ \int x^2 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int { {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a} x^{2} \,d x } \]
Timed out. \[ \int x^2 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int x^2\,\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]