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

Optimal. Leaf size=41 \[ \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x \sqrt {a x^3-b}}{b-a x^3}\right ) \]

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Rubi [F]  time = 1.23, 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 {2 b+a x^3}{\sqrt {-b+a x^3} \left (-2 b-3 x^2+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]*(-2*b - 3*x^2 + 2*a*x^3)),x]

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 6.33, size = 2865, normalized size = 69.88 \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]*(-2*b - 3*x^2 + 2*a*x^3)),x]

[Out]

(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))]*S
qrt[-b + a*x^3]) + (2*(-(((-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[-2*b - 3*#1^2 + 2*a*#1^3 & , 1]), A
rcSin[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[-2*b - 3*#1^2 + 2*a*#1^3 & , 1])*(Root[-2*b - 3*#1^2 + 2*a*#1^3 & ,
1] - Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 2])*(Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 1] - Root[-2*b - 3*#1^2 + 2*a*#
1^3 & , 3])) + ((-(((-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[-2*b - 3*#1^2 + 2*a*#1^3 & , 1]), ArcSin[Sq
rt[((-1)^(1/3)*b^(1/3) + a^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*b^(1/3))]], (-1)^(1/3)]*Root[-2*b - 3*#1^2 + 2*
a*#1^3 & , 1]^3)/(Sqrt[-b + a*x^3]*(((-1)^(1/3)*b^(1/3))/a^(1/3) + Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 1])*(Root
[-2*b - 3*#1^2 + 2*a*#1^3 & , 1] - Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 2])*(Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 1]
- Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 3])) + (2*(-(((-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[-2*b - 3*#
1^2 + 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[-2*b - 3*#1^2 + 2*a*#1^3 & , 2])*(-Root[-2
*b - 3*#1^2 + 2*a*#1^3 & , 1] + Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 2])*(Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 2] -
Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 3])) + ((-(((-1)^(1/3)*b^(1/3))/a^(1/3)) - ((-1)^(2/3)*b^(1/3))/a^(1/3))*Sqr
t[(-(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[-2*b - 3*#1^2 + 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
)]*Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 2]^3)/(Sqrt[-b + a*x^3]*(((-1)^(1/3)*b^(1/3))/a^(1/3) + Root[-2*b - 3*#1^
2 + 2*a*#1^3 & , 2])*(-Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 1] + Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 2])*(Root[-2*b
- 3*#1^2 + 2*a*#1^3 & , 2] - Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 3])) + (2*(-(((-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[-2*b - 3*#1^2 + 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[-2*b - 3*#1^2 +
2*a*#1^3 & , 3])*(-Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 1] + Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 3])*(-Root[-2*b -
3*#1^2 + 2*a*#1^3 & , 2] + Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 3])) + ((-(((-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))]*Sq
rt[((((-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[-2*b - 3*#1^2 + 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[-2*b - 3*#1^2 + 2*a*#1^3 & , 3]^3)/(Sqrt[-b + a*x^3]*(((-1)^(1/3)*b^(1/3))
/a^(1/3) + Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 3])*(-Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 1] + Root[-2*b - 3*#1^2 +
2*a*#1^3 & , 3])*(-Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 2] + Root[-2*b - 3*#1^2 + 2*a*#1^3 & , 3]))

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IntegrateAlgebraic [A]  time = 0.59, size = 41, normalized size = 1.00 \begin {gather*} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x \sqrt {-b+a x^3}}{b-a x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.51, size = 123, normalized size = 3.00 \begin {gather*} \frac {1}{12} \, \sqrt {3} \sqrt {2} \log \left (\frac {4 \, a^{2} x^{6} + 36 \, a x^{5} - 8 \, a b x^{3} + 9 \, x^{4} - 36 \, b x^{2} - 4 \, \sqrt {3} \sqrt {2} {\left (2 \, a x^{4} + 3 \, x^{3} - 2 \, b x\right )} \sqrt {a x^{3} - b} + 4 \, b^{2}}{4 \, a^{2} x^{6} - 12 \, a x^{5} - 8 \, a b x^{3} + 9 \, x^{4} + 12 \, b x^{2} + 4 \, b^{2}}\right ) \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)/(2*a*x^3-3*x^2-2*b),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*sqrt(2)*log((4*a^2*x^6 + 36*a*x^5 - 8*a*b*x^3 + 9*x^4 - 36*b*x^2 - 4*sqrt(3)*sqrt(2)*(2*a*x^4 + 3
*x^3 - 2*b*x)*sqrt(a*x^3 - b) + 4*b^2)/(4*a^2*x^6 - 12*a*x^5 - 8*a*b*x^3 + 9*x^4 + 12*b*x^2 + 4*b^2))

