Optimal. Leaf size=723 \[ -\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt{3}\right ) (19 b c-4 a f)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{1616615 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{54 a^2 \sqrt{a+b x^3} (19 b c-4 a f)}{1729 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b c-4 a f) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 a^2 \sqrt{a+b x^3} (7 b d-2 a g)}{105 b^2}+\frac{54 a^2 e x \sqrt{a+b x^3}}{935 b}+\frac{54 a^2 f x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 g x^3 \sqrt{a+b x^3}}{105 b}+\frac{2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac{2 a x \sqrt{a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845} \]
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Rubi [A] time = 1.23865, antiderivative size = 723, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {1826, 1836, 1888, 1886, 261, 1878, 218, 1877} \[ -\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt{3}\right ) (19 b c-4 a f)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1616615 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{54 a^2 \sqrt{a+b x^3} (19 b c-4 a f)}{1729 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b c-4 a f) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 a^2 \sqrt{a+b x^3} (7 b d-2 a g)}{105 b^2}+\frac{54 a^2 e x \sqrt{a+b x^3}}{935 b}+\frac{54 a^2 f x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 g x^3 \sqrt{a+b x^3}}{105 b}+\frac{2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac{2 a x \sqrt{a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845} \]
Antiderivative was successfully verified.
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Rule 1826
Rule 1836
Rule 1888
Rule 1886
Rule 261
Rule 1878
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int x \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx &=\frac{2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac{1}{2} (9 a) \int x \sqrt{a+b x^3} \left (\frac{2 c}{13}+\frac{2 d x}{15}+\frac{2 e x^2}{17}+\frac{2 f x^3}{19}+\frac{2 g x^4}{21}\right ) \, dx\\ &=\frac{2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac{2 a x \sqrt{a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac{1}{4} \left (27 a^2\right ) \int \frac{x \left (\frac{4 c}{91}+\frac{4 d x}{135}+\frac{4 e x^2}{187}+\frac{4 f x^3}{247}+\frac{4 g x^4}{315}\right )}{\sqrt{a+b x^3}} \, dx\\ &=\frac{2 a^2 g x^3 \sqrt{a+b x^3}}{105 b}+\frac{2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac{2 a x \sqrt{a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac{\left (3 a^2\right ) \int \frac{x \left (\frac{18 b c}{91}+\frac{2}{105} (7 b d-2 a g) x+\frac{18}{187} b e x^2+\frac{18}{247} b f x^3\right )}{\sqrt{a+b x^3}} \, dx}{2 b}\\ &=\frac{54 a^2 f x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 g x^3 \sqrt{a+b x^3}}{105 b}+\frac{2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac{2 a x \sqrt{a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac{\left (3 a^2\right ) \int \frac{x \left (\frac{9}{247} b (19 b c-4 a f)+\frac{1}{15} b (7 b d-2 a g) x+\frac{63}{187} b^2 e x^2\right )}{\sqrt{a+b x^3}} \, dx}{7 b^2}\\ &=\frac{54 a^2 e x \sqrt{a+b x^3}}{935 b}+\frac{54 a^2 f x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 g x^3 \sqrt{a+b x^3}}{105 b}+\frac{2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac{2 a x \sqrt{a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac{\left (6 a^2\right ) \int \frac{-\frac{63}{187} a b^2 e+\frac{45}{494} b^2 (19 b c-4 a f) x+\frac{1}{6} b^2 (7 b d-2 a g) x^2}{\sqrt{a+b x^3}} \, dx}{35 b^3}\\ &=\frac{54 a^2 e x \sqrt{a+b x^3}}{935 b}+\frac{54 a^2 f x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 g x^3 \sqrt{a+b x^3}}{105 b}+\frac{2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac{2 a x \sqrt{a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac{\left (6 a^2\right ) \int \frac{-\frac{63}{187} a b^2 e+\frac{45}{494} b^2 (19 b c-4 a f) x}{\sqrt{a+b x^3}} \, dx}{35 b^3}+\frac{\left (a^2 (7 b d-2 a g)\right ) \int \frac{x^2}{\sqrt{a+b x^3}} \, dx}{35 b}\\ &=\frac{2 a^2 (7 b d-2 a g) \sqrt{a+b x^3}}{105 b^2}+\frac{54 a^2 e x \sqrt{a+b x^3}}{935 b}+\frac{54 a^2 f x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 g x^3 \sqrt{a+b x^3}}{105 b}+\frac{2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac{2 a x \sqrt{a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac{\left (27 a^2 (19 b c-4 a f)\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{1729 b^{4/3}}-\frac{\left (27 a^{7/3} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt{3}\right ) (19 b c-4 a f)\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{1616615 b^{4/3}}\\ &=\frac{2 a^2 (7 b d-2 a g) \sqrt{a+b x^3}}{105 b^2}+\frac{54 a^2 e x \sqrt{a+b x^3}}{935 b}+\frac{54 a^2 f x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 g x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 (19 b c-4 a f) \sqrt{a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac{2 a x \sqrt{a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{7/3} (19 b c-4 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 )}{1729 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{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^{7/3} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt{3}\right ) (19 b c-4 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 )}{1616615 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 = 0.272086, size = 148, normalized size = 0.2 \[ -\frac{\sqrt{a+b x^3} \left (7980 a^2 b e x \, _2F_1\left (-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+1785 a b x^2 (4 a f-19 b c) \, _2F_1\left (-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+4 \left (a+b x^3\right )^2 \sqrt{\frac{b x^3}{a}+1} (646 a g-2261 b d-5 b x (399 e+17 x (21 f+19 g x)))\right )}{67830 b^2 \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 1383, 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{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}} x\,{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 (b g x^{8} + b f x^{7} + b e x^{6} +{\left (b d + a g\right )} x^{5} + a e x^{3} +{\left (b c + a f\right )} x^{4} + a d x^{2} + a c x\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 = 8.80445, size = 525, normalized size = 0.73 \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{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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