Optimal. Leaf size=638 \[ \frac {3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (14 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (2 a f+7 b c)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{35 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a f+7 b c) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{14 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3 \sqrt {a+b x^3} (2 a f+7 b c)}{7 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 \sqrt {a+b x^3} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^2}-\frac {3 c \sqrt {a+b x^3}}{x}-\frac {2}{3} \sqrt {a} d \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {2 a g \sqrt {a+b x^3}}{9 b} \]
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Rubi [A] time = 0.65, antiderivative size = 638, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {1826, 1835, 1832, 266, 63, 208, 1886, 261, 1878, 218, 1877} \[ \frac {3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (14 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (2 a f+7 b c)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{35 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a f+7 b c) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{14 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3 \sqrt {a+b x^3} (2 a f+7 b c)}{7 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 \sqrt {a+b x^3} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^2}-\frac {3 c \sqrt {a+b x^3}}{x}-\frac {2}{3} \sqrt {a} d \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {2 a g \sqrt {a+b x^3}}{9 b} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 218
Rule 261
Rule 266
Rule 1826
Rule 1832
Rule 1835
Rule 1877
Rule 1878
Rule 1886
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^2} \, dx &=\frac {2 \sqrt {a+b x^3} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^2}+\frac {1}{2} (3 a) \int \frac {2 c+\frac {2 d x}{3}+\frac {2 e x^2}{5}+\frac {2 f x^3}{7}+\frac {2 g x^4}{9}}{x^2 \sqrt {a+b x^3}} \, dx\\ &=-\frac {3 c \sqrt {a+b x^3}}{x}+\frac {2 \sqrt {a+b x^3} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^2}-\frac {3}{4} \int \frac {-\frac {4 a d}{3}-\frac {4 a e x}{5}-\frac {2}{7} (7 b c+2 a f) x^2-\frac {4}{9} a g x^3}{x \sqrt {a+b x^3}} \, dx\\ &=-\frac {3 c \sqrt {a+b x^3}}{x}+\frac {2 \sqrt {a+b x^3} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^2}-\frac {3}{4} \int \frac {-\frac {4 a e}{5}-\frac {2}{7} (7 b c+2 a f) x-\frac {4}{9} a g x^2}{\sqrt {a+b x^3}} \, dx+(a d) \int \frac {1}{x \sqrt {a+b x^3}} \, dx\\ &=-\frac {3 c \sqrt {a+b x^3}}{x}+\frac {2 \sqrt {a+b x^3} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^2}-\frac {3}{4} \int \frac {-\frac {4 a e}{5}-\frac {2}{7} (7 b c+2 a f) x}{\sqrt {a+b x^3}} \, dx+\frac {1}{3} (a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )+\frac {1}{3} (a g) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx\\ &=\frac {2 a g \sqrt {a+b x^3}}{9 b}-\frac {3 c \sqrt {a+b x^3}}{x}+\frac {2 \sqrt {a+b x^3} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^2}+\frac {(2 a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 b}+\frac {(3 (7 b c+2 a f)) \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}{70} \left (3 \sqrt [3]{a} \left (14 a^{2/3} e-\frac {5 \left (1-\sqrt {3}\right ) (7 b c+2 a f)}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx\\ &=\frac {2 a g \sqrt {a+b x^3}}{9 b}-\frac {3 c \sqrt {a+b x^3}}{x}+\frac {3 (7 b c+2 a f) \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 (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^2}-\frac {2}{3} \sqrt {a} d \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 c+2 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 (\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}} \sqrt [3]{a} \left (14 a^{2/3} e-\frac {5 \left (1-\sqrt {3}\right ) (7 b c+2 a f)}{\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 )}{35 \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*}
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Mathematica [C] time = 0.29, size = 211, normalized size = 0.33 \[ -\frac {c \sqrt {a+b x^3} \, _2F_1\left (-\frac {1}{2},-\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{x \sqrt {\frac {b x^3}{a}+1}}+\frac {2}{3} d \left (\sqrt {a+b x^3}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )+\frac {e x \sqrt {a+b x^3} \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\sqrt {\frac {b x^3}{a}+1}}+\frac {f x^2 \sqrt {a+b x^3} \, _2F_1\left (-\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{2 \sqrt {\frac {b x^3}{a}+1}}+\frac {2 g \left (a+b x^3\right )^{3/2}}{9 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1248, normalized size = 1.96 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.77, size = 236, normalized size = 0.37 \[ \frac {\sqrt {a} c \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \frac {2 \sqrt {a} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3} + \frac {\sqrt {a} e x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {\sqrt {a} f x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {2 a d}{3 \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 \sqrt {b} d x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} + g \left (\begin {cases} \frac {\sqrt {a} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{9 b} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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