Optimal. Leaf size=764 \[ -\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (7 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt{3}\right ) (b c-4 a f)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2240 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (b c-4 a f) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{896 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{b^2 (b d-6 a g) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{3/2}}-\frac{27 b^{7/3} \sqrt{a+b x^3} (b c-4 a f)}{448 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{27 b^2 \sqrt{a+b x^3} (b c-4 a f)}{448 a^2 x}-\frac{27 b^2 c \sqrt{a+b x^3}}{1120 a x^4}-\frac{b^2 d \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 e \sqrt{a+b x^3}}{320 a x^2}-\frac{b \sqrt{a+b x^3} \left (\frac{108 c}{x^7}+\frac{140 d}{x^6}+\frac{189 e}{x^5}+\frac{270 f}{x^4}+\frac{420 g}{x^3}\right )}{1680}-\frac{\left (a+b x^3\right )^{3/2} \left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}+\frac{420 g}{x^6}\right )}{2520} \]
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Rubi [A] time = 2.48437, antiderivative size = 764, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (7 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt{3}\right ) (b c-4 a f)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2240 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (b c-4 a f) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{896 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{b^2 (b d-6 a g) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{3/2}}-\frac{27 b^{7/3} \sqrt{a+b x^3} (b c-4 a f)}{448 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{27 b^2 \sqrt{a+b x^3} (b c-4 a f)}{448 a^2 x}-\frac{27 b^2 c \sqrt{a+b x^3}}{1120 a x^4}-\frac{b^2 d \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 e \sqrt{a+b x^3}}{320 a x^2}-\frac{b \sqrt{a+b x^3} \left (\frac{108 c}{x^7}+\frac{140 d}{x^6}+\frac{189 e}{x^5}+\frac{270 f}{x^4}+\frac{420 g}{x^3}\right )}{1680}-\frac{\left (a+b x^3\right )^{3/2} \left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}+\frac{420 g}{x^6}\right )}{2520} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^11,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((b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x**11,x)
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Mathematica [C] time = 4.58854, size = 930, normalized size = 1.22 \[ \frac{b^2 \left (-1680 g \sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3+a} \tanh ^{-1}\left (\frac{\sqrt{b x^3+a}}{\sqrt{a}}\right ) a^{3/2}+1620 \sqrt{2} \sqrt [3]{b} f \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{3 i+\sqrt{3}}} \left (-\left (-1+(-1)^{2/3}\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 )-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 )\right ) a^{4/3}+567 b^{2/3} e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\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 ) a+280 b d \sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3+a} \tanh ^{-1}\left (\frac{\sqrt{b x^3+a}}{\sqrt{a}}\right ) \sqrt{a}-405 \sqrt{2} b^{4/3} c \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{3 i+\sqrt{3}}} \left (-\left (-1+(-1)^{2/3}\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 )-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 )\right ) \sqrt [3]{a}\right )}{6720 a^2 \sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3+a}}-\frac{\sqrt{b x^3+a} \left (-1215 b^3 c x^9+3 a b^2 (162 c+x (280 d+81 x (7 e+20 f x))) x^6+4 a^2 b \left (828 c+x \left (980 d+3 x \left (700 g x^2+510 f x+399 e\right )\right )\right ) x^3+8 a^3 \left (252 c+5 x \left (84 g x^3+72 f x^2+63 e x+56 d\right )\right )\right )}{20160 a^2 x^{10}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^11,x]
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Maple [B] time = 0.015, size = 1470, normalized size = 1.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^11,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{x^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^11,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b g x^{7} + b f x^{6} + b e x^{5} +{\left (b d + a g\right )} x^{4} + a e x^{2} +{\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt{b x^{3} + a}}{x^{11}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^11,x, algorithm="fricas")
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Sympy [A] time = 41.5186, size = 576, normalized size = 0.75 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x**11,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{x^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^11,x, algorithm="giac")
[Out]