Optimal. Leaf size=657 \[ \frac{265 \sqrt{2} c^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} d^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{9 \sqrt{3} c^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{d^{5/3}}-\frac{9 c^{7/6} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{d^{5/3}}+\frac{9 c^{7/6} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^{5/3}}-\frac{265 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{14 d^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{265 c \sqrt{c+d x^3}}{7 d^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{x^2 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac{13 x^2 \sqrt{c+d x^3}}{21 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.805319, antiderivative size = 657, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.482, Rules used = {467, 581, 584, 303, 218, 1877, 486, 444, 63, 206, 2138, 2145, 205} \[ \frac{9 \sqrt{3} c^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{d^{5/3}}-\frac{9 c^{7/6} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{d^{5/3}}+\frac{9 c^{7/6} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^{5/3}}+\frac{265 \sqrt{2} c^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} d^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{265 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{14 d^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{265 c \sqrt{c+d x^3}}{7 d^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{x^2 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac{13 x^2 \sqrt{c+d x^3}}{21 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 467
Rule 581
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{x^4 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx &=\frac{x^2 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac{\int \frac{x \sqrt{c+d x^3} \left (2 c+\frac{13 d x^3}{2}\right )}{8 c-d x^3} \, dx}{3 d}\\ &=\frac{13 x^2 \sqrt{c+d x^3}}{21 d}+\frac{x^2 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac{2 \int \frac{x \left (-111 c^2 d-\frac{795}{4} c d^2 x^3\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{21 d^2}\\ &=\frac{13 x^2 \sqrt{c+d x^3}}{21 d}+\frac{x^2 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac{2 \int \left (\frac{795 c d x}{4 \sqrt{c+d x^3}}-\frac{1701 c^2 d x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}\right ) \, dx}{21 d^2}\\ &=\frac{13 x^2 \sqrt{c+d x^3}}{21 d}+\frac{x^2 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac{(265 c) \int \frac{x}{\sqrt{c+d x^3}} \, dx}{14 d}-\frac{\left (162 c^2\right ) \int \frac{x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{d}\\ &=\frac{13 x^2 \sqrt{c+d x^3}}{21 d}+\frac{x^2 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac{(27 c) \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}{2 d^2}+\frac{(265 c) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{14 d^{4/3}}-\frac{\left (27 c^{4/3}\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}{2 d^{4/3}}+\frac{\left (265 \sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} c^{4/3}\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{7 d^{4/3}}+\frac{\left (81 c^{5/3}\right ) \int \frac{x^2}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{2 d^{2/3}}\\ &=\frac{13 x^2 \sqrt{c+d x^3}}{21 d}+\frac{265 c \sqrt{c+d x^3}}{7 d^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{x^2 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac{265 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{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 )}{14 d^{5/3} \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{265 \sqrt{2} c^{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 )}{7 \sqrt [4]{3} d^{5/3} \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 (27 c^{5/3}\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 )}{d^{5/3}}+\frac{\left (27 c^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{2 d^{2/3}}-\left (54 c^{2/3} \sqrt [3]{d}\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 )\\ &=\frac{13 x^2 \sqrt{c+d x^3}}{21 d}+\frac{265 c \sqrt{c+d x^3}}{7 d^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{x^2 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac{9 \sqrt{3} c^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{d^{5/3}}-\frac{9 c^{7/6} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{d^{5/3}}-\frac{265 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{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 )}{14 d^{5/3} \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{265 \sqrt{2} c^{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 )}{7 \sqrt [4]{3} d^{5/3} \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 (27 c^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{d^{5/3}}\\ &=\frac{13 x^2 \sqrt{c+d x^3}}{21 d}+\frac{265 c \sqrt{c+d x^3}}{7 d^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{x^2 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac{9 \sqrt{3} c^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{d^{5/3}}-\frac{9 c^{7/6} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{d^{5/3}}+\frac{9 c^{7/6} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^{5/3}}-\frac{265 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{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 )}{14 d^{5/3} \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{265 \sqrt{2} c^{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 )}{7 \sqrt [4]{3} d^{5/3} \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 = 0.160137, size = 176, normalized size = 0.27 \[ -\frac{53 d x^5 \left (d x^3-8 c\right ) \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )+74 c x^2 \left (d x^3-8 c\right ) \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )+16 x^2 \left (37 c^2+35 c d x^3-2 d^2 x^6\right )}{112 d \left (d x^3-8 c\right ) \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.01, size = 1747, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} x^{4}}{{\left (d x^{3} - 8 \, c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d x^{7} + c x^{4}\right )} \sqrt{d x^{3} + c}}{d^{2} x^{6} - 16 \, c d x^{3} + 64 \, c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} x^{4}}{{\left (d x^{3} - 8 \, c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]