3.30.17 \(\int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\) [2917]

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

[Out]

1/16*(-a*x^2-10)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)/b+7/16*x*(-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)+5/48*arctan(1/3*3^(1/2)+2/3*2^(1/3)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1
/3)*3^(1/2))*2^(2/3)*3^(1/2)/b-5/48*ln(-1+2^(1/3)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3))*2^(2/3)/b+5/96
*ln(1+2^(1/3)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)+2^(2/3)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(2/3
))*2^(2/3)/b

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Rubi [F]
time = 0.34, 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 [B] Leaf count is larger than twice the leaf count of optimal. \(755\) vs. \(2(330)=660\).
time = 19.36, size = 755, normalized size = 2.29 \begin {gather*} \frac {\left (-1+a x^2\right ) \sqrt [3]{x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (30 \sqrt [3]{2} \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}-99 \sqrt [3]{2} a x^2 \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}+36 \sqrt [3]{2} a^2 x^4 \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}-81 \sqrt [3]{2} b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}} \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}+36 \sqrt [3]{2} a b x^3 \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}} \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}+10 \sqrt {3} \sqrt [3]{a} \left (-1+2 a x^2+2 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right ) \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-10 \sqrt [3]{a} \left (-1+2 a x^2+2 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right ) \log \left (-\sqrt [3]{a}+\sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}\right )-5 \sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}+\left (a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2\right )^{2/3}\right )+10 a^{4/3} x^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}+\left (a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2\right )^{2/3}\right )+10 \sqrt [3]{a} b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}+\left (a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2\right )^{2/3}\right )\right )}{48 \sqrt [3]{2} b \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (-1+a x^2+b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[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]

((-1 + a*x^2)*(x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3)*(30*2^(1/3)*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b
^2]))^(1/3) - 99*2^(1/3)*a*x^2*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3) + 36*2^(1/3)*a^2*x^4*(a*x*(a*x
+ b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3) - 81*2^(1/3)*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]*(a*x*(a*x + b*Sqrt[(a*(-1
+ a*x^2))/b^2]))^(1/3) + 36*2^(1/3)*a*b*x^3*Sqrt[(a*(-1 + a*x^2))/b^2]*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2
]))^(1/3) + 10*Sqrt[3]*a^(1/3)*(-1 + 2*a*x^2 + 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2])*ArcTan[(1 + (2*(a + (a*x + b*
Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(1/3))/a^(1/3))/Sqrt[3]] - 10*a^(1/3)*(-1 + 2*a*x^2 + 2*b*x*Sqrt[(a*(-1 + a*x^2
))/b^2])*Log[-a^(1/3) + (a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(1/3)] - 5*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)] + 10*
a^(4/3)*x^2*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)] + 10*a^(1/3)*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]*Log[a^(2/3) + a^(1/3)*(a + (a*x + b*S
qrt[(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)]))/(48*2^(1/3)*b*(a*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])^2)

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Maple [F]
time = 0.03, 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*} \text {Failed to integrate} \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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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((-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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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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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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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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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