Integrand size = 35, antiderivative size = 640 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\frac {3 c \sqrt {a+b x^3}}{2 x^2}-\frac {3 d \sqrt {a+b x^3}}{x}+\frac {3 (7 b d+2 a g) \sqrt {a+b x^3}}{7 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 \sqrt {a+b x^3} \left (105 c x-105 d x^2-35 e x^3-21 f x^4-15 g x^5\right )}{105 x^3}-\frac {2}{3} \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} (7 b d+2 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{14 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} \left (7 \sqrt [3]{b} (5 b c+4 a f)-10 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (7 b d+2 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{70 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Time = 0.53 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {1840, 1849, 1846, 272, 65, 214, 1892, 224, 1891} \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\frac {3^{3/4} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (7 \sqrt [3]{b} (4 a f+5 b c)-10 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (2 a g+7 b d)\right )}{70 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a g+7 b d) E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{14 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2}{3} \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {3 \sqrt {a+b x^3} (2 a g+7 b d)}{7 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 \sqrt {a+b x^3} \left (105 c x-105 d x^2-35 e x^3-21 f x^4-15 g x^5\right )}{105 x^3}+\frac {3 c \sqrt {a+b x^3}}{2 x^2}-\frac {3 d \sqrt {a+b x^3}}{x} \]
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Rule 65
Rule 214
Rule 224
Rule 272
Rule 1840
Rule 1846
Rule 1849
Rule 1891
Rule 1892
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a+b x^3} \left (105 c x-105 d x^2-35 e x^3-21 f x^4-15 g x^5\right )}{105 x^3}+\frac {1}{2} (3 a) \int \frac {-2 c+2 d x+\frac {2 e x^2}{3}+\frac {2 f x^3}{5}+\frac {2 g x^4}{7}}{x^3 \sqrt {a+b x^3}} \, dx \\ & = \frac {3 c \sqrt {a+b x^3}}{2 x^2}-\frac {2 \sqrt {a+b x^3} \left (105 c x-105 d x^2-35 e x^3-21 f x^4-15 g x^5\right )}{105 x^3}-\frac {3}{8} \int \frac {-8 a d-\frac {8 a e x}{3}-\frac {2}{5} (5 b c+4 a f) x^2-\frac {8}{7} a g x^3}{x^2 \sqrt {a+b x^3}} \, dx \\ & = \frac {3 c \sqrt {a+b x^3}}{2 x^2}-\frac {3 d \sqrt {a+b x^3}}{x}-\frac {2 \sqrt {a+b x^3} \left (105 c x-105 d x^2-35 e x^3-21 f x^4-15 g x^5\right )}{105 x^3}+\frac {3 \int \frac {\frac {16 a^2 e}{3}+\frac {4}{5} a (5 b c+4 a f) x+\frac {8}{7} a (7 b d+2 a g) x^2}{x \sqrt {a+b x^3}} \, dx}{16 a} \\ & = \frac {3 c \sqrt {a+b x^3}}{2 x^2}-\frac {3 d \sqrt {a+b x^3}}{x}-\frac {2 \sqrt {a+b x^3} \left (105 c x-105 d x^2-35 e x^3-21 f x^4-15 g x^5\right )}{105 x^3}+\frac {3 \int \frac {\frac {4}{5} a (5 b c+4 a f)+\frac {8}{7} a (7 b d+2 a g) x}{\sqrt {a+b x^3}} \, dx}{16 a}+(a e) \int \frac {1}{x \sqrt {a+b x^3}} \, dx \\ & = \frac {3 c \sqrt {a+b x^3}}{2 x^2}-\frac {3 d \sqrt {a+b x^3}}{x}-\frac {2 \sqrt {a+b x^3} \left (105 c x-105 d x^2-35 e x^3-21 f x^4-15 g x^5\right )}{105 x^3}+\frac {1}{3} (a e) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )+\frac {(3 (7 b d+2 a g)) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{14 \sqrt [3]{b}}+\frac {1}{140} \left (3 \left (7 (5 b c+4 a f)-\frac {10 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (7 b d+2 a g)}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx \\ & = \frac {3 c \sqrt {a+b x^3}}{2 x^2}-\frac {3 d \sqrt {a+b x^3}}{x}+\frac {3 (7 b d+2 a g) \sqrt {a+b x^3}}{7 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 \sqrt {a+b x^3} \left (105 c x-105 d x^2-35 e x^3-21 f x^4-15 g x^5\right )}{105 x^3}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} (7 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 (\sin ^{-1}\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 )}{14 b^{2/3} \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}} \left (7 (5 b c+4 a f)-\frac {10 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (7 b d+2 a g)}{\sqrt [3]{b}}\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}} F\left (\sin ^{-1}\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 )}{70 \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}}+\frac {(2 a e) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 b} \\ & = \frac {3 c \sqrt {a+b x^3}}{2 x^2}-\frac {3 d \sqrt {a+b x^3}}{x}+\frac {3 (7 b d+2 a g) \sqrt {a+b x^3}}{7 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 \sqrt {a+b x^3} \left (105 c x-105 d x^2-35 e x^3-21 f x^4-15 g x^5\right )}{105 x^3}-\frac {2}{3} \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} (7 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 (\sin ^{-1}\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 )}{14 b^{2/3} \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}} \left (7 (5 b c+4 a f)-\frac {10 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (7 b d+2 a g)}{\sqrt [3]{b}}\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}} F\left (\sin ^{-1}\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 )}{70 \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}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.78 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\frac {-3 c \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},-\frac {b x^3}{a}\right )+x \left (-6 d \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{3},\frac {2}{3},-\frac {b x^3}{a}\right )+x \left (4 e \sqrt {1+\frac {b x^3}{a}} \left (\sqrt {a+b x^3}-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )+6 f x \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+3 g x^2 \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )\right )}{6 x^2 \sqrt {1+\frac {b x^3}{a}}} \]
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Time = 1.93 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.29
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(826\) |
default | \(\text {Expression too large to display}\) | \(1529\) |
risch | \(\text {Expression too large to display}\) | \(2244\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.24 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\left [\frac {35 \, \sqrt {a} b e x^{2} \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 63 \, {\left (5 \, b c + 4 \, a f\right )} \sqrt {b} x^{2} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 90 \, {\left (7 \, b d + 2 \, a g\right )} \sqrt {b} x^{2} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (60 \, b g x^{4} + 84 \, b f x^{3} + 140 \, b e x^{2} - 210 \, b d x - 105 \, b c\right )} \sqrt {b x^{3} + a}}{210 \, b x^{2}}, \frac {70 \, \sqrt {-a} b e x^{2} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) + 63 \, {\left (5 \, b c + 4 \, a f\right )} \sqrt {b} x^{2} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 90 \, {\left (7 \, b d + 2 \, a g\right )} \sqrt {b} x^{2} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (60 \, b g x^{4} + 84 \, b f x^{3} + 140 \, b e x^{2} - 210 \, b d x - 105 \, b c\right )} \sqrt {b x^{3} + a}}{210 \, b x^{2}}\right ] \]
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Time = 2.98 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\frac {\sqrt {a} c \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {\sqrt {a} d \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \frac {2 \sqrt {a} e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3} + \frac {\sqrt {a} f x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {\sqrt {a} g x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {2 a e}{3 \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 \sqrt {b} e x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} \]
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\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{3}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^3} \, dx=\int \frac {\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^3} \,d x \]
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