Integrand size = 35, antiderivative size = 711 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {3 b^{4/3} (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {b (b d-4 a g) \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{12 a^{3/2}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (5 b c-14 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{560 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
1/12*b*(-4*a*g+b*d)*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(3/2)-1/420*(60*c/x ^7+70*d/x^6+84*e/x^5+105*f/x^4+140*g/x^3)*(b*x^3+a)^(1/2)-3/56*b*c*(b*x^3+ a)^(1/2)/a/x^4-1/12*b*d*(b*x^3+a)^(1/2)/a/x^3-3/20*b*e*(b*x^3+a)^(1/2)/a/x ^2+3/112*b*(-14*a*f+5*b*c)*(b*x^3+a)^(1/2)/a^2/x-3/112*b^(4/3)*(-14*a*f+5* b*c)*(b*x^3+a)^(1/2)/a^2/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))+3/224*3^(1/4)*b^( 4/3)*(-14*a*f+5*b*c)*(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)/a^(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)-1/560*3^(3/4)*b^(4/3)*(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)*(28*a^(2/3)*b^(1/3)*e-5*(-14*a*f+5*b*c)*(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)/a^(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 = 10.49 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx=-\frac {\sqrt {a+b x^3} \left (180 a^3 c \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},-\frac {1}{2},-\frac {4}{3},-\frac {b x^3}{a}\right )+7 x^2 \left (36 a^3 e \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {1}{2},-\frac {2}{3},-\frac {b x^3}{a}\right )+5 x \left (12 a^2 g x \left (a \sqrt {1+\frac {b x^3}{a}}+b x^3 \text {arctanh}\left (\sqrt {1+\frac {b x^3}{a}}\right )\right )+9 a^3 f \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},-\frac {1}{2},-\frac {1}{3},-\frac {b x^3}{a}\right )+8 b^2 d x^4 \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},3,\frac {5}{2},1+\frac {b x^3}{a}\right )\right )\right )\right )}{1260 a^3 x^7 \sqrt {1+\frac {b x^3}{a}}} \]
-1/1260*(Sqrt[a + b*x^3]*(180*a^3*c*Hypergeometric2F1[-7/3, -1/2, -4/3, -( (b*x^3)/a)] + 7*x^2*(36*a^3*e*Hypergeometric2F1[-5/3, -1/2, -2/3, -((b*x^3 )/a)] + 5*x*(12*a^2*g*x*(a*Sqrt[1 + (b*x^3)/a] + b*x^3*ArcTanh[Sqrt[1 + (b *x^3)/a]]) + 9*a^3*f*Hypergeometric2F1[-4/3, -1/2, -1/3, -((b*x^3)/a)] + 8 *b^2*d*x^4*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x^3)/a]))))/(a^3*x^7*Sqrt[1 + (b*x^3)/a])
Time = 1.54 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.01, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {2364, 27, 2374, 27, 2374, 27, 2374, 27, 2374, 25, 2371, 798, 73, 221, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -\frac {3}{2} b \int -\frac {140 g x^4+105 f x^3+84 e x^2+70 d x+60 c}{420 x^5 \sqrt {b x^3+a}}dx-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} b \int \frac {140 g x^4+105 f x^3+84 e x^2+70 d x+60 c}{x^5 \sqrt {b x^3+a}}dx-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{280} b \left (-\frac {\int -\frac {4 \left (280 a g x^3-15 (5 b c-14 a f) x^2+168 a e x+140 a d\right )}{x^4 \sqrt {b x^3+a}}dx}{8 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\int \frac {280 a g x^3-15 (5 b c-14 a f) x^2+168 a e x+140 a d}{x^4 \sqrt {b x^3+a}}dx}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{280} b \left (\frac {-\frac {\int -\frac {6 \left (168 e a^2-70 (b d-4 a g) x^2 a-15 (5 b c-14 a f) x a\right )}{x^3 \sqrt {b x^3+a}}dx}{6 a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {\int \frac {168 e a^2-70 (b d-4 a g) x^2 a-15 (5 b c-14 a f) x a}{x^3 \sqrt {b x^3+a}}dx}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {\int \frac {4 \left (42 b e x^2 a^2+15 (5 b c-14 a f) a^2+70 (b d-4 a g) x a^2\right )}{x^2 \sqrt {b x^3+a}}dx}{4 a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {\int \frac {42 b e x^2 a^2+15 (5 b c-14 a f) a^2+70 (b d-4 a g) x a^2}{x^2 \sqrt {b x^3+a}}dx}{a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {-\frac {\int -\frac {140 (b d-4 a g) a^3+84 b e x a^3+15 b (5 b c-14 a f) x^2 a^2}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {15 a \sqrt {a+b x^3} (5 b c-14 a f)}{x}}{a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {\frac {\int \frac {140 (b d-4 a g) a^3+84 b e x a^3+15 b (5 b c-14 a f) x^2 a^2}{x \sqrt {b x^3+a}}dx}{2 a}-\frac {15 a \sqrt {a+b x^3} (5 