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

Optimal. Leaf size=33 \[ \frac {2 \sqrt {a x^3+b}}{x}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {a x^3+b}}\right ) \]

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Rubi [F]  time = 1.22, 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 {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^2 \left (b-x^2+a x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*Sqrt[b + a*x^3])/x - (6*a^(1/3)*Sqrt[b + a*x^3])/((1 + Sqrt[3])*b^(1/3) + a^(1/3)*x) + (3*3^(1/4)*Sqrt[2 -
Sqrt[3]]*a^(1/3)*b^(1/3)*(b^(1/3) + a^(1/3)*x)*Sqrt[(b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2)/((1 + Sqrt[3])
*b^(1/3) + a^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*b^(1/3) + a^(1/3)*x)/((1 + Sqrt[3])*b^(1/3) + a^(1/3)
*x)], -7 - 4*Sqrt[3]])/(Sqrt[(b^(1/3)*(b^(1/3) + a^(1/3)*x))/((1 + Sqrt[3])*b^(1/3) + a^(1/3)*x)^2]*Sqrt[b + a
*x^3]) - (2*Sqrt[2]*3^(3/4)*a^(1/3)*b^(1/3)*(b^(1/3) + a^(1/3)*x)*Sqrt[(b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*
x^2)/((1 + Sqrt[3])*b^(1/3) + a^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*b^(1/3) + a^(1/3)*x)/((1 + Sqrt[3]
)*b^(1/3) + a^(1/3)*x)], -7 - 4*Sqrt[3]])/(Sqrt[(b^(1/3)*(b^(1/3) + a^(1/3)*x))/((1 + Sqrt[3])*b^(1/3) + a^(1/
3)*x)^2]*Sqrt[b + a*x^3]) - 2*Defer[Int][Sqrt[b + a*x^3]/(b - x^2 + a*x^3), x] + 3*a*Defer[Int][(x*Sqrt[b + a*
x^3])/(b - x^2 + a*x^3), x]

Rubi steps

\begin {align*} \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^2 \left (b-x^2+a x^3\right )} \, dx &=\int \left (-\frac {2 \sqrt {b+a x^3}}{x^2}+\frac {(-2+3 a x) \sqrt {b+a x^3}}{b-x^2+a x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {b+a x^3}}{x^2} \, dx\right )+\int \frac {(-2+3 a x) \sqrt {b+a x^3}}{b-x^2+a x^3} \, dx\\ &=\frac {2 \sqrt {b+a x^3}}{x}-(3 a) \int \frac {x}{\sqrt {b+a x^3}} \, dx+\int \left (-\frac {2 \sqrt {b+a x^3}}{b-x^2+a x^3}+\frac {3 a x \sqrt {b+a x^3}}{b-x^2+a x^3}\right ) \, dx\\ &=\frac {2 \sqrt {b+a x^3}}{x}-2 \int \frac {\sqrt {b+a x^3}}{b-x^2+a x^3} \, dx-\left (3 a^{2/3}\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}{\sqrt {b+a x^3}} \, dx+(3 a) \int \frac {x \sqrt {b+a x^3}}{b-x^2+a x^3} \, dx-\left (3 \sqrt {2 \left (2-\sqrt {3}\right )} a^{2/3} \sqrt [3]{b}\right ) \int \frac {1}{\sqrt {b+a x^3}} \, dx\\ &=\frac {2 \sqrt {b+a x^3}}{x}-\frac {6 \sqrt [3]{a} \sqrt {b+a x^3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right ) \sqrt {\frac {b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{b} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x\right )^2}} \sqrt {b+a x^3}}-\frac {2 \sqrt {2} 3^{3/4} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right ) \sqrt {\frac {b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{b} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x\right )^2}} \sqrt {b+a x^3}}-2 \int \frac {\sqrt {b+a x^3}}{b-x^2+a x^3} \, dx+(3 a) \int \frac {x \sqrt {b+a x^3}}{b-x^2+a x^3} \, dx\\ \end {align*}

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Mathematica [C]  time = 6.24, size = 2741, normalized size = 83.06 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[b + a*x^3])/x + (2*Sqrt[(b^(1/3)/a^(1/3) + x)/(b^(1/3)/a^(1/3) + ((-1)^(1/3)*b^(1/3))/a^(1/3))]*(-(((-
