Optimal. Leaf size=711 \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt{3}\right ) (5 b c-14 a f)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{560 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{3 b^{4/3} \sqrt{a+b x^3} (5 b c-14 a f)}{112 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 b c-14 a f) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{3 b \sqrt{a+b x^3} (5 b c-14 a f)}{112 a^2 x}+\frac{b (b d-4 a g) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}-\frac{1}{420} \sqrt{a+b x^3} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right )-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}-\frac{b d \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b e \sqrt{a+b x^3}}{20 a x^2} \]
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Rubi [A] time = 1.12302, antiderivative size = 711, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {14, 1825, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt{3}\right ) (5 b c-14 a f)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{560 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{3 b^{4/3} \sqrt{a+b x^3} (5 b c-14 a f)}{112 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 b c-14 a f) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{3 b \sqrt{a+b x^3} (5 b c-14 a f)}{112 a^2 x}+\frac{b (b d-4 a g) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}-\frac{1}{420} \sqrt{a+b x^3} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right )-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}-\frac{b d \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b e \sqrt{a+b x^3}}{20 a x^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1825
Rule 1835
Rule 1832
Rule 266
Rule 63
Rule 208
Rule 1878
Rule 218
Rule 1877
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^8} \, dx &=-\frac{1}{420} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right ) \sqrt{a+b x^3}-\frac{1}{2} (3 b) \int \frac{-\frac{c}{7}-\frac{d x}{6}-\frac{e x^2}{5}-\frac{f x^3}{4}-\frac{g x^4}{3}}{x^5 \sqrt{a+b x^3}} \, dx\\ &=-\frac{1}{420} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right ) \sqrt{a+b x^3}-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}+\frac{(3 b) \int \frac{\frac{4 a d}{3}+\frac{8 a e x}{5}-\frac{1}{7} (5 b c-14 a f) x^2+\frac{8}{3} a g x^3}{x^4 \sqrt{a+b x^3}} \, dx}{16 a}\\ &=-\frac{1}{420} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right ) \sqrt{a+b x^3}-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}-\frac{b d \sqrt{a+b x^3}}{12 a x^3}-\frac{b \int \frac{-\frac{48 a^2 e}{5}+\frac{6}{7} a (5 b c-14 a f) x+4 a (b d-4 a g) x^2}{x^3 \sqrt{a+b x^3}} \, dx}{32 a^2}\\ &=-\frac{1}{420} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right ) \sqrt{a+b x^3}-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}-\frac{b d \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b e \sqrt{a+b x^3}}{20 a x^2}+\frac{b \int \frac{-\frac{24}{7} a^2 (5 b c-14 a f)-16 a^2 (b d-4 a g) x-\frac{48}{5} a^2 b e x^2}{x^2 \sqrt{a+b x^3}} \, dx}{128 a^3}\\ &=-\frac{1}{420} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right ) \sqrt{a+b x^3}-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}-\frac{b d \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b e \sqrt{a+b x^3}}{20 a x^2}+\frac{3 b (5 b c-14 a f) \sqrt{a+b x^3}}{112 a^2 x}-\frac{b \int \frac{32 a^3 (b d-4 a g)+\frac{96}{5} a^3 b e x+\frac{24}{7} a^2 b (5 b c-14 a f) x^2}{x \sqrt{a+b x^3}} \, dx}{256 a^4}\\ &=-\frac{1}{420} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right ) \sqrt{a+b x^3}-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}-\frac{b d \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b e \sqrt{a+b x^3}}{20 a x^2}+\frac{3 b (5 b c-14 a f) \sqrt{a+b x^3}}{112 a^2 x}-\frac{b \int \frac{\frac{96}{5} a^3 b e+\frac{24}{7} a^2 b (5 b c-14 a f) x}{\sqrt{a+b x^3}} \, dx}{256 a^4}-\frac{(b (b d-4 a g)) \int \frac{1}{x \sqrt{a+b x^3}} \, dx}{8 a}\\ &=-\frac{1}{420} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right ) \sqrt{a+b x^3}-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}-\frac{b d \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b e \sqrt{a+b x^3}}{20 a x^2}+\frac{3 b (5 b c-14 a f) \sqrt{a+b x^3}}{112 a^2 x}-\frac{\left (3 b^{5/3} (5 b c-14 a f)\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{224 a^2}-\frac{\left (3 b^{5/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt{3}\right ) (5 b c-14 a f)\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{1120 a^{5/3}}-\frac{(b (b d-4 a g)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{24 a}\\ &=-\frac{1}{420} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right ) \sqrt{a+b x^3}-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}-\frac{b d \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b e \sqrt{a+b x^3}}{20 a x^2}+\frac{3 b (5 b c-14 a f) \sqrt{a+b x^3}}{112 a^2 x}-\frac{3 b^{4/3} (5 b c-14 a f) \sqrt{a+b x^3}}{112 a^2 \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} (5 b c-14 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 )}{224 a^{5/3} \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^{4/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt{3}\right ) (5 b c-14 a f)\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 )}{560 a^{5/3} \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 d-4 a g) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{12 a}\\ &=-\frac{1}{420} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right ) \sqrt{a+b x^3}-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}-\frac{b d \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b e \sqrt{a+b x^3}}{20 a x^2}+\frac{3 b (5 b c-14 a f) \sqrt{a+b x^3}}{112 a^2 x}-\frac{3 b^{4/3} (5 b c-14 a f) \sqrt{a+b x^3}}{112 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{b (b d-4 a g) \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} (5 b c-14 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 )}{224 a^{5/3} \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^{4/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt{3}\right ) (5 b c-14 a f)\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 )}{560 a^{5/3} \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 = 0.452092, size = 213, normalized size = 0.3 \[ -\frac{\sqrt{a+b x^3} \left (7 x^2 \left (5 x \left (9 a^3 f \, _2F_1\left (-\frac{4}{3},-\frac{1}{2};-\frac{1}{3};-\frac{b x^3}{a}\right )+12 a^2 g x \left (a \sqrt{\frac{b x^3}{a}+1}+b x^3 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )\right )+8 b^2 d x^4 \left (a+b x^3\right ) \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{b x^3}{a}+1\right )\right )+36 a^3 e \, _2F_1\left (-\frac{5}{3},-\frac{1}{2};-\frac{2}{3};-\frac{b x^3}{a}\right )\right )+180 a^3 c \, _2F_1\left (-\frac{7}{3},-\frac{1}{2};-\frac{4}{3};-\frac{b x^3}{a}\right )\right )}{1260 a^3 x^7 \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 1376, normalized size = 1.9 \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 )} \sqrt{b x^{3} + a}}{x^{8}}\,{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 (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a}}{x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.0289, size = 308, normalized size = 0.43 \begin{align*} \frac{\sqrt{a} c \Gamma \left (- \frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{3}, - \frac{1}{2} \\ - \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac{4}{3}\right )} + \frac{\sqrt{a} e \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} f \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{a d}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{\sqrt{b} d}{4 x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{\sqrt{b} g \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} - \frac{b^{\frac{3}{2}} d}{12 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{b g \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} + \frac{b^{2} d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{12 a^{\frac{3}{2}}} \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 )} \sqrt{b x^{3} + a}}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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