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

Optimal. Leaf size=351 \[ \frac {a x \sqrt {a^2 x^2-b} \left (x \sqrt {a^2 x^2-b}+a x^2\right )^{2/3}}{2 b}+\frac {\left (5 b-4 a^2 x^2\right ) \left (x \sqrt {a^2 x^2-b}+a x^2\right )^{2/3}}{8 b}-\frac {7 b^{2/3} \log \left (\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}}{\sqrt [3]{b}}-1\right )}{12\ 2^{2/3} a^{2/3}}+\frac {7 b^{2/3} \log \left (\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}}{\sqrt [3]{b}}+\frac {2^{2/3} a^{2/3} \left (x \sqrt {a^2 x^2-b}+a x^2\right )^{2/3}}{b^{2/3}}+1\right )}{24\ 2^{2/3} a^{2/3}}-\frac {7 b^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}}{\sqrt {3} \sqrt [3]{b}}+\frac {1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3} a^{2/3}} \]

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Rubi [F]  time = 0.31, 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 {\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[Sqrt[-b + a^2*x^2]/(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3),x]

[Out]

Defer[Int][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 {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx &=\int \frac {\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 [C]  time = 24.91, size = 12131, normalized size = 34.56 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 3.50, size = 351, normalized size = 1.00 \begin {gather*} \frac {\left (5 b-4 a^2 x^2\right ) \left (a x^2+x \sqrt {-b+a^2 x^2}\right )^{2/3}}{8 b}+\frac {a x \sqrt {-b+a^2 x^2} \left (a x^2+x \sqrt {-b+a^2 x^2}\right )^{2/3}}{2 b}-\frac {7 b^{2/3} \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 )}{4\ 2^{2/3} \sqrt {3} a^{2/3}}-\frac {7 b^{2/3} \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 )}{12\ 2^{2/3} a^{2/3}}+\frac {7 b^{2/3} \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 )}{24\ 2^{2/3} a^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((5*b - 4*a^2*x^2)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(2/3))/(8*b) + (a*x*Sqrt[-b + a^2*x^2]*(a*x^2 + x*Sqrt[-b +
a^2*x^2])^(2/3))/(2*b) - (7*b^(2/3)*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))])/(4*2^(2/3)*Sqrt[3]*a^(2/3)) - (7*b^(2/3)*Log[-1 + (2^(1/3)*a^(1/3)*(a*x^2 + x*Sqrt[-b +
a^2*x^2])^(1/3))/b^(1/3)])/(12*2^(2/3)*a^(2/3)) + (7*b^(2/3)*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)])/(24*2^(2/3)*a^(2/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((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 {\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((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(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 {\sqrt {a^2\,x^2-b}}{{\left (x\,\sqrt {a^2\,x^2-b}+a\,x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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