1)^(1/3)*b^(1/3))/a^(1/3)) + x)*Sqrt[(((-1)^(2/3)*b^(1/3))/a^(1/3) + x)/(((-1)^(1/3)*b^(1/3))/a^(1/3) + ((-1)^
(2/3)*b^(1/3))/a^(1/3))]*EllipticF[ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) - a^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^(
1/3))]], (-1)^(1/3)])/(Sqrt[(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + x)/(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/
3)*b^(1/3))/a^(1/3))]*Sqrt[b + a*x^3]) - (4*(((-1)^(1/3)*b^(1/3))/a^(1/3) + ((-1)^(2/3)*b^(1/3))/a^(1/3))*b*Sq
rt[(b^(1/3)/a^(1/3) + x)/(b^(1/3)/a^(1/3) + ((-1)^(1/3)*b^(1/3))/a^(1/3))]*Sqrt[((-(((-1)^(2/3)*b^(1/3))/a^(1/
3)) - x)*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + x))/(((-1)^(1/3)*b^(1/3))/a^(1/3) + ((-1)^(2/3)*b^(1/3))/a^(1/3))^
2]*EllipticPi[((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/3))/((-1)^(1/3)*b^(1/3) - a^(1/3)*Root[b - #1^2 + a*#1^3 &
, 1]), ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) - a^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)])/(a*Sq
rt[b + a*x^3]*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + Root[b - #1^2 + a*#1^3 & , 1])*(Root[b - #1^2 + a*#1^3 & , 1]
- Root[b - #1^2 + a*#1^3 & , 2])*(Root[b - #1^2 + a*#1^3 & , 1] - Root[b - #1^2 + a*#1^3 & , 3])) + (2*(((-1)
^(1/3)*b^(1/3))/a^(1/3) + ((-1)^(2/3)*b^(1/3))/a^(1/3))*Sqrt[(b^(1/3)/a^(1/3) + x)/(b^(1/3)/a^(1/3) + ((-1)^(1
/3)*b^(1/3))/a^(1/3))]*Sqrt[((-(((-1)^(2/3)*b^(1/3))/a^(1/3)) - x)*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + x))/(((-
1)^(1/3)*b^(1/3))/a^(1/3) + ((-1)^(2/3)*b^(1/3))/a^(1/3))^2]*EllipticPi[((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/
3))/((-1)^(1/3)*b^(1/3) - a^(1/3)*Root[b - #1^2 + a*#1^3 & , 1]), ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) - a^(1/3)*x)
/(((-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)]*Root[b - #1^2 + a*#1^3 & , 1]^3)/(Sqrt[b + a*x^3]*(-(((-1)
^(1/3)*b^(1/3))/a^(1/3)) + Root[b - #1^2 + a*#1^3 & , 1])*(Root[b - #1^2 + a*#1^3 & , 1] - Root[b - #1^2 + a*#
1^3 & , 2])*(Root[b - #1^2 + a*#1^3 & , 1] - Root[b - #1^2 + a*#1^3 & , 3])) - (4*(((-1)^(1/3)*b^(1/3))/a^(1/3
) + ((-1)^(2/3)*b^(1/3))/a^(1/3))*b*Sqrt[(b^(1/3)/a^(1/3) + x)/(b^(1/3)/a^(1/3) + ((-1)^(1/3)*b^(1/3))/a^(1/3)
)]*Sqrt[((-(((-1)^(2/3)*b^(1/3))/a^(1/3)) - x)*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + x))/(((-1)^(1/3)*b^(1/3))/a^
(1/3) + ((-1)^(2/3)*b^(1/3))/a^(1/3))^2]*EllipticPi[((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/3))/((-1)^(1/3)*b^(1
/3) - a^(1/3)*Root[b - #1^2 + a*#1^3 & , 2]), ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) - a^(1/3)*x)/(((-1)^(1/3) + (-1)
^(2/3))*b^(1/3))]], (-1)^(1/3)])/(a*Sqrt[b + a*x^3]*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + Root[b - #1^2 + a*#1^3
& , 2])*(-Root[b - #1^2 + a*#1^3 & , 1] + Root[b - #1^2 + a*#1^3 & , 2])*(Root[b - #1^2 + a*#1^3 & , 2] - Root
[b - #1^2 + a*#1^3 & , 3])) + (2*(((-1)^(1/3)*b^(1/3))/a^(1/3) + ((-1)^(2/3)*b^(1/3))/a^(1/3))*Sqrt[(b^(1/3)/a
