Optimal. Leaf size=692 \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-5 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g+2 b d\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{40 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{16 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{27 b^{4/3} e \sqrt{a+b x^3}}{8 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{1}{60} \left (a+b x^3\right )^{3/2} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right )-\frac{b \sqrt{a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac{b (4 a f+b c) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{b c \sqrt{a+b x^3}}{4 x^3}+\frac{27 b d \sqrt{a+b x^3}}{20 x^2}-\frac{27 b e \sqrt{a+b x^3}}{8 x} \]
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Rubi [A] time = 1.00417, 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 = {14, 1825, 1826, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-5 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g+2 b d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{40 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{16 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{27 b^{4/3} e \sqrt{a+b x^3}}{8 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{1}{60} \left (a+b x^3\right )^{3/2} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right )-\frac{b \sqrt{a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac{b (4 a f+b c) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{b c \sqrt{a+b x^3}}{4 x^3}+\frac{27 b d \sqrt{a+b x^3}}{20 x^2}-\frac{27 b e \sqrt{a+b x^3}}{8 x} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1825
Rule 1826
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^7} \, dx &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac{1}{2} (9 b) \int \frac{\sqrt{a+b x^3} \left (-\frac{c}{6}-\frac{d x}{5}-\frac{e x^2}{4}-\frac{f x^3}{3}-\frac{g x^4}{2}\right )}{x^4} \, dx\\ &=-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac{b \sqrt{a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac{1}{4} (27 a b) \int \frac{\frac{c}{9}+\frac{2 d x}{5}-\frac{e x^2}{2}-\frac{2 f x^3}{9}-\frac{g x^4}{5}}{x^4 \sqrt{a+b x^3}} \, dx\\ &=\frac{b c \sqrt{a+b x^3}}{4 x^3}-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac{b \sqrt{a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac{1}{8} (9 b) \int \frac{-\frac{12 a d}{5}+3 a e x+\frac{1}{3} (b c+4 a f) x^2+\frac{6}{5} a g x^3}{x^3 \sqrt{a+b x^3}} \, dx\\ &=\frac{b c \sqrt{a+b x^3}}{4 x^3}+\frac{27 b d \sqrt{a+b x^3}}{20 x^2}-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac{b \sqrt{a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac{(9 b) \int \frac{-12 a^2 e-\frac{4}{3} a (b c+4 a f) x-\frac{12}{5} a (b d+2 a g) x^2}{x^2 \sqrt{a+b x^3}} \, dx}{32 a}\\ &=\frac{b c \sqrt{a+b x^3}}{4 x^3}+\frac{27 b d \sqrt{a+b x^3}}{20 x^2}-\frac{27 b e \sqrt{a+b x^3}}{8 x}-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac{b \sqrt{a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac{(9 b) \int \frac{\frac{8}{3} a^2 (b c+4 a f)+\frac{24}{5} a^2 (b d+2 a g) x+12 a^2 b e x^2}{x \sqrt{a+b x^3}} \, dx}{64 a^2}\\ &=\frac{b c \sqrt{a+b x^3}}{4 x^3}+\frac{27 b d \sqrt{a+b x^3}}{20 x^2}-\frac{27 b e \sqrt{a+b x^3}}{8 x}-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac{b \sqrt{a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac{(9 b) \int \frac{\frac{24}{5} a^2 (b d+2 a g)+12 a^2 b e x}{\sqrt{a+b x^3}} \, dx}{64 a^2}+\frac{1}{8} (3 b (b c+4 a f)) \int \frac{1}{x \sqrt{a+b x^3}} \, dx\\ &=\frac{b c \sqrt{a+b x^3}}{4 x^3}+\frac{27 b d \sqrt{a+b x^3}}{20 x^2}-\frac{27 b e \sqrt{a+b x^3}}{8 x}-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac{b \sqrt{a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac{1}{16} \left (27 b^{5/3} e\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx+\frac{1}{8} (b (b c+4 a f)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )+\frac{1}{80} \left (27 b \left (2 b d-5 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx\\ &=\frac{b c \sqrt{a+b x^3}}{4 x^3}+\frac{27 b d \sqrt{a+b x^3}}{20 x^2}-\frac{27 b e \sqrt{a+b x^3}}{8 x}+\frac{27 b^{4/3} e \sqrt{a+b x^3}}{8 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac{b \sqrt{a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} b^{4/3} 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 )}{16 \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^{2/3} \left (2 b d-5 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+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 )}{40 \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{1}{4} (b c+4 a f) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )\\ &=\frac{b c \sqrt{a+b x^3}}{4 x^3}+\frac{27 b d \sqrt{a+b x^3}}{20 x^2}-\frac{27 b e \sqrt{a+b x^3}}{8 x}+\frac{27 b^{4/3} e \sqrt{a+b x^3}}{8 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{1}{60} \left (\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac{b \sqrt{a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac{b (b c+4 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} b^{4/3} 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 )}{16 \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^{2/3} \left (2 b d-5 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+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 )}{40 \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 = 0.507349, size = 240, normalized size = 0.35 \[ \frac{-\frac{12 a^2 d \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right )}{x^5}-\frac{15 a^2 e \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{3}{2},-\frac{4}{3};-\frac{1}{3};-\frac{b x^3}{a}\right )}{x^4}+\frac{8 b f \left (a+b x^3\right )^3 \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x^3}{a}+1\right )}{a^2}-\frac{30 a^2 g \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{3}{2},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right )}{x^2}-15 b^2 c \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )-\frac{10 c \left (a+b x^3\right )^2}{x^6}-\frac{15 b c \left (a+b x^3\right )}{x^3}}{60 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 1196, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^{7}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.5393, size = 524, normalized size = 0.76 \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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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