Optimal. Leaf size=550 \[ -\frac{\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-8 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-8 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{16 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{\sqrt{a+b x^3} (5 A b-8 a B)}{8 a^2 x}-\frac{\sqrt [3]{b} \sqrt{a+b x^3} (5 A b-8 a B)}{8 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{A \sqrt{a+b x^3}}{4 a x^4} \]
[Out]
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Rubi [A] time = 0.6744, antiderivative size = 550, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-8 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-8 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{16 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{\sqrt{a+b x^3} (5 A b-8 a B)}{8 a^2 x}-\frac{\sqrt [3]{b} \sqrt{a+b x^3} (5 A b-8 a B)}{8 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{A \sqrt{a+b x^3}}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^3)/(x^5*Sqrt[a + b*x^3]),x]
[Out]
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Rubi in Sympy [A] time = 48.8695, size = 486, normalized size = 0.88 \[ - \frac{A \sqrt{a + b x^{3}}}{4 a x^{4}} - \frac{\sqrt [3]{b} \sqrt{a + b x^{3}} \left (5 A b - 8 B a\right )}{8 a^{2} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )} + \frac{\sqrt{a + b x^{3}} \left (5 A b - 8 B a\right )}{8 a^{2} x} + \frac{\sqrt [4]{3} \sqrt [3]{b} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (5 A b - 8 B a\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{16 a^{\frac{5}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [3]{b} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (5 A b - 8 B a\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{24 a^{\frac{5}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)/x**5/(b*x**3+a)**(1/2),x)
[Out]
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Mathematica [C] time = 0.582742, size = 249, normalized size = 0.45 \[ \frac{\sqrt{a+b x^3} \left (5 A b x^3-2 a \left (A+4 B x^3\right )\right )}{8 a^2 x^4}-\frac{\sqrt [6]{-1} \sqrt [3]{-b} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}-1\right )} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} (8 a B-5 A b) \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{8 \sqrt [4]{3} a^{4/3} \sqrt{a+b x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(A + B*x^3)/(x^5*Sqrt[a + b*x^3]),x]
[Out]
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Maple [B] time = 0.013, size = 929, normalized size = 1.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)/x^5/(b*x^3+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{3} + A}{\sqrt{b x^{3} + a} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/(sqrt(b*x^3 + a)*x^5),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{3} + A}{\sqrt{b x^{3} + a} x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/(sqrt(b*x^3 + a)*x^5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.6525, size = 88, normalized size = 0.16 \[ \frac{A \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{2} \\ - \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt{a} x^{4} \Gamma \left (- \frac{1}{3}\right )} + \frac{B \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{2} \\ \frac{2}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt{a} x \Gamma \left (\frac{2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)/x**5/(b*x**3+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{3} + A}{\sqrt{b x^{3} + a} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/(sqrt(b*x^3 + a)*x^5),x, algorithm="giac")
[Out]