^(1/3) + x)/(b^(1/3)/a^(1/3) + ((-1)^(1/3)*b^(1/3))/a^(1/3))]*Sqrt[((-(((-1)^(2/3)*b^(1/3))/a^(1/3)) - x)*(-((
(-1)^(1/3)*b^(1/3))/a^(1/3)) + x))/(((-1)^(1/3)*b^(1/3))/a^(1/3) + ((-1)^(2/3)*b^(1/3))/a^(1/3))^2]*EllipticPi
[((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/3))/((-1)^(1/3)*b^(1/3) - a^(1/3)*Root[b - #1^2 + a*#1^3 & , 2]), ArcSi
n[Sqrt[((-1)^(1/3)*b^(1/3) - a^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)]*Root[b - #1^2 + a*#
1^3 & , 2]^3)/(Sqrt[b + a*x^3]*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + Root[b - #1^2 + a*#1^3 & , 2])*(-Root[b - #1
^2 + a*#1^3 & , 1] + Root[b - #1^2 + a*#1^3 & , 2])*(Root[b - #1^2 + a*#1^3 & , 2] - Root[b - #1^2 + a*#1^3 &
, 3])) - (4*(((-1)^(1/3)*b^(1/3))/a^(1/3) + ((-1)^(2/3)*b^(1/3))/a^(1/3))*b*Sqrt[(b^(1/3)/a^(1/3) + x)/(b^(1/3
)/a^(1/3) + ((-1)^(1/3)*b^(1/3))/a^(1/3))]*Sqrt[((-(((-1)^(2/3)*b^(1/3))/a^(1/3)) - x)*(-(((-1)^(1/3)*b^(1/3))
/a^(1/3)) + x))/(((-1)^(1/3)*b^(1/3))/a^(1/3) + ((-1)^(2/3)*b^(1/3))/a^(1/3))^2]*EllipticPi[((-1)^(1/3)*b^(1/3
) + (-1)^(2/3)*b^(1/3))/((-1)^(1/3)*b^(1/3) - a^(1/3)*Root[b - #1^2 + a*#1^3 & , 3]), ArcSin[Sqrt[((-1)^(1/3)*
b^(1/3) - a^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)])/(a*Sqrt[b + a*x^3]*(-(((-1)^(1/3)*b^(
1/3))/a^(1/3)) + Root[b - #1^2 + a*#1^3 & , 3])*(-Root[b - #1^2 + a*#1^3 & , 1] + Root[b - #1^2 + a*#1^3 & , 3
])*(-Root[b - #1^2 + a*#1^3 & , 2] + Root[b - #1^2 + a*#1^3 & , 3])) + (2*(((-1)^(1/3)*b^(1/3))/a^(1/3) + ((-1
)^(2/3)*b^(1/3))/a^(1/3))*Sqrt[(b^(1/3)/a^(1/3) + x)/(b^(1/3)/a^(1/3) + ((-1)^(1/3)*b^(1/3))/a^(1/3))]*Sqrt[((
-(((-1)^(2/3)*b^(1/3))/a^(1/3)) - x)*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + x))/(((-1)^(1/3)*b^(1/3))/a^(1/3) + ((
-1)^(2/3)*b^(1/3))/a^(1/3))^2]*EllipticPi[((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/3))/((-1)^(1/3)*b^(1/3) - a^(1
/3)*Root[b - #1^2 + a*#1^3 & , 3]), ArcSin[Sqrt[((-1)^(1/3)*b^(1/3) - a^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^
(1/3))]], (-1)^(1/3)]*Root[b - #1^2 + a*#1^3 & , 3]^3)/(Sqrt[b + a*x^3]*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + Roo
t[b - #1^2 + a*#1^3 & , 3])*(-Root[b - #1^2 + a*#1^3 & , 1] + Root[b - #1^2 + a*#1^3 & , 3])*(-Root[b - #1^2 +
a*#1^3 & , 2] + Root[b - #1^2 + a*#1^3 & , 3]))

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IntegrateAlgebraic [A]  time = 0.52, size = 33, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b+a x^3}}{x}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {b+a x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b + a*x^3])/x - 2*ArcTanh[x/Sqrt[b + a*x^3]]

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fricas [A]  time = 27.70, size = 56, normalized size = 1.70 \begin {gather*} \frac {x \log \left (\frac {a x^{3} + x^{2} - 2 \, \sqrt {a x^{3} + b} x + b}{a x^{3} - x^{2} + b}\right ) + 2 \, \sqrt {a x^{3} + b}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*log((a*x^3 + x^2 - 2*sqrt(a*x^3 + b)*x + b)/(a*x^3 - x^2 + b)) + 2*sqrt(a*x^3 + b))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.48, size = 841, normalized size = 25.48

method	result	size
