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

Optimal. Leaf size=330 \[ \frac {7}{16} x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+\frac {\left (-a x^2-10\right ) \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{16 b}-\frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-1\right )}{24 \sqrt [3]{2} b}+\frac {5 \log \left (2^{2/3} \left (b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+1\right )}{48 \sqrt [3]{2} b}+\frac {5 \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{8 \sqrt [3]{2} \sqrt {3} b} \]

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Rubi [F]  time = 0.47, 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 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 20.84, size = 10907, normalized size = 33.05 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 3.68, size = 333, normalized size = 1.01 \begin {gather*} \frac {\left (-10-a x^2\right ) \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{16 b}+\frac {7}{16} x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}+\frac {5 \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 )}{8 \sqrt [3]{2} \sqrt {3} b}-\frac {5 \log \left (-b+\sqrt [3]{2} b \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{24 \sqrt [3]{2} b}+\frac {5 \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 )}{48 \sqrt [3]{2} b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-10 - a*x^2)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3))/(16*b) + (7*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2
]*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3))/16 + (5*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]])/(8*2^(1/3)*Sqrt[3]*b) - (5*Log[-b + 2^(1/3)*b*(a*x^2 + b*x*Sqrt[-(
a/b^2) + (a^2*x^2)/b^2])^(1/3)])/(24*2^(1/3)*b) + (5*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)])/(48*2^(1/3)*b)

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

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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