3.28.14 \(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx\)

Optimal. Leaf size=248 \[ -\frac {\log \left (\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}}{\sqrt [3]{b}}+\frac {2^{2/3} a^{2/3} \left (x \sqrt {a^2 x^2-b}+a x^2\right )^{2/3}}{b^{2/3}}+1\right )}{2\ 2^{2/3} a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}}{\sqrt [3]{b}}-1\right )}{2^{2/3} a^{2/3} \sqrt [3]{b}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}}{\sqrt {3} \sqrt [3]{b}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} a^{2/3} \sqrt [3]{b}} \]

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Rubi [F]  time = 0.33, 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 \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(Sqrt[-b + a^2*x^2]*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3)),x]

[Out]

Defer[Int][1/(Sqrt[-b + a^2*x^2]*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3)), x]

Rubi steps

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

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Mathematica [A]  time = 13.65, size = 225, normalized size = 0.91 \begin {gather*} \frac {\sqrt {a^2 x^2-b} \left (a x \left (\sqrt {a^2 x^2-b}+a x\right )\right )^{4/3} \left (-2 \log \left (\sqrt {a^2 x^2-b}+a x\right )+3 \log \left (\sqrt [3]{b}-\sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^2+b}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^2+b}}{\sqrt [3]{b}}+1}{\sqrt {3}}\right )\right )}{2\ 2^{2/3} a^2 \sqrt [3]{b} x \sqrt [3]{x \left (\sqrt {a^2 x^2-b}+a x\right )} \left (a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-b + a^2*x^2]*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3)),x]

[Out]

(Sqrt[-b + a^2*x^2]*(a*x*(a*x + Sqrt[-b + a^2*x^2]))^(4/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(b + (a*x + Sqrt[-b + a^2
*x^2])^2)^(1/3))/b^(1/3))/Sqrt[3]] - 2*Log[a*x + Sqrt[-b + a^2*x^2]] + 3*Log[b^(1/3) - (b + (a*x + Sqrt[-b + a
^2*x^2])^2)^(1/3)]))/(2*2^(2/3)*a^2*b^(1/3)*x*(x*(a*x + Sqrt[-b + a^2*x^2]))^(1/3)*(-b + a*x*(a*x + Sqrt[-b +
a^2*x^2])))

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IntegrateAlgebraic [A]  time = 2.45, size = 248, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}}{\sqrt {3} \sqrt [3]{b}}\right )}{2^{2/3} a^{2/3} \sqrt [3]{b}}+\frac {\log \left (-1+\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}}{\sqrt [3]{b}}\right )}{2^{2/3} a^{2/3} \sqrt [3]{b}}-\frac {\log \left (1+\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}}{\sqrt [3]{b}}+\frac {2^{2/3} a^{2/3} \left (a x^2+x \sqrt {-b+a^2 x^2}\right )^{2/3}}{b^{2/3}}\right )}{2\ 2^{2/3} a^{2/3} \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3)),x]

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*2^(1/3)*a^(1/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3))/(Sqrt[3]*b^(1/3))])/(2^(2
/3)*a^(2/3)*b^(1/3)) + Log[-1 + (2^(1/3)*a^(1/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3))/b^(1/3)]/(2^(2/3)*a^(2/
3)*b^(1/3)) - Log[1 + (2^(1/3)*a^(1/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3))/b^(1/3) + (2^(2/3)*a^(2/3)*(a*x^2
+ x*Sqrt[-b + a^2*x^2])^(2/3))/b^(2/3)]/(2*2^(2/3)*a^(2/3)*b^(1/3))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a^{2} x^{2}-b}\, \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (x\,\sqrt {a^2\,x^2-b}+a\,x^2\right )}^{1/3}\,\sqrt {a^2\,x^2-b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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