default	\(-\frac {2 i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}}{-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{3 a \sqrt {a \,x^{3}+b}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+3 b \right ) \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{\left (-a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{-3 \left (-a^{2} b \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i a \left (2 x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{2 \left (-a^{2} b \right )^{\frac {1}{3}}}}\, \left (-i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a +i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a +\left (-a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}+\underline {\hspace {1.25 ex}}\alpha \left (-a^{2} b \right )^{\frac {2}{3}} a -\left (-a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a +2 a^{2} b -\left (-a^{2} b \right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}}{3}, \frac {-i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -3 \left (-a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b -i \sqrt {3}\, a b -3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +3 \left (-a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{2 a b}, \sqrt {\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \left (3 \underline {\hspace {1.25 ex}}\alpha a -2\right ) \sqrt {a \,x^{3}+b}}\right )}{a^{2} b}+\frac {2 \sqrt {a \,x^{3}+b}}{x}\)	\(841\)
risch	\(-\frac {2 i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}}{-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{3 a \sqrt {a \,x^{3}+b}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+3 b \right ) \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{\left (-a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{-3 \left (-a^{2} b \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i a \left (2 x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{2 \left (-a^{2} b \right )^{\frac {1}{3}}}}\, \left (-i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a +i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a +\left (-a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}+\underline {\hspace {1.25 ex}}\alpha \left (-a^{2} b \right )^{\frac {2}{3}} a -\left (-a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a +2 a^{2} b -\left (-a^{2} b \right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}}{3}, \frac {-i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -3 \left (-a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b -i \sqrt {3}\, a b -3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +3 \left (-a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{2 a b}, \sqrt {\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \left (3 \underline {\hspace {1.25 ex}}\alpha a -2\right ) \sqrt {a \,x^{3}+b}}\right )}{a^{2} b}+\frac {2 \sqrt {a \,x^{3}+b}}{x}\)	\(841\)
