3.9.73 \(\int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx\)

Optimal. Leaf size=66 \[ \frac {4 \left (2 x^2-1\right ) \sqrt {x^2 \left (\sqrt {x^2-1}+x\right )}}{15 x}-\frac {2}{15} \sqrt {x^2-1} \sqrt {x^2 \left (\sqrt {x^2-1}+x\right )} \]

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Rubi [F]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[x^3 + x^2*Sqrt[-1 + x^2]],x]

[Out]

Defer[Int][Sqrt[x^3 + x^2*Sqrt[-1 + x^2]], x]

Rubi steps

\begin {align*} \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx &=\int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 89, normalized size = 1.35 \begin {gather*} \frac {2 x \sqrt {x^2-1} \left (6 x^4-6 x^2-3 \sqrt {x^2-1} x+6 \sqrt {x^2-1} x^3+2\right )}{15 \sqrt {x^2 \left (\sqrt {x^2-1}+x\right )} \left (x^2+\sqrt {x^2-1} x-1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x^3 + x^2*Sqrt[-1 + x^2]],x]

[Out]

(2*x*Sqrt[-1 + x^2]*(2 - 6*x^2 + 6*x^4 - 3*x*Sqrt[-1 + x^2] + 6*x^3*Sqrt[-1 + x^2]))/(15*Sqrt[x^2*(x + Sqrt[-1
+ x^2])]*(-1 + x^2 + x*Sqrt[-1 + x^2]))

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IntegrateAlgebraic [A]  time = 3.20, size = 66, normalized size = 1.00 \begin {gather*} -\frac {2}{15} \sqrt {-1+x^2} \sqrt {x^2 \left (x+\sqrt {-1+x^2}\right )}+\frac {4 \left (-1+2 x^2\right ) \sqrt {x^2 \left (x+\sqrt {-1+x^2}\right )}}{15 x} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[x^3 + x^2*Sqrt[-1 + x^2]],x]

[Out]

(-2*Sqrt[-1 + x^2]*Sqrt[x^2*(x + Sqrt[-1 + x^2])])/15 + (4*(-1 + 2*x^2)*Sqrt[x^2*(x + Sqrt[-1 + x^2])])/(15*x)

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fricas [A]  time = 0.55, size = 39, normalized size = 0.59 \begin {gather*} \frac {2 \, \sqrt {x^{3} + \sqrt {x^{2} - 1} x^{2}} {\left (4 \, x^{2} - \sqrt {x^{2} - 1} x - 2\right )}}{15 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x^2*(x^2-1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

2/15*sqrt(x^3 + sqrt(x^2 - 1)*x^2)*(4*x^2 - sqrt(x^2 - 1)*x - 2)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{3} + \sqrt {x^{2} - 1} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x^2*(x^2-1)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(x^3 + sqrt(x^2 - 1)*x^2), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \sqrt {x^{3}+x^{2} \sqrt {x^{2}-1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+x^2*(x^2-1)^(1/2))^(1/2),x)

[Out]

int((x^3+x^2*(x^2-1)^(1/2))^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{3} + \sqrt {x^{2} - 1} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x^2*(x^2-1)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^3 + sqrt(x^2 - 1)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {x^2\,\sqrt {x^2-1}+x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(x^2 - 1)^(1/2) + x^3)^(1/2),x)

[Out]

int((x^2*(x^2 - 1)^(1/2) + x^3)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{3} + x^{2} \sqrt {x^{2} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+x**2*(x**2-1)**(1/2))**(1/2),x)

[Out]

Integral(sqrt(x**3 + x**2*sqrt(x**2 - 1)), x)

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