Optimal. Leaf size=676 \[ -\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2}{3} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {54 a^2 e \sqrt {a+b x^3}}{91 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 a^2 f \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 g x \sqrt {a+b x^3}}{935 b}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-182 a g+1547 b d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{85085 b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 a \sqrt {a+b x^3} \left (85085 c x+41769 d x^2+25245 e x^3+17017 f x^4+12285 g x^5\right )}{255255 x}+\frac {2 \left (a+b x^3\right )^{3/2} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395 x} \]
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Rubi [A] time = 0.71, antiderivative size = 676, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {1826, 1832, 266, 63, 208, 1888, 1886, 261, 1878, 218, 1877} \[ \frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-182 a g+1547 b d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{85085 b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 e \sqrt {a+b x^3}}{91 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2}{3} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {2 a^2 f \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 g x \sqrt {a+b x^3}}{935 b}+\frac {2 a \sqrt {a+b x^3} \left (85085 c x+41769 d x^2+25245 e x^3+17017 f x^4+12285 g x^5\right )}{255255 x}+\frac {2 \left (a+b x^3\right )^{3/2} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395 x} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 218
Rule 261
Rule 266
Rule 1826
Rule 1832
Rule 1877
Rule 1878
Rule 1886
Rule 1888
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx &=\frac {2 \left (a+b x^3\right )^{3/2} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395 x}+\frac {1}{2} (9 a) \int \frac {\sqrt {a+b x^3} \left (\frac {2 c}{9}+\frac {2 d x}{11}+\frac {2 e x^2}{13}+\frac {2 f x^3}{15}+\frac {2 g x^4}{17}\right )}{x} \, dx\\ &=\frac {2 \left (a+b x^3\right )^{3/2} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395 x}+\frac {2 a \sqrt {a+b x^3} \left (85085 c x+41769 d x^2+25245 e x^3+17017 f x^4+12285 g x^5\right )}{255255 x}+\frac {1}{4} \left (27 a^2\right ) \int \frac {\frac {4 c}{27}+\frac {4 d x}{55}+\frac {4 e x^2}{91}+\frac {4 f x^3}{135}+\frac {4 g x^4}{187}}{x \sqrt {a+b x^3}} \, dx\\ &=\frac {2 \left (a+b x^3\right )^{3/2} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395 x}+\frac {2 a \sqrt {a+b x^3} \left (85085 c x+41769 d x^2+25245 e x^3+17017 f x^4+12285 g x^5\right )}{255255 x}+\frac {1}{4} \left (27 a^2\right ) \int \frac {\frac {4 d}{55}+\frac {4 e x}{91}+\frac {4 f x^2}{135}+\frac {4 g x^3}{187}}{\sqrt {a+b x^3}} \, dx+\left (a^2 c\right ) \int \frac {1}{x \sqrt {a+b x^3}} \, dx\\ &=\frac {54 a^2 g x \sqrt {a+b x^3}}{935 b}+\frac {2 \left (a+b x^3\right )^{3/2} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395 x}+\frac {2 a \sqrt {a+b x^3} \left (85085 c x+41769 d x^2+25245 e x^3+17017 f x^4+12285 g x^5\right )}{255255 x}+\frac {\left (27 a^2\right ) \int \frac {\frac {2}{187} (17 b d-2 a g)+\frac {10 b e x}{91}+\frac {2}{27} b f x^2}{\sqrt {a+b x^3}} \, dx}{10 b}+\frac {1}{3} \left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {54 a^2 g x \sqrt {a+b x^3}}{935 b}+\frac {2 \left (a+b x^3\right )^{3/2} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395 x}+\frac {2 a \sqrt {a+b x^3} \left (85085 c x+41769 d x^2+25245 e x^3+17017 f x^4+12285 g x^5\right )}{255255 x}+\frac {\left (27 a^2\right ) \int \frac {\frac {2}{187} (17 b d-2 a g)+\frac {10 b e x}{91}}{\sqrt {a+b x^3}} \, dx}{10 b}+\frac {\left (2 a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 b}+\frac {1}{5} \left (a^2 f\right ) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx\\ &=\frac {2 a^2 f \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 g x \sqrt {a+b x^3}}{935 b}+\frac {2 \left (a+b x^3\right )^{3/2} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395 x}+\frac {2 a \sqrt {a+b x^3} \left (85085 c x+41769 d x^2+25245 e x^3+17017 f x^4+12285 g x^5\right )}{255255 x}-\frac {2}{3} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {\left (27 a^2 e\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{91 \sqrt [3]{b}}+\frac {\left (27 a^2 \left (1547 b d-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-182 a g\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{85085 b}\\ &=\frac {2 a^2 f \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 g x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 e \sqrt {a+b x^3}}{91 b^{2/3} \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 (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395 x}+\frac {2 a \sqrt {a+b x^3} \left (85085 c x+41769 d x^2+25245 e x^3+17017 f x^4+12285 g x^5\right )}{255255 x}-\frac {2}{3} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} 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 )}{91 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 {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (1547 b d-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-182 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 )}{85085 b^{4/3} \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}}\\ \end {align*}
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Mathematica [C] time = 0.58, size = 215, normalized size = 0.32 \[ \frac {4 \sqrt {\frac {b x^3}{a}+1} \left (\sqrt {a+b x^3} \left (a^2 (51 f+45 g x)+2 a b \left (170 c+51 f x^3+45 g x^4\right )+b^2 x^3 \left (85 c+51 f x^3+45 g x^4\right )\right )-255 a^{3/2} b c \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )-90 a x \sqrt {a+b x^3} (2 a g-17 b d) \, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+765 a b e x^2 \sqrt {a+b x^3} \, _2F_1\left (-\frac {3}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{1530 b \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x}, x\right ) \]
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 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1188, normalized size = 1.76 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.74, size = 473, normalized size = 0.70 \[ - \frac {2 a^{\frac {3}{2}} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3} + \frac {a^{\frac {3}{2}} d 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}} e 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}} g 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^{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 e 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 g 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 {2 a^{2} c}{3 \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 a \sqrt {b} c x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} + a f \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 c \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 f \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 ) \]
Verification of antiderivative is not currently implemented for this CAS.
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