Optimal. Leaf size=694 \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right ) \left (91 \sqrt [3]{b} (4 a f+11 b c)-110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (2 a g+13 b d)\right )}{10010 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a g+13 b d) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{182 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2}{3} a^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{27 a \sqrt{a+b x^3} (2 a g+13 b d)}{91 b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{2 a \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}+\frac{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{27 a d \sqrt{a+b x^3}}{7 x} \]
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Rubi [A] time = 0.887517, antiderivative size = 694, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {1826, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) \left (91 \sqrt [3]{b} (4 a f+11 b c)-110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (2 a g+13 b d)\right )}{10010 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a g+13 b d) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{182 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2}{3} a^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{27 a \sqrt{a+b x^3} (2 a g+13 b d)}{91 b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{2 a \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}+\frac{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{27 a d \sqrt{a+b x^3}}{7 x} \]
Antiderivative was successfully verified.
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Rule 1826
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^3} \, dx &=\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}+\frac{1}{2} (9 a) \int \frac{\sqrt{a+b x^3} \left (\frac{2 c}{5}+\frac{2 d x}{7}+\frac{2 e x^2}{9}+\frac{2 f x^3}{11}+\frac{2 g x^4}{13}\right )}{x^3} \, dx\\ &=-\frac{2 a \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}+\frac{1}{4} \left (27 a^2\right ) \int \frac{-\frac{4 c}{5}+\frac{4 d x}{7}+\frac{4 e x^2}{27}+\frac{4 f x^3}{55}+\frac{4 g x^4}{91}}{x^3 \sqrt{a+b x^3}} \, dx\\ &=\frac{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{2 a \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}-\frac{1}{16} (27 a) \int \frac{-\frac{16 a d}{7}-\frac{16 a e x}{27}-\frac{4}{55} (11 b c+4 a f) x^2-\frac{16}{91} a g x^3}{x^2 \sqrt{a+b x^3}} \, dx\\ &=\frac{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{27 a d \sqrt{a+b x^3}}{7 x}-\frac{2 a \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}+\frac{27}{32} \int \frac{\frac{32 a^2 e}{27}+\frac{8}{55} a (11 b c+4 a f) x+\frac{16}{91} a (13 b d+2 a g) x^2}{x \sqrt{a+b x^3}} \, dx\\ &=\frac{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{27 a d \sqrt{a+b x^3}}{7 x}-\frac{2 a \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}+\frac{27}{32} \int \frac{\frac{8}{55} a (11 b c+4 a f)+\frac{16}{91} a (13 b d+2 a g) x}{\sqrt{a+b x^3}} \, dx+\left (a^2 e\right ) \int \frac{1}{x \sqrt{a+b x^3}} \, dx\\ &=\frac{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{27 a d \sqrt{a+b x^3}}{7 x}-\frac{2 a \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}+\frac{1}{3} \left (a^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )+\frac{(27 a (13 b d+2 a g)) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{182 \sqrt [3]{b}}+\frac{\left (27 a \left (91 (11 b c+4 a f)-\frac{110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (13 b d+2 a g)}{\sqrt [3]{b}}\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{20020}\\ &=\frac{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{27 a d \sqrt{a+b x^3}}{7 x}+\frac{27 a (13 b d+2 a g) \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 \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} (13 b d+2 a g) \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 )}{182 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{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (91 (11 b c+4 a f)-\frac{110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (13 b d+2 a g)}{\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 )}{10010 \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}}+\frac{\left (2 a^2 e\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{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{27 a d \sqrt{a+b x^3}}{7 x}+\frac{27 a (13 b d+2 a g) \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 \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}-\frac{2}{3} a^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} (13 b d+2 a g) \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 )}{182 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{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (91 (11 b c+4 a f)-\frac{110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (13 b d+2 a g)}{\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 )}{10010 \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*}
Mathematica [C] time = 0.355531, size = 232, normalized size = 0.33 \[ \frac{4 e x^2 \sqrt{\frac{b x^3}{a}+1} \left (\sqrt{a+b x^3} \left (4 a+b x^3\right )-3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )\right )-9 a c \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right )-18 a d x \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right )+18 a f x^3 \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+9 a g x^4 \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{18 x^2 \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 1613, normalized size = 2.3 \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^{3}}\,{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^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.5008, size = 462, normalized size = 0.67 \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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