Optimal. Leaf size=742 \[ \frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (43010 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-1729 (23 b d-8 a g)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{37182145 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 a^2 \sqrt{a+b x^3} (7 b c-2 a f)}{105 b^2}+\frac{54 a^2 x \sqrt{a+b x^3} (23 b d-8 a g)}{21505 b^2}-\frac{216 a^3 e \sqrt{a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{108 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{54 a^2 e x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 f x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435} \]
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Rubi [A] time = 1.55222, antiderivative size = 742, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {1826, 1836, 1888, 1594, 1886, 261, 1878, 218, 1877} \[ \frac{2 a^2 \sqrt{a+b x^3} (7 b c-2 a f)}{105 b^2}+\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (43010 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-1729 (23 b d-8 a g)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{37182145 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{54 a^2 x \sqrt{a+b x^3} (23 b d-8 a g)}{21505 b^2}-\frac{216 a^3 e \sqrt{a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{108 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{54 a^2 e x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 f x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435} \]
Antiderivative was successfully verified.
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Rule 1826
Rule 1836
Rule 1888
Rule 1594
Rule 1886
Rule 261
Rule 1878
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int x^2 \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^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{1}{2} (9 a) \int x^2 \sqrt{a+b x^3} \left (\frac{2 c}{15}+\frac{2 d x}{17}+\frac{2 e x^2}{19}+\frac{2 f x^3}{21}+\frac{2 g x^4}{23}\right ) \, dx\\ &=\frac{2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac{1}{4} \left (27 a^2\right ) \int \frac{x^2 \left (\frac{4 c}{135}+\frac{4 d x}{187}+\frac{4 e x^2}{247}+\frac{4 f x^3}{315}+\frac{4 g x^4}{391}\right )}{\sqrt{a+b x^3}} \, dx\\ &=\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac{\left (27 a^2\right ) \int \frac{x^2 \left (\frac{22 b c}{135}+\frac{2}{391} (23 b d-8 a g) x+\frac{22}{247} b e x^2+\frac{22}{315} b f x^3\right )}{\sqrt{a+b x^3}} \, dx}{22 b}\\ &=\frac{2 a^2 f x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac{\left (3 a^2\right ) \int \frac{x^2 \left (\frac{11}{105} b (7 b c-2 a f)+\frac{9}{391} b (23 b d-8 a g) x+\frac{99}{247} b^2 e x^2\right )}{\sqrt{a+b x^3}} \, dx}{11 b^2}\\ &=\frac{54 a^2 e x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 f x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac{\left (6 a^2\right ) \int \frac{-\frac{198}{247} a b^2 e x+\frac{11}{30} b^2 (7 b c-2 a f) x^2+\frac{63}{782} b^2 (23 b d-8 a g) x^3}{\sqrt{a+b x^3}} \, dx}{77 b^3}\\ &=\frac{54 a^2 e x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 f x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac{\left (6 a^2\right ) \int \frac{x \left (-\frac{198}{247} a b^2 e+\frac{11}{30} b^2 (7 b c-2 a f) x+\frac{63}{782} b^2 (23 b d-8 a g) x^2\right )}{\sqrt{a+b x^3}} \, dx}{77 b^3}\\ &=\frac{54 a^2 (23 b d-8 a g) x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 e x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 f x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac{\left (12 a^2\right ) \int \frac{-\frac{63}{782} a b^2 (23 b d-8 a g)-\frac{495}{247} a b^3 e x+\frac{11}{12} b^3 (7 b c-2 a f) x^2}{\sqrt{a+b x^3}} \, dx}{385 b^4}\\ &=\frac{54 a^2 (23 b d-8 a g) x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 e x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 f x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac{\left (12 a^2\right ) \int \frac{-\frac{63}{782} a b^2 (23 b d-8 a g)-\frac{495}{247} a b^3 e x}{\sqrt{a+b x^3}} \, dx}{385 b^4}+\frac{\left (a^2 (7 b c-2 a f)\right ) \int \frac{x^2}{\sqrt{a+b x^3}} \, dx}{35 b}\\ &=\frac{2 a^2 (7 b c-2 a f) \sqrt{a+b x^3}}{105 b^2}+\frac{54 a^2 (23 b d-8 a g) x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 e x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 f x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}-\frac{\left (108 a^3 e\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 (54 a^3 \left (39767 b d-43010 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-13832 a g\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{37182145 b^2}\\ &=\frac{2 a^2 (7 b c-2 a f) \sqrt{a+b x^3}}{105 b^2}+\frac{54 a^2 (23 b d-8 a g) x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 e x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 f x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}-\frac{216 a^3 e \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^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac{108 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} 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 )}{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{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left (39767 b d-43010 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-13832 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 )}{37182145 b^{7/3} \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 = 0.349519, size = 162, normalized size = 0.22 \[ \frac{2 \left (1995 a^3 x \sqrt{\frac{b x^3}{a}+1} (8 a g-23 b d) \, _2F_1\left (-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )-41055 a^3 b e x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+\left (a+b x^3\right )^3 (-38 a (391 f+420 g x)+52003 b c+5 b x (9177 d+17 x (483 e+19 x (23 f+21 g x))))\right )}{780045 b^2 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 1269, normalized size = 1.7 \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*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} c}{15 \, b} + \int{\left (b g x^{9} + b f x^{8} + b e x^{7} + a f x^{5} +{\left (b d + a g\right )} x^{6} + a e x^{4} + a d x^{3}\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 (b g x^{9} + b f x^{8} + b e x^{7} +{\left (b d + a g\right )} x^{6} + a e x^{4} +{\left (b c + a f\right )} x^{5} + a d x^{3} + a c x^{2}\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 = 9.44158, size = 525, normalized size = 0.71 \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^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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