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

[Out]

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

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maple [C]  time = 0.20, size = 790, normalized size = 19.27

method	result	size
default	\(\frac {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 (2 a \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2}-2 b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2}-2 b \right ) \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}}}}\, \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 {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}}}}\, \left (-2 i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+2 i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a +3 i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a -2 \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-3 i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}-2 \underline {\hspace {1.25 ex}}\alpha \left (a^{2} b \right )^{\frac {2}{3}} a +3 \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a +4 a^{2} b +3 \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 {2 i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,a^{2} b -3 i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -6 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a -4 i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b -3 i \sqrt {3}\, a b +6 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +9 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -9 a b}{6 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 (\underline {\hspace {1.25 ex}}\alpha a -1\right ) \sqrt {a \,x^{3}-b}}\right )}{12 a^{2} b}\)	\(790\)
elliptic	\(\frac {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 (2 a \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2}-2 b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2}-2 b \right ) \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}}}}\, \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 {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}}}}\, \left (-2 i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+2 i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a +3 i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a -2 \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-3 i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}-2 \underline {\hspace {1.25 ex}}\alpha \left (a^{2} b \right )^{\frac {2}{3}} a +3 \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a +4 a^{2} b +3 \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 {2 i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,a^{2} b -3 i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -6 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a -4 i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b -3 i \sqrt {3}\, a b +6 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +9 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -9 a b}{6 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 (\underline {\hspace {1.25 ex}}\alpha a -1\right ) \sqrt {a \,x^{3}-b}}\right )}{12 a^{2} b}\)	\(790\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*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))+1/12*I/a^2/b*2^(1/2)*sum((-_alpha^2-2*b)/_al
pha/(_alpha*a-1)*(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)*(-2*I*(a^2*b)^(1/3)*3^(1/2)*_alpha^2*a^2+2*I*(a
^2*b)^(2/3)*3^(1/2)*_alpha*a+3*I*(a^2*b)^(1/3)*3^(1/2)*_alpha*a-2*(a^2*b)^(1/3)*_alpha^2*a^2-3*I*(a^2*b)^(2/3)
*3^(1/2)-2*_alpha*(a^2*b)^(2/3)*a+3*(a^2*b)^(1/3)*_alpha*a+4*a^2*b+3*(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),-1/6/a*(2*I*(a^2*b)^(2/3
)*3^(1/2)*_alpha^2*a+2*I*3^(1/2)*_alpha*a^2*b-3*I*(a^2*b)^(2/3)*3^(1/2)*_alpha-6*(a^2*b)^(2/3)*_alpha^2*a-4*I*
(a^2*b)^(1/3)*3^(1/2)*a*b-3*I*3^(1/2)*a*b+6*_alpha*a^2*b+9*(a^2*b)^(2/3)*_alpha-9*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(2*_Z^3*a-3*_Z^2-2*b))

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

[Out]

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

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mupad [B]  time = 1.64, size = 56, normalized size = 1.37 \begin {gather*} \frac {\sqrt {6}\,\ln \left (\frac {2\,b-2\,a\,x^3-3\,x^2+2\,\sqrt {6}\,x\,\sqrt {a\,x^3-b}}{-2\,a\,x^3+3\,x^2+2\,b}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6^(1/2)*log((2*b - 2*a*x^3 - 3*x^2 + 2*6^(1/2)*x*(a*x^3 - b)^(1/2))/(2*b - 2*a*x^3 + 3*x^2)))/6

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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)/(2*a*x**3-3*x**2-2*b),x)

[Out]

Timed out

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