Optimal. Leaf size=340 \[ -\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{55 b^{4/3} c^2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{6 a \sqrt{c x^2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{55 b c^2}+\frac{2}{11} x^4 \sqrt{a+b \left (c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.502507, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{55 b^{4/3} c^2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{6 a \sqrt{c x^2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{55 b c^2}+\frac{2}{11} x^4 \sqrt{a+b \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^3*Sqrt[a + b*(c*x^2)^(3/2)],x]
[Out]
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Rubi in Sympy [A] time = 24.6502, size = 301, normalized size = 0.89 \[ - \frac{4 \cdot 3^{\frac{3}{4}} a^{2} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} + b^{\frac{2}{3}} c x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{55 b^{\frac{4}{3}} c^{2} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}} + \frac{6 a \sqrt{c x^{2}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{55 b c^{2}} + \frac{2 x^{4} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(a+b*(c*x**2)**(3/2))**(1/2),x)
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Mathematica [C] time = 0.256877, size = 132, normalized size = 0.39 \[ \frac{-6 a^2 \sqrt{c x^2} \sqrt{\frac{a+b \left (c x^2\right )^{3/2}}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b \left (c x^2\right )^{3/2}}{a}\right )+6 a^2 \sqrt{c x^2}+16 a b c^2 x^4+10 b^2 c^3 x^6 \sqrt{c x^2}}{55 b c^2 \sqrt{a+b \left (c x^2\right )^{3/2}}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*Sqrt[a + b*(c*x^2)^(3/2)],x]
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Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int{x}^{3}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(a+b*(c*x^2)^(3/2))^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x^2)^(3/2)*b + a)*x^3,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\sqrt{c x^{2}} b c x^{2} + a} x^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x^2)^(3/2)*b + a)*x^3,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{3} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(a+b*(c*x**2)**(3/2))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x^2)^(3/2)*b + a)*x^3,x, algorithm="giac")
[Out]