Integrand size = 32, antiderivative size = 694 \[ \int \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {2 a^2 e \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {54 a^2 (19 b d-4 a g) \sqrt {a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 b d-4 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{1616615 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
2/692835*(b*x^3+a)^(3/2)*(36465*g*x^5+40755*f*x^4+46189*e*x^3+53295*d*x^2+ 62985*c*x)+2/15*a^2*e*(b*x^3+a)^(1/2)/b+54/935*a^2*f*x*(b*x^3+a)^(1/2)/b+5 4/1729*a^2*g*x^2*(b*x^3+a)^(1/2)/b+2/4849845*a*(176715*g*x^5+233415*f*x^4+ 323323*e*x^3+479655*d*x^2+793611*c*x)*(b*x^3+a)^(1/2)+54/1729*a^2*(-4*a*g+ 19*b*d)*(b*x^3+a)^(1/2)/b^(5/3)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))-27/1729*3^ (1/4)*a^(7/3)*(-4*a*g+19*b*d)*(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)+18/1616615*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)*(1729*b^(1/3)*(-2*a*f+17*b*c)-935*a^(1/3)*(-4 *a*g+19*b*d)*(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^(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)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.19 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.20 \[ \int \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {\sqrt {a+b x^3} \left (4 \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} (323 e+15 x (19 f+17 g x))-570 a (-17 b c+2 a f) x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )-255 a (-19 b d+4 a g) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{9690 b \sqrt {1+\frac {b x^3}{a}}} \]
(Sqrt[a + b*x^3]*(4*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*(323*e + 15*x*(19*f + 17*g*x)) - 570*a*(-17*b*c + 2*a*f)*x*Hypergeometric2F1[-3/2, 1/3, 4/3, - ((b*x^3)/a)] - 255*a*(-19*b*d + 4*a*g)*x^2*Hypergeometric2F1[-3/2, 2/3, 5/ 3, -((b*x^3)/a)]))/(9690*b*Sqrt[1 + (b*x^3)/a])
Time = 1.19 (sec) , antiderivative size = 684, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2392, 27, 2392, 27, 2427, 27, 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.
| \(\displaystyle \int \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2392 |
| \(\displaystyle \frac {9}{2} a \int \frac {2 \sqrt {b x^3+a} \left (36465 g x^4+40755 f x^3+46189 e x^2+53295 d x+62985 c\right )}{692835}dx+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \int \sqrt {b x^3+a} \left (36465 g x^4+40755 f x^3+46189 e x^2+53295 d x+62985 c\right )dx}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 2392 |
| \(\displaystyle \frac {3 a \left (\frac {3}{2} a \int \frac {2 \left (176715 g x^4+233415 f x^3+323323 e x^2+479655 d x+793611 c\right )}{63 \sqrt {b x^3+a}}dx+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \left (\frac {1}{21} a \int \frac {176715 g x^4+233415 f x^3+323323 e x^2+479655 d x+793611 c}{\sqrt {b x^3+a}}dx+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {3 a \left (\frac {1}{21} a \left (\frac {2 \int \frac {7 \left (233415 b f x^3+323323 b e x^2+25245 (19 b d-4 a g) x+793611 b c\right )}{2 \sqrt {b x^3+a}}dx}{7 b}+\frac {50490 g x^2 \sqrt {a+b x^3}}{b}\right )+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \left (\frac {1}{21} a \left (\frac {\int \frac {233415 b f x^3+323323 b e x^2+25245 (19 b d-4 a g) x+793611 b c}{\sqrt {b x^3+a}}dx}{b}+\frac {50490 g x^2 \sqrt {a+b x^3}}{b}\right )+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {3 a \left (\frac {1}{21} a \left (\frac {\frac {2 \int \frac {5 \left (323323 b^2 e x^2+25245 b (19 b d-4 a g) x+46683 b (17 b c-2 a f)\right )}{2 \sqrt {b x^3+a}}dx}{5 b}+93366 f x \sqrt {a+b x^3}}{b}+\frac {50490 g x^2 \sqrt {a+b x^3}}{b}\right )+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 a \left (\frac {1}{21} a \left (\frac {\frac {\int \frac {323323 b^2 e x^2+25245 b (19 b d-4 a g) x+46683 b (17 b c-2 a f)}{\sqrt {b x^3+a}}dx}{b}+93366 f x \sqrt {a+b x^3}}{b}+\frac {50490 g x^2 \sqrt {a+b x^3}}{b}\right )+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 2425 |
