3.2941 \(\int \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=306 \[ \frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a x \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} \sqrt{c x^2} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{2}{5} x \sqrt{a+b \left (c x^2\right )^{3/2}} \]

[Out]

(2*x*Sqrt[a + b*(c*x^2)^(3/2)])/5 + (2*3^(3/4)*Sqrt[2 + Sqrt[3]]*a*x*(a^(1/3) +
b^(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b^(2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2]
)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])]
, -7 - 4*Sqrt[3]])/(5*b^(1/3)*Sqrt[c*x^2]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[
c*x^2]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*Sqrt[a + b*(c*x^2)^(3/
2)])

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Rubi [A]  time = 0.318783, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a x \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} \sqrt{c x^2} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{2}{5} x \sqrt{a+b \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*(c*x^2)^(3/2)],x]

[Out]

(2*x*Sqrt[a + b*(c*x^2)^(3/2)])/5 + (2*3^(3/4)*Sqrt[2 + Sqrt[3]]*a*x*(a^(1/3) +
b^(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b^(2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2]
)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])]
, -7 - 4*Sqrt[3]])/(5*b^(1/3)*Sqrt[c*x^2]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[
c*x^2]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*Sqrt[a + b*(c*x^2)^(3/
2)])

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Rubi in Sympy [A]  time = 13.4397, size = 270, normalized size = 0.88 \[ \frac{2 \cdot 3^{\frac{3}{4}} a x \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} + b^{\frac{2}{3}} c x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{5 \sqrt [3]{b} \sqrt{c x^{2}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}} + \frac{2 x \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**2)**(3/2))**(1/2),x)

[Out]

2*3**(3/4)*a*x*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*sqrt(c*x**2) + b**(2/3)*c*x**2
)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**2))**2)*sqrt(sqrt(3) + 2)*(a**(1/
3) + b**(1/3)*sqrt(c*x**2))*elliptic_f(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)
*sqrt(c*x**2))/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**2))), -7 - 4*sqrt(3)
)/(5*b**(1/3)*sqrt(c*x**2)*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)*sqrt(c*x**2))/(a**
(1/3)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**2))**2)*sqrt(a + b*(c*x**2)**(3/2))) +
2*x*sqrt(a + b*(c*x**2)**(3/2))/5

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Mathematica [A]  time = 0.0150059, size = 0, normalized size = 0. \[ \int \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*(c*x^2)^(3/2)],x]

[Out]

Integrate[Sqrt[a + b*(c*x^2)^(3/2)], x]

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int \sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^2)^(3/2))^(1/2),x)

[Out]

int((a+b*(c*x^2)^(3/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a),x, algorithm="maxima")

[Out]

integrate(sqrt((c*x^2)^(3/2)*b + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\sqrt{c x^{2}} b c x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a),x, algorithm="fricas")

[Out]

integral(sqrt(sqrt(c*x^2)*b*c*x^2 + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**2)**(3/2))**(1/2),x)

[Out]

Integral(sqrt(a + b*(c*x**2)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a),x, algorithm="giac")

[Out]

integrate(sqrt((c*x^2)^(3/2)*b + a), x)