Optimal. Leaf size=692 \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (182 a^{2/3} \sqrt [3]{b} e-55 \left (1-\sqrt{3}\right ) (2 a f+13 b c)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{5005 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a f+13 b c) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{182 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2}{3} a^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2 a^2 g \sqrt{a+b x^3}}{15 b}+\frac{27 a \sqrt{a+b x^3} (2 a f+13 b c)}{91 b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 a \sqrt{a+b x^3} \left (19305 c x+5005 d x^2+2457 e x^3+1485 f x^4+1001 g x^5\right )}{15015 x^2}+\frac{2 \left (a+b x^3\right )^{3/2} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045 x^2}-\frac{27 a c \sqrt{a+b x^3}}{7 x} \]
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Rubi [A] time = 0.790546, antiderivative size = 692, 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, 1835, 1832, 266, 63, 208, 1886, 261, 1878, 218, 1877} \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (182 a^{2/3} \sqrt [3]{b} e-55 \left (1-\sqrt{3}\right ) (2 a f+13 b c)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{5005 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a f+13 b c) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{182 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2}{3} a^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2 a^2 g \sqrt{a+b x^3}}{15 b}+\frac{27 a \sqrt{a+b x^3} (2 a f+13 b c)}{91 b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 a \sqrt{a+b x^3} \left (19305 c x+5005 d x^2+2457 e x^3+1485 f x^4+1001 g x^5\right )}{15015 x^2}+\frac{2 \left (a+b x^3\right )^{3/2} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045 x^2}-\frac{27 a c \sqrt{a+b x^3}}{7 x} \]
Antiderivative was successfully verified.
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Rule 1826
Rule 1835
Rule 1832
Rule 266
Rule 63
Rule 208
Rule 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^2} \, dx &=\frac{2 \left (a+b x^3\right )^{3/2} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045 x^2}+\frac{1}{2} (9 a) \int \frac{\sqrt{a+b x^3} \left (\frac{2 c}{7}+\frac{2 d x}{9}+\frac{2 e x^2}{11}+\frac{2 f x^3}{13}+\frac{2 g x^4}{15}\right )}{x^2} \, dx\\ &=\frac{2 a \sqrt{a+b x^3} \left (19305 c x+5005 d x^2+2457 e x^3+1485 f x^4+1001 g x^5\right )}{15015 x^2}+\frac{2 \left (a+b x^3\right )^{3/2} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045 x^2}+\frac{1}{4} \left (27 a^2\right ) \int \frac{\frac{4 c}{7}+\frac{4 d x}{27}+\frac{4 e x^2}{55}+\frac{4 f x^3}{91}+\frac{4 g x^4}{135}}{x^2 \sqrt{a+b x^3}} \, dx\\ &=-\frac{27 a c \sqrt{a+b x^3}}{7 x}+\frac{2 a \sqrt{a+b x^3} \left (19305 c x+5005 d x^2+2457 e x^3+1485 f x^4+1001 g x^5\right )}{15015 x^2}+\frac{2 \left (a+b x^3\right )^{3/2} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045 x^2}-\frac{1}{8} (27 a) \int \frac{-\frac{8 a d}{27}-\frac{8 a e x}{55}-\frac{4}{91} (13 b c+2 a f) x^2-\frac{8}{135} a g x^3}{x \sqrt{a+b x^3}} \, dx\\ &=-\frac{27 a c \sqrt{a+b x^3}}{7 x}+\frac{2 a \sqrt{a+b x^3} \left (19305 c x+5005 d x^2+2457 e x^3+1485 f x^4+1001 g x^5\right )}{15015 x^2}+\frac{2 \left (a+b x^3\right )^{3/2} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045 x^2}-\frac{1}{8} (27 a) \int \frac{-\frac{8 a e}{55}-\frac{4}{91} (13 b c+2 a f) x-\frac{8}{135} a g x^2}{\sqrt{a+b x^3}} \, dx+\left (a^2 d\right ) \int \frac{1}{x \sqrt{a+b x^3}} \, dx\\ &=-\frac{27 a c \sqrt{a+b x^3}}{7 x}+\frac{2 a \sqrt{a+b x^3} \left (19305 c x+5005 d x^2+2457 e x^3+1485 f x^4+1001 g x^5\right )}{15015 x^2}+\frac{2 \left (a+b x^3\right )^{3/2} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045 x^2}-\frac{1}{8} (27 a) \int \frac{-\frac{8 a e}{55}-\frac{4}{91} (13 b c+2 a f) x}{\sqrt{a+b x^3}} \, dx+\frac{1}{3} \left (a^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )+\frac{1}{5} \left (a^2 g\right ) \int \frac{x^2}{\sqrt{a+b x^3}} \, dx\\ &=\frac{2 a^2 g \sqrt{a+b x^3}}{15 b}-\frac{27 a c \sqrt{a+b x^3}}{7 x}+\frac{2 a \sqrt{a+b x^3} \left (19305 c x+5005 d x^2+2457 e x^3+1485 f x^4+1001 g x^5\right )}{15015 x^2}+\frac{2 \left (a+b x^3\right )^{3/2} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045 x^2}+\frac{\left (2 a^2 d\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 (13 b c+2 a f)) \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^{4/3} \left (182 a^{2/3} e-\frac{55 \left (1-\sqrt{3}\right ) (13 b c+2 a f)}{\sqrt [3]{b}}\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{10010}\\ &=\frac{2 a^2 g \sqrt{a+b x^3}}{15 b}-\frac{27 a c \sqrt{a+b x^3}}{7 x}+\frac{27 a (13 b c+2 a f) \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 (19305 c x+5005 d x^2+2457 e x^3+1485 f x^4+1001 g x^5\right )}{15015 x^2}+\frac{2 \left (a+b x^3\right )^{3/2} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045 x^2}-\frac{2}{3} a^{3/2} d \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 c+2 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 )}{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^{4/3} \left (182 a^{2/3} e-\frac{55 \left (1-\sqrt{3}\right ) (13 b c+2 a f)}{\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 )}{5005 \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 = 0.340123, size = 224, normalized size = 0.32 \[ \frac{2}{9} d \left (\sqrt{a+b x^3} \left (4 a+b x^3\right )-3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )\right )-\frac{a c \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right )}{x \sqrt{\frac{b x^3}{a}+1}}+\frac{a e x \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\sqrt{\frac{b x^3}{a}+1}}+\frac{a f x^2 \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 \sqrt{\frac{b x^3}{a}+1}}+\frac{2 g \left (a+b x^3\right )^{5/2}}{15 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 1317, 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 )}{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{x^{2}}\,{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^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.2973, size = 474, normalized size = 0.68 \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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