Optimal. Leaf size=676 \[ \frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-935 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-182 a g+1547 b d\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{85085 b^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{54 a^2 e \sqrt{a+b x^3}}{91 b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{91 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2}{3} a^{3/2} c \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2 a^2 f \sqrt{a+b x^3}}{15 b}+\frac{54 a^2 g x \sqrt{a+b x^3}}{935 b}+\frac{2 a \sqrt{a+b x^3} \left (85085 c x+41769 d x^2+25245 e x^3+17017 f x^4+12285 g x^5\right )}{255255 x}+\frac{2 \left (a+b x^3\right )^{3/2} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395 x} \]
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Rubi [A] time = 0.707064, antiderivative size = 676, normalized size of antiderivative = 1., 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 1826
Rule 1832
Rule 266
Rule 63
Rule 208
Rule 1888
Rule 1886
Rule 261
Rule 1878
Rule 218
Rule 1877
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*}
Mathematica [C] time = 0.439483, 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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Maple [B] time = 0.009, size = 1188, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.3669, size = 473, normalized size = 0.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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