3.28.14 \(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx\) [2714]

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

[Out]

1/2*3^(1/2)*arctan(1/3*3^(1/2)+2/3*2^(1/3)*a^(1/3)*(a*x^2+x*(a^2*x^2-b)^(1/2))^(1/3)*3^(1/2)/b^(1/3))*2^(1/3)/
a^(2/3)/b^(1/3)+1/2*ln(-1+2^(1/3)*a^(1/3)*(a*x^2+x*(a^2*x^2-b)^(1/2))^(1/3)/b^(1/3))*2^(1/3)/a^(2/3)/b^(1/3)-1
/4*ln(1+2^(1/3)*a^(1/3)*(a*x^2+x*(a^2*x^2-b)^(1/2))^(1/3)/b^(1/3)+2^(2/3)*a^(2/3)*(a*x^2+x*(a^2*x^2-b)^(1/2))^
(2/3)/b^(2/3))*2^(1/3)/a^(2/3)/b^(1/3)

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Rubi [F]
time = 0.22, 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 = 0.44, size = 276, normalized size = 1.11 \begin {gather*} \frac {\sqrt [3]{x} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [3]{b}}}{\sqrt {3}}\right )+2 \log \left (-\sqrt [3]{b}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )-\log \left (b^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}+2^{2/3} a^{2/3} x^{2/3} \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}\right )\right )}{2\ 2^{2/3} a^{2/3} \sqrt [3]{b} \sqrt [3]{x \left (a x+\sqrt {-b+a^2 x^2}\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]

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

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Maple [F]
time = 0.03, 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*} \text {Failed to integrate} \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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Fricas [F(-1)] Timed out
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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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