3.2940 \(\int x^3 \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=340 \[ -\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{55 b^{4/3} c^2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{6 a \sqrt{c x^2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{55 b c^2}+\frac{2}{11} x^4 \sqrt{a+b \left (c x^2\right )^{3/2}} \]

[Out]

(2*x^4*Sqrt[a + b*(c*x^2)^(3/2)])/11 + (6*a*Sqrt[c*x^2]*Sqrt[a + b*(c*x^2)^(3/2)
])/(55*b*c^2) - (4*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*(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]])/(
55*b^(4/3)*c^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 = 0.502507, 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]

(2*x^4*Sqrt[a + b*(c*x^2)^(3/2)])/11 + (6*a*Sqrt[c*x^2]*Sqrt[a + b*(c*x^2)^(3/2)
])/(55*b*c^2) - (4*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*(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]])/(
55*b^(4/3)*c^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 = 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)

[Out]

-4*3**(3/4)*a**2*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))/(55*b**(4/3)*c**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))) + 6*a*s
qrt(c*x**2)*sqrt(a + b*(c*x**2)**(3/2))/(55*b*c**2) + 2*x**4*sqrt(a + b*(c*x**2)
**(3/2))/11

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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]

[Out]

(16*a*b*c^2*x^4 + 6*a^2*Sqrt[c*x^2] + 10*b^2*c^3*x^6*Sqrt[c*x^2] - 6*a^2*Sqrt[c*
x^2]*Sqrt[(a + b*(c*x^2)^(3/2))/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*(c*x^2)
^(3/2))/a)])/(55*b*c^2*Sqrt[a + b*(c*x^2)^(3/2)])

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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)

[Out]

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")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\sqrt{c x^{2}} b c x^{2} + a} x^{3}, x\right ) \]

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")

[Out]

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

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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]

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

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

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]

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