Optimal. Leaf size=714 \[ -\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g+7 b d\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{2240 a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{b^2 (b c-6 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{3/2}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}+\frac{27 b^{7/3} e \sqrt{a+b x^3}}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{b \sqrt{a+b x^3} \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right )}{1680}-\frac{\left (a+b x^3\right )^{3/2} \left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right )}{2520} \]
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Rubi [A] time = 1.12299, antiderivative size = 714, 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} \[ \frac{b^2 (b c-6 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{3/2}}-\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g+7 b d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2240 a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}+\frac{27 b^{7/3} e \sqrt{a+b x^3}}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{b \sqrt{a+b x^3} \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right )}{1680}-\frac{\left (a+b x^3\right )^{3/2} \left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right )}{2520} \]
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{\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx &=-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac{1}{2} (9 b) \int \frac{\sqrt{a+b x^3} \left (-\frac{c}{9}-\frac{d x}{8}-\frac{e x^2}{7}-\frac{f x^3}{6}-\frac{g x^4}{5}\right )}{x^7} \, dx\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac{1}{4} \left (27 b^2\right ) \int \frac{\frac{c}{54}+\frac{d x}{40}+\frac{e x^2}{28}+\frac{f x^3}{18}+\frac{g x^4}{10}}{x^4 \sqrt{a+b x^3}} \, dx\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac{\left (9 b^2\right ) \int \frac{-\frac{3 a d}{20}-\frac{3 a e x}{14}+\frac{1}{18} (b c-6 a f) x^2-\frac{3}{5} a g x^3}{x^3 \sqrt{a+b x^3}} \, dx}{8 a}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac{\left (9 b^2\right ) \int \frac{\frac{6 a^2 e}{7}-\frac{2}{9} a (b c-6 a f) x-\frac{3}{20} a (b d-16 a g) x^2}{x^2 \sqrt{a+b x^3}} \, dx}{32 a^2}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac{\left (9 b^2\right ) \int \frac{\frac{4}{9} a^2 (b c-6 a f)+\frac{3}{10} a^2 (b d-16 a g) x-\frac{6}{7} a^2 b e x^2}{x \sqrt{a+b x^3}} \, dx}{64 a^3}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac{\left (9 b^2\right ) \int \frac{\frac{3}{10} a^2 (b d-16 a g)-\frac{6}{7} a^2 b e x}{\sqrt{a+b x^3}} \, dx}{64 a^3}-\frac{\left (b^2 (b c-6 a f)\right ) \int \frac{1}{x \sqrt{a+b x^3}} \, dx}{16 a}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac{\left (27 b^{8/3} e\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{224 a}-\frac{\left (b^2 (b c-6 a f)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{48 a}-\frac{\left (27 b^2 \left (7 b d+20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{4480 a}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}+\frac{27 b^{7/3} e \sqrt{a+b x^3}}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{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 )}{224 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{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (7 b d+20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-112 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 )}{2240 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 (b c-6 a f)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{24 a}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}+\frac{27 b^{7/3} e \sqrt{a+b x^3}}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac{b^2 (b c-6 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{3/2}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{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 )}{224 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{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (7 b d+20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-112 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 )}{2240 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*}
Mathematica [C] time = 0.644999, size = 226, normalized size = 0.32 \[ -\frac{\sqrt{a+b x^3} \left (2 x \left (7 x \left (5 a^3 f \left (3 b^2 x^6 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )+a \left (2 a+5 b x^3\right ) \sqrt{\frac{b x^3}{a}+1}\right )+12 a^5 g x \, _2F_1\left (-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right )-8 b^3 c x^6 \left (a+b x^3\right )^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{b x^3}{a}+1\right )\right )+60 a^5 e \, _2F_1\left (-\frac{7}{3},-\frac{3}{2};-\frac{4}{3};-\frac{b x^3}{a}\right )\right )+105 a^5 d \, _2F_1\left (-\frac{8}{3},-\frac{3}{2};-\frac{5}{3};-\frac{b x^3}{a}\right )\right )}{840 a^4 x^8 \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 1273, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^{10}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 23.7195, size = 573, normalized size = 0.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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