elliptic	\(-\frac {2 i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}}{-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{3 a \sqrt {a \,x^{3}+b}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+3 b \right ) \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{\left (-a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{-3 \left (-a^{2} b \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i a \left (2 x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{2 \left (-a^{2} b \right )^{\frac {1}{3}}}}\, \left (-i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a +i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a +\left (-a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}+\underline {\hspace {1.25 ex}}\alpha \left (-a^{2} b \right )^{\frac {2}{3}} a -\left (-a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a +2 a^{2} b -\left (-a^{2} b \right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a^{2} b \right )^{\frac {1}{3}}}}}{3}, \frac {-i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -3 \left (-a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b -i \sqrt {3}\, a b -3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +3 \left (-a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{2 a b}, \sqrt {\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \left (3 \underline {\hspace {1.25 ex}}\alpha a -2\right ) \sqrt {a \,x^{3}+b}}\right )}{a^{2} b}+\frac {2 \sqrt {a \,x^{3}+b}}{x}\)	\(841\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*I*3^(1/2)/a*(-a^2*b)^(1/3)*(I*(x+1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*3^(1/2)*a/(-a^2*b)^
(1/3))^(1/2)*((x-1/a*(-a^2*b)^(1/3))/(-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3)))^(1/2)*(-I*(x+1/2/
a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*3^(1/2)*a/(-a^2*b)^(1/3))^(1/2)/(a*x^3+b)^(1/2)*EllipticF(1/3
*3^(1/2)*(I*(x+1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*3^(1/2)*a/(-a^2*b)^(1/3))^(1/2),(I*3^(1/2)
/a*(-a^2*b)^(1/3)/(-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3)))^(1/2))-I/a^2/b*2^(1/2)*sum((-_alpha^
2+3*b)/_alpha/(3*_alpha*a-2)*(-a^2*b)^(1/3)*(1/2*I*a*(2*x+1/a*((-a^2*b)^(1/3)-I*3^(1/2)*(-a^2*b)^(1/3)))/(-a^2
*b)^(1/3))^(1/2)*(a*(x-1/a*(-a^2*b)^(1/3))/(-3*(-a^2*b)^(1/3)+I*3^(1/2)*(-a^2*b)^(1/3)))^(1/2)*(-1/2*I*a*(2*x+
1/a*((-a^2*b)^(1/3)+I*3^(1/2)*(-a^2*b)^(1/3)))/(-a^2*b)^(1/3))^(1/2)/(a*x^3+b)^(1/2)*(-I*(-a^2*b)^(1/3)*3^(1/2
)*_alpha^2*a^2+I*(-a^2*b)^(2/3)*3^(1/2)*_alpha*a+I*(-a^2*b)^(1/3)*3^(1/2)*_alpha*a+(-a^2*b)^(1/3)*_alpha^2*a^2
-I*(-a^2*b)^(2/3)*3^(1/2)+_alpha*(-a^2*b)^(2/3)*a-(-a^2*b)^(1/3)*_alpha*a+2*a^2*b-(-a^2*b)^(2/3))*EllipticPi(1
/3*3^(1/2)*(I*(x+1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*3^(1/2)*a/(-a^2*b)^(1/3))^(1/2),1/2/a*(-
I*(-a^2*b)^(2/3)*3^(1/2)*_alpha^2*a+I*3^(1/2)*_alpha*a^2*b+I*(-a^2*b)^(2/3)*3^(1/2)*_alpha-3*(-a^2*b)^(2/3)*_a
lpha^2*a-2*I*(-a^2*b)^(1/3)*3^(1/2)*a*b-I*3^(1/2)*a*b-3*_alpha*a^2*b+3*(-a^2*b)^(2/3)*_alpha+3*a*b)/b,(I*3^(1/
2)/a*(-a^2*b)^(1/3)/(-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*a-_Z^2+b
))+2*(a*x^3+b)^(1/2)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.92, size = 43, normalized size = 1.30 \begin {gather*} \ln \left (\frac {x-\sqrt {a\,x^3+b}}{x+\sqrt {a\,x^3+b}}\right )+\frac {2\,\sqrt {a\,x^3+b}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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