| \(\displaystyle \frac {3 a \left (\frac {1}{21} a \left (\frac {\frac {323323 b^2 e \int \frac {x^2}{\sqrt {b x^3+a}}dx+\int \frac {46683 b (17 b c-2 a f)+25245 b (19 b d-4 a g) x}{\sqrt {b x^3+a}}dx}{b}+93366 f x \sqrt {a+b x^3}}{b}+\frac {50490 g x^2 \sqrt {a+b x^3}}{b}\right )+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 793 |
| \(\displaystyle \frac {3 a \left (\frac {1}{21} a \left (\frac {\frac {\int \frac {46683 b (17 b c-2 a f)+25245 b (19 b d-4 a g) x}{\sqrt {b x^3+a}}dx+\frac {646646}{3} b e \sqrt {a+b x^3}}{b}+93366 f x \sqrt {a+b x^3}}{b}+\frac {50490 g x^2 \sqrt {a+b x^3}}{b}\right )+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {3 a \left (\frac {1}{21} a \left (\frac {\frac {27 b^{2/3} \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+25245 b^{2/3} (19 b d-4 a g) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {646646}{3} b e \sqrt {a+b x^3}}{b}+93366 f x \sqrt {a+b x^3}}{b}+\frac {50490 g x^2 \sqrt {a+b x^3}}{b}\right )+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {3 a \left (\frac {1}{21} a \left (\frac {\frac {25245 b^{2/3} (19 b d-4 a g) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {646646}{3} b e \sqrt {a+b x^3}}{b}+93366 f x \sqrt {a+b x^3}}{b}+\frac {50490 g x^2 \sqrt {a+b x^3}}{b}\right )+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {3 a \left (\frac {1}{21} a \left (\frac {\frac {\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+25245 b^{2/3} (19 b d-4 a g) \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )+\frac {646646}{3} b e \sqrt {a+b x^3}}{b}+93366 f x \sqrt {a+b x^3}}{b}+\frac {50490 g x^2 \sqrt {a+b x^3}}{b}\right )+\frac {2}{63} \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )\right )}{230945}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}\) |
(2*(a + b*x^3)^(3/2)*(62985*c*x + 53295*d*x^2 + 46189*e*x^3 + 40755*f*x^4 + 36465*g*x^5))/692835 + (3*a*((2*Sqrt[a + b*x^3]*(793611*c*x + 479655*d*x ^2 + 323323*e*x^3 + 233415*f*x^4 + 176715*g*x^5))/63 + (a*((50490*g*x^2*Sq rt[a + b*x^3])/b + (93366*f*x*Sqrt[a + b*x^3] + ((646646*b*e*Sqrt[a + b*x^ 3])/3 + 25245*b^(2/3)*(19*b*d - 4*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 + Sq rt[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]]*b^(1/3)*(1729*b^(1/3)* (17*b*c - 2*a*f) - 935*(1 - Sqrt[3])*a^(1/3)*(19*b*d - 4*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))/21))/230945
3.5.61.3.1 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[(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 + 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 = 1.70 (sec) , antiderivative size = 1024, normalized size of antiderivative = 1.48
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1024\) |
| risch | \(\text {Expression too large to display}\) | \(1138\) |
| default | \(\text {Expression too large to display}\) | \(1629\) |
