Optimal. Leaf size=770 \[ \frac{4 \sqrt{2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}-\frac{6 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} E\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{12 a \sqrt{a+b \sqrt{c x^3}}}{7 b^{2/3} \sqrt [3]{c} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}+\frac{4}{7} x \sqrt{a+b \sqrt{c x^3}} \]
[Out]
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Rubi [A] time = 1.04694, antiderivative size = 770, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{4 \sqrt{2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}-\frac{6 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} E\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{12 a \sqrt{a+b \sqrt{c x^3}}}{7 b^{2/3} \sqrt [3]{c} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}+\frac{4}{7} x \sqrt{a+b \sqrt{c x^3}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[c*x^3]],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b \sqrt{c x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c*x**3)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0126361, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sqrt{c x^3}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[a + b*Sqrt[c*x^3]],x]
[Out]
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Maple [A] time = 0.173, size = 860, normalized size = 1.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c*x^3)^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{c x^{3}} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^3)*b + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\sqrt{c x^{3}} b + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^3)*b + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b \sqrt{c x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c*x**3)**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{c x^{3}} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^3)*b + a),x, algorithm="giac")
[Out]