Optimal. Leaf size=796 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{b x^3+a}}{\sqrt{a}}\right ) b^3}{24 a^{3/2}}+\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (b d-4 a g) \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt{\frac{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{7/3}}{896 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{b x^3+a}}+\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (b d-4 a g)\right ) \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt{\frac{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt{3}\right ) b^{7/3}}{49280 a^2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{b x^3+a}}-\frac{27 (b d-4 a g) \sqrt{b x^3+a} b^{7/3}}{448 a^2 \left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )}+\frac{27 (b d-4 a g) \sqrt{b x^3+a} b^2}{448 a^2 x}+\frac{27 (7 b c-22 a f) \sqrt{b x^3+a} b^2}{7040 a^2 x^2}-\frac{e \sqrt{b x^3+a} b^2}{24 a x^3}-\frac{27 d \sqrt{b x^3+a} b^2}{1120 a x^4}-\frac{27 c \sqrt{b x^3+a} b^2}{1760 a x^5}-\frac{\left (\frac{945 c}{x^8}+\frac{2970 g}{x^4}+\frac{2079 f}{x^5}+\frac{1540 e}{x^6}+\frac{1188 d}{x^7}\right ) \sqrt{b x^3+a} b}{18480}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{3960 g}{x^7}+\frac{3465 f}{x^8}+\frac{3080 e}{x^9}+\frac{2772 d}{x^{10}}\right ) \left (b x^3+a\right )^{3/2}}{27720} \]
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Rubi [A] time = 1.53375, antiderivative size = 796, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {14, 1825, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{b x^3+a}}{\sqrt{a}}\right ) b^3}{24 a^{3/2}}+\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (b d-4 a g) \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt{\frac{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{7/3}}{896 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{b x^3+a}}+\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (b d-4 a g)\right ) \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt{\frac{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{7/3}}{49280 a^2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{b x^3+a}}-\frac{27 (b d-4 a g) \sqrt{b x^3+a} b^{7/3}}{448 a^2 \left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )}+\frac{27 (b d-4 a g) \sqrt{b x^3+a} b^2}{448 a^2 x}+\frac{27 (7 b c-22 a f) \sqrt{b x^3+a} b^2}{7040 a^2 x^2}-\frac{e \sqrt{b x^3+a} b^2}{24 a x^3}-\frac{27 d \sqrt{b x^3+a} b^2}{1120 a x^4}-\frac{27 c \sqrt{b x^3+a} b^2}{1760 a x^5}-\frac{\left (\frac{945 c}{x^8}+\frac{2970 g}{x^4}+\frac{2079 f}{x^5}+\frac{1540 e}{x^6}+\frac{1188 d}{x^7}\right ) \sqrt{b x^3+a} b}{18480}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{3960 g}{x^7}+\frac{3465 f}{x^8}+\frac{3080 e}{x^9}+\frac{2772 d}{x^{10}}\right ) \left (b x^3+a\right )^{3/2}}{27720} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1825
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^{12}} \, dx &=-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac{1}{2} (9 b) \int \frac{\sqrt{a+b x^3} \left (-\frac{c}{11}-\frac{d x}{10}-\frac{e x^2}{9}-\frac{f x^3}{8}-\frac{g x^4}{7}\right )}{x^9} \, dx\\ &=-\frac{b \left (\frac{945 c}{x^8}+\frac{1188 d}{x^7}+\frac{1540 e}{x^6}+\frac{2079 f}{x^5}+\frac{2970 g}{x^4}\right ) \sqrt{a+b x^3}}{18480}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac{1}{4} \left (27 b^2\right ) \int \frac{\frac{c}{88}+\frac{d x}{70}+\frac{e x^2}{54}+\frac{f x^3}{40}+\frac{g x^4}{28}}{x^6 \sqrt{a+b x^3}} \, dx\\ &=-\frac{b \left (\frac{945 c}{x^8}+\frac{1188 d}{x^7}+\frac{1540 e}{x^6}+\frac{2079 f}{x^5}+\frac{2970 g}{x^4}\right ) \sqrt{a+b x^3}}{18480}-\frac{27 b^2 c \sqrt{a+b x^3}}{1760 a x^5}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac{\left (27 b^2\right ) \int \frac{-\frac{a d}{7}-\frac{5 a e x}{27}+\frac{1}{88} (7 b c-22 a f) x^2-\frac{5}{14} a g x^3}{x^5 \sqrt{a+b x^3}} \, dx}{40 a}\\ &=-\frac{b \left (\frac{945 c}{x^8}+\frac{1188 d}{x^7}+\frac{1540 e}{x^6}+\frac{2079 f}{x^5}+\frac{2970 g}{x^4}\right ) \sqrt{a+b x^3}}{18480}-\frac{27 b^2 c \sqrt{a+b x^3}}{1760 a x^5}-\frac{27 b^2 d \sqrt{a+b x^3}}{1120 a x^4}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac{\left (27 b^2\right ) \int \frac{\frac{40 a^2 e}{27}-\frac{1}{11} a (7 b c-22 a f) x-\frac{5}{7} a (b d-4 a g) x^2}{x^4 \sqrt{a+b x^3}} \, dx}{320 a^2}\\ &=-\frac{b \left (\frac{945 c}{x^8}+\frac{1188 d}{x^7}+\frac{1540 e}{x^6}+\frac{2079 f}{x^5}+\frac{2970 g}{x^4}\right ) \sqrt{a+b x^3}}{18480}-\frac{27 b^2 c \sqrt{a+b x^3}}{1760 a x^5}-\frac{27 b^2 d \sqrt{a+b x^3}}{1120 a x^4}-\frac{b^2 e \sqrt{a+b x^3}}{24 a x^3}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac{\left (9 b^2\right ) \int \frac{\frac{6}{11} a^2 (7 b c-22 a f)+\frac{30}{7} a^2 (b d-4 a g) x+\frac{40}{9} a^2 b e x^2}{x^3 \sqrt{a+b x^3}} \, dx}{640 a^3}\\ &=-\frac{b \left (\frac{945 c}{x^8}+\frac{1188 d}{x^7}+\frac{1540 e}{x^6}+\frac{2079 f}{x^5}+\frac{2970 g}{x^4}\right ) \sqrt{a+b x^3}}{18480}-\frac{27 b^2 c \sqrt{a+b x^3}}{1760 a x^5}-\frac{27 b^2 d \sqrt{a+b x^3}}{1120 a x^4}-\frac{b^2 e \sqrt{a+b x^3}}{24 a x^3}+\frac{27 b^2 (7 b c-22 a f) \sqrt{a+b x^3}}{7040 a^2 x^2}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac{\left (9 b^2\right ) \int \frac{-\frac{120}{7} a^3 (b d-4 a g)-\frac{160}{9} a^3 b e x+\frac{6}{11} a^2 b (7 b c-22 a f) x^2}{x^2 \sqrt{a+b x^3}} \, dx}{2560 a^4}\\ &=-\frac{b \left (\frac{945 c}{x^8}+\frac{1188 d}{x^7}+\frac{1540 e}{x^6}+\frac{2079 f}{x^5}+\frac{2970 g}{x^4}\right ) \sqrt{a+b x^3}}{18480}-\frac{27 b^2 c \sqrt{a+b x^3}}{1760 a x^5}-\frac{27 b^2 d \sqrt{a+b x^3}}{1120 a x^4}-\frac{b^2 e \sqrt{a+b x^3}}{24 a x^3}+\frac{27 b^2 (7 b c-22 a f) \sqrt{a+b x^3}}{7040 a^2 x^2}+\frac{27 b^2 (b d-4 a g) \sqrt{a+b x^3}}{448 a^2 x}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac{\left (9 b^2\right ) \int \frac{\frac{320}{9} a^4 b e-\frac{12}{11} a^3 b (7 b c-22 a f) x+\frac{120}{7} a^3 b (b d-4 a g) x^2}{x \sqrt{a+b x^3}} \, dx}{5120 a^5}\\ &=-\frac{b \left (\frac{945 c}{x^8}+\frac{1188 d}{x^7}+\frac{1540 e}{x^6}+\frac{2079 f}{x^5}+\frac{2970 g}{x^4}\right ) \sqrt{a+b x^3}}{18480}-\frac{27 b^2 c \sqrt{a+b x^3}}{1760 a x^5}-\frac{27 b^2 d \sqrt{a+b x^3}}{1120 a x^4}-\frac{b^2 e \sqrt{a+b x^3}}{24 a x^3}+\frac{27 b^2 (7 b c-22 a f) \sqrt{a+b x^3}}{7040 a^2 x^2}+\frac{27 b^2 (b d-4 a g) \sqrt{a+b x^3}}{448 a^2 x}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac{\left (9 b^2\right ) \int \frac{-\frac{12}{11} a^3 b (7 b c-22 a f)+\frac{120}{7} a^3 b (b d-4 a g) x}{\sqrt{a+b x^3}} \, dx}{5120 a^5}-\frac{\left (b^3 e\right ) \int \frac{1}{x \sqrt{a+b x^3}} \, dx}{16 a}\\ &=-\frac{b \left (\frac{945 c}{x^8}+\frac{1188 d}{x^7}+\frac{1540 e}{x^6}+\frac{2079 f}{x^5}+\frac{2970 g}{x^4}\right ) \sqrt{a+b x^3}}{18480}-\frac{27 b^2 c \sqrt{a+b x^3}}{1760 a x^5}-\frac{27 b^2 d \sqrt{a+b x^3}}{1120 a x^4}-\frac{b^2 e \sqrt{a+b x^3}}{24 a x^3}+\frac{27 b^2 (7 b c-22 a f) \sqrt{a+b x^3}}{7040 a^2 x^2}+\frac{27 