Optimal. Leaf size=651 \[ -\frac{9 \sqrt{3} d^{4/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{128 c^{5/6}}+\frac{9 d^{4/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{128 c^{5/6}}-\frac{9 d^{4/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{128 c^{5/6}}+\frac{3^{3/4} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{8 \sqrt{2} c^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{32 c^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{3 d^{4/3} \sqrt{c+d x^3}}{16 c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{3 d \sqrt{c+d x^3}}{16 c x}-\frac{\sqrt{c+d x^3}}{32 x^4} \]
[Out]
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Rubi [A] time = 1.9875, antiderivative size = 651, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 14, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.518 \[ -\frac{9 \sqrt{3} d^{4/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{128 c^{5/6}}+\frac{9 d^{4/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{128 c^{5/6}}-\frac{9 d^{4/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{128 c^{5/6}}+\frac{3^{3/4} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{8 \sqrt{2} c^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{32 c^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{3 d^{4/3} \sqrt{c+d x^3}}{16 c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{3 d \sqrt{c+d x^3}}{16 c x}-\frac{\sqrt{c+d x^3}}{32 x^4} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3)^(3/2)/(x^5*(8*c - d*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 26.2599, size = 54, normalized size = 0.08 \[ - \frac{\sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{4}{3},- \frac{3}{2},1,- \frac{1}{3},- \frac{d x^{3}}{c},\frac{d x^{3}}{8 c} \right )}}{32 x^{4} \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c)**(3/2)/x**5/(-d*x**3+8*c),x)
[Out]
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Mathematica [C] time = 0.403511, size = 363, normalized size = 0.56 \[ \frac{\frac{96 d^3 x^5 F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (d x^3-8 c\right ) \left (3 d x^3 \left (F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+64 c F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{3225 c d^2 x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (3 d x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+40 c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}-\frac{5 \left (c^2+7 c d x^3+6 d^2 x^6\right )}{2 c x^4}}{80 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(c + d*x^3)^(3/2)/(x^5*(8*c - d*x^3)),x]
[Out]
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Maple [C] time = 0.038, size = 1810, normalized size = 2.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^(3/2)/x^5/(-d*x^3+8*c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (d x^{3} - 8 \, c\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^5),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{d x^{8} - 8 \, c x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^5),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**3+c)**(3/2)/x**5/(-d*x**3+8*c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (d x^{3} - 8 \, c\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^5),x, algorithm="giac")
[Out]