Optimal. Leaf size=639 \[ \frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right ) \left (91 \sqrt [3]{b} (11 b c-2 a f)-55 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (13 b d-4 a g)\right )}{5005 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (13 b d-4 a g) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{91 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{6 a \sqrt{a+b x^3} (13 b d-4 a g)}{91 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 \sqrt{a+b x^3} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045}+\frac{2 a e \sqrt{a+b x^3}}{9 b}+\frac{6 a f x \sqrt{a+b x^3}}{55 b}+\frac{6 a g x^2 \sqrt{a+b x^3}}{91 b} \]
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Rubi [A] time = 0.719342, antiderivative size = 639, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {1853, 1888, 1886, 261, 1878, 218, 1877} \[ \frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) \left (91 \sqrt [3]{b} (11 b c-2 a f)-55 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (13 b d-4 a g)\right )}{5005 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (13 b d-4 a g) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{91 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{6 a \sqrt{a+b x^3} (13 b d-4 a g)}{91 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 \sqrt{a+b x^3} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045}+\frac{2 a e \sqrt{a+b x^3}}{9 b}+\frac{6 a f x \sqrt{a+b x^3}}{55 b}+\frac{6 a g x^2 \sqrt{a+b x^3}}{91 b} \]
Antiderivative was successfully verified.
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Rule 1853
Rule 1888
Rule 1886
Rule 261
Rule 1878
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \sqrt{a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx &=\frac{2 \sqrt{a+b x^3} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045}+\frac{1}{2} (3 a) \int \frac{\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}}{\sqrt{a+b x^3}} \, dx\\ &=\frac{6 a g x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 \sqrt{a+b x^3} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045}+\frac{(3 a) \int \frac{\frac{7 b c}{5}+\frac{1}{13} (13 b d-4 a g) x+\frac{7}{9} b e x^2+\frac{7}{11} b f x^3}{\sqrt{a+b x^3}} \, dx}{7 b}\\ &=\frac{6 a f x \sqrt{a+b x^3}}{55 b}+\frac{6 a g x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 \sqrt{a+b x^3} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045}+\frac{(6 a) \int \frac{\frac{7}{22} b (11 b c-2 a f)+\frac{5}{26} b (13 b d-4 a g) x+\frac{35}{18} b^2 e x^2}{\sqrt{a+b x^3}} \, dx}{35 b^2}\\ &=\frac{6 a f x \sqrt{a+b x^3}}{55 b}+\frac{6 a g x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 \sqrt{a+b x^3} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045}+\frac{(6 a) \int \frac{\frac{7}{22} b (11 b c-2 a f)+\frac{5}{26} b (13 b d-4 a g) x}{\sqrt{a+b x^3}} \, dx}{35 b^2}+\frac{1}{3} (a e) \int \frac{x^2}{\sqrt{a+b x^3}} \, dx\\ &=\frac{2 a e \sqrt{a+b x^3}}{9 b}+\frac{6 a f x \sqrt{a+b x^3}}{55 b}+\frac{6 a g x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 \sqrt{a+b x^3} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045}+\frac{(3 a (13 b d-4 a g)) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{91 b^{4/3}}+\frac{\left (3 a \left (91 \sqrt [3]{b} (11 b c-2 a f)-55 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (13 b d-4 a g)\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{5005 b^{4/3}}\\ &=\frac{2 a e \sqrt{a+b x^3}}{9 b}+\frac{6 a f x \sqrt{a+b x^3}}{55 b}+\frac{6 a g x^2 \sqrt{a+b x^3}}{91 b}+\frac{6 a (13 b d-4 a g) \sqrt{a+b x^3}}{91 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 \sqrt{a+b x^3} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} (13 b d-4 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 )}{91 b^{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{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (91 \sqrt [3]{b} (11 b c-2 a f)-55 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (13 b d-4 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 )}{5005 b^{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 = 0.148057, size = 135, normalized size = 0.21 \[ \frac{\sqrt{a+b x^3} \left (234 x (11 b c-2 a f) \, _2F_1\left (-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+99 x^2 (13 b d-4 a g) \, _2F_1\left (-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+4 \left (a+b x^3\right ) \sqrt{\frac{b x^3}{a}+1} (143 e+9 x (13 f+11 g x))\right )}{2574 b \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 1557, normalized size = 2.4 \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{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a}\,{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 ({\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.90194, size = 194, normalized size = 0.3 \begin{align*} \frac{\sqrt{a} c x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \frac{\sqrt{a} d x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} + \frac{\sqrt{a} f x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{\sqrt{a} g x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{8}{3}\right )} + e \left (\begin{cases} \frac{\sqrt{a} x^{3}}{3} & \text{for}\: b = 0 \\\frac{2 \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b} & \text{otherwise} \end{cases}\right ) \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{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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