b^2 (b d-4 a g) \sqrt{a+b x^3}}{448 a^2 x}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac{\left (b^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{48 a}-\frac{\left (27 b^{8/3} (b d-4 a g)\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{896 a^2}+\frac{\left (27 b^{8/3} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (b d-4 a g)\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{98560 a^2}\\ &=-\frac{b \left (\frac{945 c}{x^8}+\frac{1188 d}{x^7}+\frac{1540 e}{x^6}+\frac{2079 f}{x^5}+\frac{2970 g}{x^4}\right ) \sqrt{a+b x^3}}{18480}-\frac{27 b^2 c \sqrt{a+b x^3}}{1760 a x^5}-\frac{27 b^2 d \sqrt{a+b x^3}}{1120 a x^4}-\frac{b^2 e \sqrt{a+b x^3}}{24 a x^3}+\frac{27 b^2 (7 b c-22 a f) \sqrt{a+b x^3}}{7040 a^2 x^2}+\frac{27 b^2 (b d-4 a g) \sqrt{a+b x^3}}{448 a^2 x}-\frac{27 b^{7/3} (b d-4 a g) \sqrt{a+b x^3}}{448 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{7/3} (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 )}{896 a^{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{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{7/3} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (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 )}{49280 a^2 \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 (b^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{24 a}\\ &=-\frac{b \left (\frac{945 c}{x^8}+\frac{1188 d}{x^7}+\frac{1540 e}{x^6}+\frac{2079 f}{x^5}+\frac{2970 g}{x^4}\right ) \sqrt{a+b x^3}}{18480}-\frac{27 b^2 c \sqrt{a+b x^3}}{1760 a x^5}-\frac{27 b^2 d \sqrt{a+b x^3}}{1120 a x^4}-\frac{b^2 e \sqrt{a+b x^3}}{24 a x^3}+\frac{27 b^2 (7 b c-22 a f) \sqrt{a+b x^3}}{7040 a^2 x^2}+\frac{27 b^2 (b d-4 a g) \sqrt{a+b x^3}}{448 a^2 x}-\frac{27 b^{7/3} (b d-4 a g) \sqrt{a+b x^3}}{448 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{2772 d}{x^{10}}+\frac{3080 e}{x^9}+\frac{3465 f}{x^8}+\frac{3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac{b^3 e \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{3/2}}+\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{7/3} (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 )}{896 a^{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{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{7/3} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (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 )}{49280 a^2 \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 = 0.400484, size = 194, normalized size = 0.24 \[ \frac{\sqrt{a+b x^3} \left (11 x^3 \left (-105 a^5 f \, _2F_1\left (-\frac{8}{3},-\frac{3}{2};-\frac{5}{3};-\frac{b x^3}{a}\right )-120 a^5 g x \, _2F_1\left (-\frac{7}{3},-\frac{3}{2};-\frac{4}{3};-\frac{b x^3}{a}\right )+112 b^3 e x^8 \left (a+b x^3\right )^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{b x^3}{a}+1\right )\right )-840 a^5 c \, _2F_1\left (-\frac{11}{3},-\frac{3}{2};-\frac{8}{3};-\frac{b x^3}{a}\right )-924 a^5 d x \, _2F_1\left (-\frac{10}{3},-\frac{3}{2};-\frac{7}{3};-\frac{b x^3}{a}\right )\right )}{9240 a^4 x^{11} \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 1773, normalized size = 2.2 \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^{12}}\,{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^{12}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.8427, size = 541, 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^{12}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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