b c-14 a f)}{x}}{a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 2371 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {\frac {140 a^3 (b d-4 a g) \int \frac {1}{x \sqrt {b x^3+a}}dx+\int \frac {84 b e a^3+15 b (5 b c-14 a f) x a^2}{\sqrt {b x^3+a}}dx}{2 a}-\frac {15 a \sqrt {a+b x^3} (5 b c-14 a f)}{x}}{a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {\frac {\frac {140}{3} a^3 (b d-4 a g) \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3+\int \frac {84 b e a^3+15 b (5 b c-14 a f) x a^2}{\sqrt {b x^3+a}}dx}{2 a}-\frac {15 a \sqrt {a+b x^3} (5 b c-14 a f)}{x}}{a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {\frac {\frac {280 a^3 (b d-4 a g) \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{3 b}+\int \frac {84 b e a^3+15 b (5 b c-14 a f) x a^2}{\sqrt {b x^3+a}}dx}{2 a}-\frac {15 a \sqrt {a+b x^3} (5 b c-14 a f)}{x}}{a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {\frac {\int \frac {84 b e a^3+15 b (5 b c-14 a f) x a^2}{\sqrt {b x^3+a}}dx-\frac {280}{3} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (b d-4 a g)}{2 a}-\frac {15 a \sqrt {a+b x^3} (5 b c-14 a f)}{x}}{a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {\frac {3 a^{7/3} b^{2/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+15 a^2 b^{2/3} (5 b c-14 a f) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx-\frac {280}{3} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (b d-4 a g)}{2 a}-\frac {15 a \sqrt {a+b x^3} (5 b c-14 a f)}{x}}{a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {\frac {15 a^2 b^{2/3} (5 b c-14 a f) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{7/3} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {280}{3} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (b d-4 a g)}{2 a}-\frac {15 a \sqrt {a+b x^3} (5 b c-14 a f)}{x}}{a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {1}{280} b \left (\frac {\frac {-\frac {\frac {\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{7/3} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {280}{3} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (b d-4 a g)+15 a^2 b^{2/3} (5 b c-14 a f) \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{2 a}-\frac {15 a \sqrt {a+b x^3} (5 b c-14 a f)}{x}}{a}-\frac {84 a e \sqrt {a+b x^3}}{x^2}}{a}-\frac {140 d \sqrt {a+b x^3}}{3 x^3}}{2 a}-\frac {15 c \sqrt {a+b x^3}}{a x^4}\right )-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )\) |
-1/420*(((60*c)/x^7 + (70*d)/x^6 + (84*e)/x^5 + (105*f)/x^4 + (140*g)/x^3) *Sqrt[a + b*x^3]) + (b*((-15*c*Sqrt[a + b*x^3])/(a*x^4) + ((-140*d*Sqrt[a + b*x^3])/(3*x^3) + ((-84*a*e*Sqrt[a + b*x^3])/x^2 - ((-15*a*(5*b*c - 14*a *f)*Sqrt[a + b*x^3])/x + ((-280*a^(5/2)*(b*d - 4*a*g)*ArcTanh[Sqrt[a + b*x ^3]/Sqrt[a]])/3 + 15*a^2*b^(2/3)*(5*b*c - 14*a*f)*((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])) + (2*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^(7/3)*b^(1 /3)*(28*a^(2/3)*b^(1/3)*e - 5*(1 - Sqrt[3])*(5*b*c - 14*a*f))*(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]))/(2*a))/a)/a)/(2*a)))/280
3.5.56.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a + b*x^n)^p, x] - Simp[b*n*p Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a, b} , x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1 , 0]
Int[(Pq_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Simp[Coeff[Pq, x, 0] Int[1/(x*Sqrt[a + b*x^n]), x], x] + Int[ExpandToSum[(Pq - Coeff[Pq, x, 0])/x, x]/Sqrt[a + b*x^n], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IG tQ[n, 0] && NeQ[Coeff[Pq, x, 0], 0]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wit h[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c *(m + 1))), x] + Simp[1/(2*a*c*(m + 1)) Int[(c*x)^(m + 1)*ExpandToSum[2*a *(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b* x^n)^p, x], x] /; NeQ[Pq0, 0]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]
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]
Time = 1.82 (sec) , antiderivative size = 901, normalized size of antiderivative = 1.27
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(901\) |
| risch | \(\text {Expression too large to display}\) | \(1283\) |
| default | \(\text {Expression too large to display}\) | \(1376\) |
