Optimal. Leaf size=225 \[ -\frac{2 b x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{5 \sqrt [4]{3} a^{4/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}-\frac{2 \sqrt{a x+b x^4}}{5 a x^3} \]
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Rubi [A] time = 0.364562, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{2 b x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{5 \sqrt [4]{3} a^{4/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}-\frac{2 \sqrt{a x+b x^4}}{5 a x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*Sqrt[a*x + b*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 21.6407, size = 207, normalized size = 0.92 \[ - \frac{2 \sqrt{a x + b x^{4}}}{5 a x^{3}} - \frac{2 \cdot 3^{\frac{3}{4}} b \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt{a x + b x^{4}} F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{15 a^{\frac{4}{3}} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (a + b x^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**4+a*x)**(1/2),x)
[Out]
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Mathematica [C] time = 0.525653, size = 172, normalized size = 0.76 \[ -\frac{-6 \sqrt [3]{-a} \left (a+b x^3\right )+4 i 3^{3/4} b^{4/3} x^4 \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-a}}{\sqrt [3]{b} x}-1\right )} \sqrt{\frac{\frac{(-a)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{-a} x}{\sqrt [3]{b}}+x^2}{x^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{15 (-a)^{4/3} x^2 \sqrt{x \left (a+b x^3\right )}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^3*Sqrt[a*x + b*x^4]),x]
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Maple [C] time = 0.035, size = 696, normalized size = 3.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^4+a*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a x} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^4 + a*x)*x^3),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x^{4} + a x} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^4 + a*x)*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{x \left (a + b x^{3}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**4+a*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a x} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^4 + a*x)*x^3),x, algorithm="giac")
[Out]