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

Optimal. Leaf size=266 \[ \frac {3 b \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{2 a}+\frac {b \log \left (\sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-1\right )}{2 \sqrt [3]{2} a}-\frac {b \log \left (2^{2/3} \left (b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+1\right )}{4 \sqrt [3]{2} a}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} a} \]

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

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{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 [3]{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 = 1.44, size = 331, normalized size = 1.24 \begin {gather*} \frac {a x \sqrt [3]{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 (-\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}+\left (\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a\right )^{2/3}\right )+6 \sqrt [3]{2} \sqrt [3]{a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}+2 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}\right )-2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )\right )}{4 \sqrt [3]{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 )^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3)/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]))^(1/3)*(-1 + a*x^2 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2])*(6*2^(1/3)*(
a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3) - 2*Sqrt[3]*a^(1/3)*ArcTan[(1 + (2*(a + (a*x + b*Sqrt[(a*(-1 +
a*x^2))/b^2])^2)^(1/3))/a^(1/3))/Sqrt[3]] + 2*a^(1/3)*Log[a^(1/3) - (a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])
^2)^(1/3)] - a^(1/3)*Log[a^(2/3) + a^(1/3)*(a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(1/3) + (a + (a*x + b*
Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(2/3)]))/(4*2^(1/3)*Sqrt[(a*(-1 + a*x^2))/b^2]*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^
2))/b^2]))^(4/3))

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IntegrateAlgebraic [A]  time = 3.79, size = 266, normalized size = 1.00 \begin {gather*} \frac {3 b \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{2 a}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{2} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} a}+\frac {b \log \left (-1+\sqrt [3]{2} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{2 \sqrt [3]{2} a}-\frac {b \log \left (1+\sqrt [3]{2} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}+2^{2/3} \left (a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )^{2/3}\right )}{4 \sqrt [3]{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])^(1/3)/Sqrt[-(a/b^2) + (a^2*x^2)/b^2],x]

[Out]

(3*b*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3))/(2*a) - (Sqrt[3]*b*ArcTan[1/Sqrt[3] + (2*2^(1/3)*(a*x
^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3))/Sqrt[3]])/(2*2^(1/3)*a) + (b*Log[-1 + 2^(1/3)*(a*x^2 + b*x*Sqr
t[-(a/b^2) + (a^2*x^2)/b^2])^(1/3)])/(2*2^(1/3)*a) - (b*Log[1 + 2^(1/3)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)
/b^2])^(1/3) + 2^(2/3)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(2/3)])/(4*2^(1/3)*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))^(1/3)/(-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}{3}}}{\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))^(1/3)/(-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)^(1/3)/sqrt(a^2*x^2/b^2 - a/b^2), x)

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

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))^(1/3)/(-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))^(1/3)/(-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}{3}}}{\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))^(1/3)/(-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)^(1/3)/sqrt(a^2*x^2/b^2 - a/b^2), x)

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

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