-1/7*c*(b*x^3+a)^(1/2)/x^7-1/6*d*(b*x^3+a)^(1/2)/x^6-1/5*e*(b*x^3+a)^(1/2) /x^5-1/56*(14*a*f+3*b*c)/a*(b*x^3+a)^(1/2)/x^4-1/12/a*(4*a*g+b*d)*(b*x^3+a )^(1/2)/x^3-3/20*b*e*(b*x^3+a)^(1/2)/a/x^2-3/112*(14*a*f-5*b*c)*b/a^2*(b*x ^3+a)^(1/2)/x+1/20*I/a*b*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)*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))-1/112*I*(14*a*f-5*b*c)*b/a^2*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)*(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)^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx=\left [-\frac {252 \, a b^{\frac {3}{2}} e x^{7} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 35 \, {\left (b^{2} d - 4 \, a b g\right )} \sqrt {a} x^{7} \log \left (\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} \sqrt {b} x^{7} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (252 \, a b e x^{5} - 45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} x^{6} + 336 \, a^{2} e x^{2} + 140 \, {\left (a b d + 4 \, a^{2} g\right )} x^{4} + 280 \, a^{2} d x + 30 \, {\left (3 \, a b c + 14 \, a^{2} f\right )} x^{3} + 240 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{1680 \, a^{2} x^{7}}, -\frac {252 \, a b^{\frac {3}{2}} e x^{7} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 70 \, {\left (b^{2} d - 4 \, a b g\right )} \sqrt {-a} x^{7} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) - 45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} \sqrt {b} x^{7} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (252 \, a b e x^{5} - 45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} x^{6} + 336 \, a^{2} e x^{2} + 140 \, {\left (a b d + 4 \, a^{2} g\right )} x^{4} + 280 \, a^{2} d x + 30 \, {\left (3 \, a b c + 14 \, a^{2} f\right )} x^{3} + 240 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{1680 \, a^{2} x^{7}}\right ] \]
[-1/1680*(252*a*b^(3/2)*e*x^7*weierstrassPInverse(0, -4*a/b, x) + 35*(b^2* d - 4*a*b*g)*sqrt(a)*x^7*log((b^2*x^6 + 8*a*b*x^3 - 4*(b*x^3 + 2*a)*sqrt(b *x^3 + a)*sqrt(a) + 8*a^2)/x^6) - 45*(5*b^2*c - 14*a*b*f)*sqrt(b)*x^7*weie rstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (252*a*b*e*x^5 - 45*(5*b^2*c - 14*a*b*f)*x^6 + 336*a^2*e*x^2 + 140*(a*b*d + 4*a^2*g)*x^4 + 280*a^2*d*x + 30*(3*a*b*c + 14*a^2*f)*x^3 + 240*a^2*c)*sqrt(b*x^3 + a)) /(a^2*x^7), -1/1680*(252*a*b^(3/2)*e*x^7*weierstrassPInverse(0, -4*a/b, x) + 70*(b^2*d - 4*a*b*g)*sqrt(-a)*x^7*arctan(1/2*(b*x^3 + 2*a)*sqrt(b*x^3 + a)*sqrt(-a)/(a*b*x^3 + a^2)) - 45*(5*b^2*c - 14*a*b*f)*sqrt(b)*x^7*weiers trassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (252*a*b*e*x^5 - 45*(5*b^2*c - 14*a*b*f)*x^6 + 336*a^2*e*x^2 + 140*(a*b*d + 4*a^2*g)*x^4 + 280*a^2*d*x + 30*(3*a*b*c + 14*a^2*f)*x^3 + 240*a^2*c)*sqrt(b*x^3 + a))/( a^2*x^7)]
Time = 4.84 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx=\frac {\sqrt {a} c \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {\sqrt {a} e \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {\sqrt {a} f \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} - \frac {a d}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b} d}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b} g \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} d}{12 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b g \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3 \sqrt {a}} + \frac {b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{12 a^{\frac {3}{2}}} \]
sqrt(a)*c*gamma(-7/3)*hyper((-7/3, -1/2), (-4/3,), b*x**3*exp_polar(I*pi)/ a)/(3*x**7*gamma(-4/3)) + sqrt(a)*e*gamma(-5/3)*hyper((-5/3, -1/2), (-2/3, ), b*x**3*exp_polar(I*pi)/a)/(3*x**5*gamma(-2/3)) + sqrt(a)*f*gamma(-4/3)* hyper((-4/3, -1/2), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**4*gamma(-1/3) ) - a*d/(6*sqrt(b)*x**(15/2)*sqrt(a/(b*x**3) + 1)) - sqrt(b)*d/(4*x**(9/2) *sqrt(a/(b*x**3) + 1)) - sqrt(b)*g*sqrt(a/(b*x**3) + 1)/(3*x**(3/2)) - b** (3/2)*d/(12*a*x**(3/2)*sqrt(a/(b*x**3) + 1)) - b*g*asinh(sqrt(a)/(sqrt(b)* x**(3/2)))/(3*sqrt(a)) + b**2*d*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(12*a**( 3/2))
\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{8}} \,d x } \]
\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{8}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx=\int \frac {\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^8} \,d x \]