Integrand size = 35, antiderivative size = 659 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \sqrt {a+b x^3}-\frac {b c \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b d \sqrt {a+b x^3}}{20 a x^2}-\frac {3 b e \sqrt {a+b x^3}}{8 a x}+\frac {3 b^{4/3} e \sqrt {a+b x^3}}{8 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {b (b c-4 a f) \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{12 a^{3/2}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{16 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \left (2 b d+5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-20 a g\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{40 a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Time = 0.66 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {14, 1839, 1849, 1846, 272, 65, 214, 1892, 224, 1891} \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=-\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-20 a g+2 b d\right )}{40 a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{16 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (b c-4 a f)}{12 a^{3/2}}+\frac {3 b^{4/3} e \sqrt {a+b x^3}}{8 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{60} \sqrt {a+b x^3} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )-\frac {b c \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b d \sqrt {a+b x^3}}{20 a x^2}-\frac {3 b e \sqrt {a+b x^3}}{8 a x} \]
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Rule 14
Rule 65
Rule 214
Rule 224
Rule 272
Rule 1839
Rule 1846
Rule 1849
Rule 1891
Rule 1892
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \sqrt {a+b x^3}-\frac {1}{2} (3 b) \int \frac {-\frac {c}{6}-\frac {d x}{5}-\frac {e x^2}{4}-\frac {f x^3}{3}-\frac {g x^4}{2}}{x^4 \sqrt {a+b x^3}} \, dx \\ & = -\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \sqrt {a+b x^3}-\frac {b c \sqrt {a+b x^3}}{12 a x^3}+\frac {b \int \frac {\frac {6 a d}{5}+\frac {3 a e x}{2}-\frac {1}{2} (b c-4 a f) x^2+3 a g x^3}{x^3 \sqrt {a+b x^3}} \, dx}{4 a} \\ & = -\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \sqrt {a+b x^3}-\frac {b c \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b d \sqrt {a+b x^3}}{20 a x^2}-\frac {b \int \frac {-6 a^2 e+2 a (b c-4 a f) x+\frac {6}{5} a (b d-10 a g) x^2}{x^2 \sqrt {a+b x^3}} \, dx}{16 a^2} \\ & = -\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \sqrt {a+b x^3}-\frac {b c \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b d \sqrt {a+b x^3}}{20 a x^2}-\frac {3 b e \sqrt {a+b x^3}}{8 a x}+\frac {b \int \frac {-4 a^2 (b c-4 a f)-\frac {12}{5} a^2 (b d-10 a g) x+6 a^2 b e x^2}{x \sqrt {a+b x^3}} \, dx}{32 a^3} \\ & = -\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \sqrt {a+b x^3}-\frac {b c \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b d \sqrt {a+b x^3}}{20 a x^2}-\frac {3 b e \sqrt {a+b x^3}}{8 a x}+\frac {b \int \frac {-\frac {12}{5} a^2 (b d-10 a g)+6 a^2 b e x}{\sqrt {a+b x^3}} \, dx}{32 a^3}-\frac {(b (b c-4 a f)) \int \frac {1}{x \sqrt {a+b x^3}} \, dx}{8 a} \\ & = -\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \sqrt {a+b x^3}-\frac {b c \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b d \sqrt {a+b x^3}}{20 a x^2}-\frac {3 b e \sqrt {a+b x^3}}{8 a x}+\frac {\left (3 b^{5/3} e\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{16 a}-\frac {(b (b c-4 a f)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{24 a}-\frac {\left (3 b \left (2 b d+5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-20 a g\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{80 a} \\ & = -\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \sqrt {a+b x^3}-\frac {b c \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b d \sqrt {a+b x^3}}{20 a x^2}-\frac {3 b e \sqrt {a+b x^3}}{8 a x}+\frac {3 b^{4/3} e \sqrt {a+b x^3}}{8 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} 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 (\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 )}{16 a^{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}} b^{2/3} \left (2 b d+5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-20 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}} 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 )}{40 a \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 {(b c-4 a f) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{12 a} \\ & = -\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \sqrt {a+b x^3}-\frac {b c \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b d \sqrt {a+b x^3}}{20 a x^2}-\frac {3 b e \sqrt {a+b x^3}}{8 a x}+\frac {3 b^{4/3} e \sqrt {a+b x^3}}{8 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {b (b c-4 a f) \tanh ^{-1}\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} 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 (\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 )}{16 a^{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}} b^{2/3} \left (2 b d+5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-20 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}} 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 )}{40 a \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 = 10.43 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=-\frac {\sqrt {a+b x^3} \left (36 a^3 d \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {1}{2},-\frac {2}{3},-\frac {b x^3}{a}\right )+5 x \left (9 a^3 e \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},-\frac {1}{2},-\frac {1}{3},-\frac {b x^3}{a}\right )+2 x \left (6 a^2 f \left (a \sqrt {1+\frac {b x^3}{a}}+b x^3 \text {arctanh}\left (\sqrt {1+\frac {b x^3}{a}}\right )\right )+9 a^3 g x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},-\frac {b x^3}{a}\right )+4 b^2 c x^3 \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},3,\frac {5}{2},1+\frac {b x^3}{a}\right )\right )\right )\right )}{180 a^3 x^5 \sqrt {1+\frac {b x^3}{a}}} \]
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Time = 1.82 (sec) , antiderivative size = 883, normalized size of antiderivative = 1.34
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(883\) |
risch | \(\text {Expression too large to display}\) | \(1102\) |
default | \(\text {Expression too large to display}\) | \(1180\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\left [-\frac {90 \, a b^{\frac {3}{2}} e x^{6} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 5 \, {\left (b^{2} c - 4 \, a b f\right )} \sqrt {a} x^{6} \log \left (\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 36 \, {\left (a b d - 10 \, a^{2} g\right )} \sqrt {b} x^{6} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 2 \, {\left (45 \, a b e x^{5} + 30 \, a^{2} e x^{2} + 6 \, {\left (3 \, a b d + 10 \, a^{2} g\right )} x^{4} + 24 \, a^{2} d x + 10 \, {\left (a b c + 4 \, a^{2} f\right )} x^{3} + 20 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{240 \, a^{2} x^{6}}, -\frac {45 \, a b^{\frac {3}{2}} e x^{6} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 5 \, {\left (b^{2} c - 4 \, a b f\right )} \sqrt {-a} x^{6} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) + 18 \, {\left (a b d - 10 \, a^{2} g\right )} \sqrt {b} x^{6} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (45 \, a b e x^{5} + 30 \, a^{2} e x^{2} + 6 \, {\left (3 \, a b d + 10 \, a^{2} g\right )} x^{4} + 24 \, a^{2} d x + 10 \, {\left (a b c + 4 \, a^{2} f\right )} x^{3} + 20 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{120 \, a^{2} x^{6}}\right ] \]
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Time = 4.76 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\frac {\sqrt {a} d \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {\sqrt {a} e \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt {a} g \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} - \frac {a c}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b} c}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b} f \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} c}{12 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3 \sqrt {a}} + \frac {b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{12 a^{\frac {3}{2}}} \]
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\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{7}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{7}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx=\int \frac {\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^7} \,d x \]
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