Optimal. Leaf size=687 \[ -\frac{5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt{3}\right )}{432 \sqrt{2} \sqrt [4]{3} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{5 d^{4/3} \sqrt{c+d x^3}}{864 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{25 d^{4/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{27648 \sqrt{3} c^{23/6}}+\frac{25 d^{4/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{82944 c^{23/6}}-\frac{25 d^{4/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{82944 c^{23/6}}+\frac{5 \sqrt{2-\sqrt{3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{5 d \sqrt{c+d x^3}}{864 c^4 x}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4} \]
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Rubi [A] time = 0.927277, antiderivative size = 687, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.482, Rules used = {472, 583, 584, 303, 218, 1877, 486, 444, 63, 206, 2138, 2145, 205} \[ -\frac{5 d^{4/3} \sqrt{c+d x^3}}{864 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{25 d^{4/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{27648 \sqrt{3} c^{23/6}}+\frac{25 d^{4/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{82944 c^{23/6}}-\frac{25 d^{4/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{82944 c^{23/6}}-\frac{5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{432 \sqrt{2} \sqrt [4]{3} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{5 \sqrt{2-\sqrt{3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{5 d \sqrt{c+d x^3}}{864 c^4 x}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4} \]
Antiderivative was successfully verified.
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Rule 472
Rule 583
Rule 584
Rule 303
Rule 218
Rule 1877
Rule 486
Rule 444
Rule 63
Rule 206
Rule 2138
Rule 2145
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (8 c-d x^3\right )^2 \sqrt{c+d x^3}} \, dx &=\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}+\frac{\int \frac{31 c d+\frac{11 d^2 x^3}{2}}{x^5 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{216 c^2 d}\\ &=-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac{\int \frac{320 c^2 d^2-\frac{155}{2} c d^3 x^3}{x^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{6912 c^4 d}\\ &=-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4}+\frac{5 d \sqrt{c+d x^3}}{864 c^4 x}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}+\frac{\int \frac{x \left (-980 c^3 d^3+160 c^2 d^4 x^3\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{55296 c^6 d}\\ &=-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4}+\frac{5 d \sqrt{c+d x^3}}{864 c^4 x}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}+\frac{\int \left (-\frac{160 c^2 d^3 x}{\sqrt{c+d x^3}}+\frac{300 c^3 d^3 x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}\right ) \, dx}{55296 c^6 d}\\ &=-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4}+\frac{5 d \sqrt{c+d x^3}}{864 c^4 x}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac{\left (5 d^2\right ) \int \frac{x}{\sqrt{c+d x^3}} \, dx}{1728 c^4}+\frac{\left (25 d^2\right ) \int \frac{x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{4608 c^3}\\ &=-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4}+\frac{5 d \sqrt{c+d x^3}}{864 c^4 x}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac{(25 d) \int \frac{2 \sqrt [3]{c} d^{2/3}-2 d x-\frac{d^{4/3} x^2}{\sqrt [3]{c}}}{\left (4+\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}\right ) \sqrt{c+d x^3}} \, dx}{55296 c^4}-\frac{\left (5 d^{5/3}\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{1728 c^4}+\frac{\left (25 d^{5/3}\right ) \int \frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (2-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt{c+d x^3}} \, dx}{55296 c^{11/3}}-\frac{\left (5 \sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} d^{5/3}\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{864 c^{11/3}}-\frac{\left (25 d^{7/3}\right ) \int \frac{x^2}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{18432 c^{10/3}}\\ &=-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4}+\frac{5 d \sqrt{c+d x^3}}{864 c^4 x}-\frac{5 d^{4/3} \sqrt{c+d x^3}}{864 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}+\frac{5 \sqrt{2-\sqrt{3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{432 \sqrt{2} \sqrt [4]{3} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{\left (25 d^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{9-c x^2} \, dx,x,\frac{\left (1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\sqrt{c+d x^3}}\right )}{27648 c^{10/3}}-\frac{\left (25 d^{7/3}\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{55296 c^{10/3}}+\frac{\left (25 d^{10/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{2 d^2}{c}-6 d^2 x^2} \, dx,x,\frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{c+d x^3}}\right )}{13824 c^{13/3}}\\ &=-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4}+\frac{5 d \sqrt{c+d x^3}}{864 c^4 x}-\frac{5 d^{4/3} \sqrt{c+d x^3}}{864 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac{25 d^{4/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{27648 \sqrt{3} c^{23/6}}+\frac{25 d^{4/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{82944 c^{23/6}}+\frac{5 \sqrt{2-\sqrt{3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{432 \sqrt{2} \sqrt [4]{3} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\left (25 d^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{27648 c^{10/3}}\\ &=-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4}+\frac{5 d \sqrt{c+d x^3}}{864 c^4 x}-\frac{5 d^{4/3} \sqrt{c+d x^3}}{864 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac{25 d^{4/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{27648 \sqrt{3} c^{23/6}}+\frac{25 d^{4/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{82944 c^{23/6}}-\frac{25 d^{4/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{82944 c^{23/6}}+\frac{5 \sqrt{2-\sqrt{3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{432 \sqrt{2} \sqrt [4]{3} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}\\ \end{align*}
Mathematica [C] time = 0.137667, size = 196, normalized size = 0.29 \[ \frac{245 c d^2 x^6 \left (d x^3-8 c\right ) \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-16 \left (d^3 x^9 \left (d x^3-8 c\right ) \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )+2 c \left (-135 c^2 d x^3+216 c^3-311 c d^2 x^6+40 d^3 x^9\right )\right )}{221184 c^5 x^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.013, size = 2240, normalized size = 3.3 \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{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )}^{2} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )}^{2} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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