3.21.39 \(\int \frac {\sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\)

Optimal. Leaf size=145 \[ \frac {2 b \sqrt [4]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{a}-\frac {b \tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}\right )}{\sqrt [4]{2} a}-\frac {b \tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}\right )}{\sqrt [4]{2} a} \]

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Rubi [F]  time = 0.50, 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 [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/4)/Sqrt[-(a/b^2) + (a^2*x^2)/b^2],x]

[Out]

Defer[Int][(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/4)/Sqrt[-(a/b^2) + (a^2*x^2)/b^2], x]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx &=\int \frac {\sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.13, size = 231, normalized size = 1.59 \begin {gather*} -\frac {a x \sqrt [4]{x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \left (-2 \sqrt [4]{2} \sqrt [4]{a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}+\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}}{\sqrt [4]{a}}\right )+\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}}{\sqrt [4]{a}}\right )\right )}{\sqrt [4]{2} \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \left (a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )\right )^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/4)/Sqrt[-(a/b^2) + (a^2*x^2)/b^2],x]

[Out]

-((a*x*(x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/4)*(-1 + a*x^2 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2])*(-2*2^(1/4
)*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/4) + a^(1/4)*ArcTan[(a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^
2)^(1/4)/a^(1/4)] + a^(1/4)*ArcTanh[(a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(1/4)/a^(1/4)]))/(2^(1/4)*Sqr
t[(a*(-1 + a*x^2))/b^2]*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(5/4)))

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IntegrateAlgebraic [A]  time = 3.71, size = 206, normalized size = 1.42 \begin {gather*} \frac {2 b \sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{a}-\frac {b \tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt [4]{2} a}+\frac {b \log \left (-1+\sqrt [4]{2} \sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{2 \sqrt [4]{2} a}-\frac {b \log \left (a+\sqrt [4]{2} a \sqrt [4]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{2 \sqrt [4]{2} a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/4)/Sqrt[-(a/b^2) + (a^2*x^2)/b^2],x]

[Out]

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

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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*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/4)/(-a/b^2+a^2*x^2/b^2)^(1/2),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 {{\left (a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x\right )}^{\frac {1}{4}}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/4)/(-a/b^2+a^2*x^2/b^2)^(1/2),x, algorithm="giac")

[Out]

integrate((a*x^2 + sqrt(a^2*x^2/b^2 - a/b^2)*b*x)^(1/4)/sqrt(a^2*x^2/b^2 - a/b^2), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\right )^{\frac {1}{4}}}{\sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/4)/(-a/b^2+a^2*x^2/b^2)^(1/2),x)

[Out]

int((a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/4)/(-a/b^2+a^2*x^2/b^2)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x\right )}^{\frac {1}{4}}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/4)/(-a/b^2+a^2*x^2/b^2)^(1/2),x, algorithm="maxima")

[Out]

integrate((a*x^2 + sqrt(a^2*x^2/b^2 - a/b^2)*b*x)^(1/4)/sqrt(a^2*x^2/b^2 - a/b^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}\right )}^{1/4}}{\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2 + b*x*((a^2*x^2)/b^2 - a/b^2)^(1/2))^(1/4)/((a^2*x^2)/b^2 - a/b^2)^(1/2),x)

[Out]

int((a*x^2 + b*x*((a^2*x^2)/b^2 - a/b^2)^(1/2))^(1/4)/((a^2*x^2)/b^2 - a/b^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}{\sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**2+b*x*(-a/b**2+a**2*x**2/b**2)**(1/2))**(1/4)/(-a/b**2+a**2*x**2/b**2)**(1/2),x)

[Out]

Integral((x*(a*x + b*sqrt(a**2*x**2/b**2 - a/b**2)))**(1/4)/sqrt(a*(a*x**2 - 1)/b**2), x)

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