2/19*g*b*x^8*(b*x^3+a)^(1/2)+2/17*b*f*x^7*(b*x^3+a)^(1/2)+2/15*b*e*x^6*(b* x^3+a)^(1/2)+2/13*(22/19*a*b*g+b^2*d)/b*x^5*(b*x^3+a)^(1/2)+2/11*(20/17*a* f*b+b^2*c)/b*x^4*(b*x^3+a)^(1/2)+4/15*a*e*x^3*(b*x^3+a)^(1/2)+2/7*(a^2*g+2 *a*b*d-10/13*a/b*(22/19*a*b*g+b^2*d))/b*x^2*(b*x^3+a)^(1/2)+2/5*(a^2*f+2*a *b*c-8/11*a/b*(20/17*a*f*b+b^2*c))/b*x*(b*x^3+a)^(1/2)+2/15*a^2*e*(b*x^3+a )^(1/2)/b-2/3*I*(a^2*c-2/5*a/b*(a^2*f+2*a*b*c-8/11*a/b*(20/17*a*f*b+b^2*c) ))*3^(1/2)/b*(-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))-2 /3*I*(a^2*d-4/7*a/b*(a^2*g+2*a*b*d-10/13*a/b*(22/19*a*b*g+b^2*d)))*3^(1/2) /b*(-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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.29 \[ \int \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {2 \, {\left (140049 \, {\left (17 \, a^{2} b c - 2 \, a^{3} f\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 75735 \, {\left (19 \, a^{2} b d - 4 \, a^{3} g\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (255255 \, b^{3} g x^{8} + 285285 \, b^{3} f x^{7} + 323323 \, b^{3} e x^{6} + 646646 \, a b^{2} e x^{3} + 19635 \, {\left (19 \, b^{3} d + 22 \, a b^{2} g\right )} x^{5} + 25935 \, {\left (17 \, b^{3} c + 20 \, a b^{2} f\right )} x^{4} + 323323 \, a^{2} b e + 2805 \, {\left (304 \, a b^{2} d + 27 \, a^{2} b g\right )} x^{2} + 5187 \, {\left (238 \, a b^{2} c + 27 \, a^{2} b f\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{4849845 \, b^{2}} \]
2/4849845*(140049*(17*a^2*b*c - 2*a^3*f)*sqrt(b)*weierstrassPInverse(0, -4 *a/b, x) - 75735*(19*a^2*b*d - 4*a^3*g)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (255255*b^3*g*x^8 + 285285*b^3*f*x^7 + 323323*b^3*e*x^6 + 646646*a*b^2*e*x^3 + 19635*(19*b^3*d + 22*a*b^2*g)*x ^5 + 25935*(17*b^3*c + 20*a*b^2*f)*x^4 + 323323*a^2*b*e + 2805*(304*a*b^2* d + 27*a^2*b*g)*x^2 + 5187*(238*a*b^2*c + 27*a^2*b*f)*x)*sqrt(b*x^3 + a))/ b^2
Time = 3.62 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.64 \[ \int \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {a^{\frac {3}{2}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {3}{2}} d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {a^{\frac {3}{2}} f 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 {a^{\frac {3}{2}} g 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} b c 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} b d 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} b f 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 )} + \frac {\sqrt {a} b g x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} + a e \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 ) + b e \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 ) \]
a**(3/2)*c*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/ a)/(3*gamma(4/3)) + a**(3/2)*d*x**2*gamma(2/3)*hyper((-1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(5/3)) + a**(3/2)*f*x**4*gamma(4/3)*hype r((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + a**(3/2) *g*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)*b*c*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)*b*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)) + sqrt(a)*b*f *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*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)) + a*e*Piecewise((sqrt(a)*x**3/3, Eq (b, 0)), (2*(a + b*x**3)**(3/2)/(9*b), True)) + b*e*Piecewise((-4*a**2*sqr t(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))
\[ \int \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int { {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int { {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int {\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]