Optimal. Leaf size=637 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-10 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g+5 b d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{10 \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \sqrt [3]{b} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} \left (5 c x+15 d x^2-15 e x^3-5 f x^4-3 g x^5\right )}{15 x^4}-\frac{(2 a f+b c) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}}+\frac{c \sqrt{a+b x^3}}{3 x^3}+\frac{3 d \sqrt{a+b x^3}}{2 x^2}-\frac{3 e \sqrt{a+b x^3}}{x}+\frac{3 \sqrt [3]{b} e \sqrt{a+b x^3}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x} \]
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Rubi [A] time = 1.59714, antiderivative size = 637, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-10 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g+5 b d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{10 \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \sqrt [3]{b} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} \left (5 c x+15 d x^2-15 e x^3-5 f x^4-3 g x^5\right )}{15 x^4}-\frac{(2 a f+b c) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}}+\frac{c \sqrt{a+b x^3}}{3 x^3}+\frac{3 d \sqrt{a+b x^3}}{2 x^2}-\frac{3 e \sqrt{a+b x^3}}{x}+\frac{3 \sqrt [3]{b} e \sqrt{a+b x^3}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^4,x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x**4+f*x**3+e*x**2+d*x+c)*(b*x**3+a)**(1/2)/x**4,x)
[Out]
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Mathematica [C] time = 2.64752, size = 769, normalized size = 1.21 \[ -\frac{3 b^{2/3} d \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{\sqrt [3]{-1} \sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{2 \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{a+b x^3}}+\sqrt{a+b x^3} \left (\frac{2 f}{3}-\frac{10 c+3 x \left (5 d+10 e x-4 g x^3\right )}{30 x^3}\right )-\frac{b c \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}}-\frac{3 \sqrt{2} \sqrt [3]{a} \sqrt [3]{b} e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}} \left (F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )+\left ((-1)^{2/3}-1\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )}{\sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{a+b x^3}}-\frac{2}{3} \sqrt{a} f \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )-\frac{6 a g \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{\sqrt [3]{-1} \sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{5 \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{a+b x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^4,x]
[Out]
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Maple [B] time = 0.015, size = 1114, normalized size = 1.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2)/x^4,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*sqrt(b*x^3 + a)/x^4,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*sqrt(b*x^3 + a)/x^4,x, algorithm="fricas")
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Sympy [A] time = 9.14721, size = 265, normalized size = 0.42 \[ \frac{\sqrt{a} d \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, - \frac{1}{2} \\ \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac{1}{3}\right )} + \frac{\sqrt{a} e \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac{2}{3}\right )} - \frac{2 \sqrt{a} f \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3} + \frac{\sqrt{a} g x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \frac{2 a f}{3 \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{\sqrt{b} c \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} + \frac{2 \sqrt{b} f x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{b c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x**4+f*x**3+e*x**2+d*x+c)*(b*x**3+a)**(1/2)/x**4,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*sqrt(b*x^3 + a)/x^4,x, algorithm="giac")
[Out]