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

Optimal. Leaf size=26 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a x^3+b}}\right )}{\sqrt {c}} \]

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Rubi [F]  time = 0.91, 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 (b+c 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]*(b + c*x^2 + a*x^3)),x]

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 6.32, size = 2725, normalized size = 104.81 \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]*(b + c*x^2 + a*x^3)),x]

[Out]

(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)])/(S
qrt[(-(((-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))]*Sq
rt[b + a*x^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 + c*#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*Sqrt[b + a*x^3]*(-(((-1)
^(1/3)*b^(1/3))/a^(1/3)) + Root[b + c*#1^2 + a*#1^3 & , 1])*(Root[b + c*#1^2 + a*#1^3 & , 1] - Root[b + c*#1^2
+ a*#1^3 & , 2])*(Root[b + c*#1^2 + a*#1^3 & , 1] - Root[b + c*#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 + c*#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 + c*#1^2 + a*#1^3 & , 1]^3)/(Sqrt[b + a*x^3]*(-(((-1)^(1/3)*b
^(1/3))/a^(1/3)) + Root[b + c*#1^2 + a*#1^3 & , 1])*(Root[b + c*#1^2 + a*#1^3 & , 1] - Root[b + c*#1^2 + a*#1^
3 & , 2])*(Root[b + c*#1^2 + a*#1^3 & , 1] - Root[b + c*#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 + c*#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 + c*#1^2 + a
*#1^3 & , 2])*(-Root[b + c*#1^2 + a*#1^3 & , 1] + Root[b + c*#1^2 + a*#1^3 & , 2])*(Root[b + c*#1^2 + a*#1^3 &
, 2] - Root[b + c*#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))*S
qrt[(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 + c*#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)]*Roo
t[b + c*#1^2 + a*#1^3 & , 2]^3)/(Sqrt[b + a*x^3]*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + Root[b + c*#1^2 + a*#1^3 &
, 2])*(-Root[b + c*#1^2 + a*#1^3 & , 1] + Root[b + c*#1^2 + a*#1^3 & , 2])*(Root[b + c*#1^2 + a*#1^3 & , 2] -
Root[b + c*#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]*E
llipticPi[((-1)^(1/3)*b^(1/3) + (-1)^(2/3)*b^(1/3))/((-1)^(1/3)*b^(1/3) - a^(1/3)*Root[b + c*#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 + c*#1^2 + a*#1^3 & , 3])*(-Root[b + c*#1^2 + a*#1^3 & ,
1] + Root[b + c*#1^2 + a*#1^3 & , 3])*(-Root[b + c*#1^2 + a*#1^3 & , 2] + Root[b + c*#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 + c*#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 + c*#1^2 + a*#1^3 & , 3]^3)/(Sqrt[b + a
*x^3]*(-(((-1)^(1/3)*b^(1/3))/a^(1/3)) + Root[b + c*#1^2 + a*#1^3 & , 3])*(-Root[b + c*#1^2 + a*#1^3 & , 1] +
Root[b + c*#1^2 + a*#1^3 & , 3])*(-Root[b + c*#1^2 + a*#1^3 & , 2] + Root[b + c*#1^2 + a*#1^3 & , 3]))

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IntegrateAlgebraic [A]  time = 0.57, size = 26, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b+a x^3}}\right )}{\sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.52, size = 169, normalized size = 6.50 \begin {gather*} \left [-\frac {\sqrt {-c} \log \left (\frac {a^{2} x^{6} - 6 \, a c x^{5} + c^{2} x^{4} + 2 \, a b x^{3} - 6 \, b c x^{2} + b^{2} - 4 \, {\left (a x^{4} - c x^{3} + b x\right )} \sqrt {a x^{3} + b} \sqrt {-c}}{a^{2} x^{6} + 2 \, a c x^{5} + c^{2} x^{4} + 2 \, a b x^{3} + 2 \, b c x^{2} + b^{2}}\right )}{2 \, c}, \frac {\arctan \left (\frac {{\left (a x^{3} - c x^{2} + b\right )} \sqrt {a x^{3} + b} \sqrt {c}}{2 \, {\left (a c x^{4} + b c x\right )}}\right )}{\sqrt {c}}\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)/(a*x^3+c*x^2+b),x, algorithm="fricas")

[Out]

[-1/2*sqrt(-c)*log((a^2*x^6 - 6*a*c*x^5 + c^2*x^4 + 2*a*b*x^3 - 6*b*c*x^2 + b^2 - 4*(a*x^4 - c*x^3 + b*x)*sqrt
(a*x^3 + b)*sqrt(-c))/(a^2*x^6 + 2*a*c*x^5 + c^2*x^4 + 2*a*b*x^3 + 2*b*c*x^2 + b^2))/c, arctan(1/2*(a*x^3 - c*
x^2 + b)*sqrt(a*x^3 + b)*sqrt(c)/(a*c*x^4 + b*c*x))/sqrt(c)]

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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 (a x^{3} + c x^{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)/(a*x^3+c*x^2+b),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.20, size = 841, normalized size = 32.35

method	result	size
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}+c \,\textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} c +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 c +i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, c +\left (-a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+\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 c +\left (-a^{2} b \right )^{\frac {2}{3}} c +2 a^{2} b \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 \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha c +i \sqrt {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 ^{2} a -2 i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b +i \sqrt {3}\, a b c -3 \left (-a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha c -3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b -3 a b c}{2 a b c}, \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 c \right ) \sqrt {a \,x^{3}+b}}\right )}{a^{2} b c}\)	\(841\)
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}+c \,\textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} c -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 c +i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, c +\left (-a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+\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 c +\left (-a^{2} b \right )^{\frac {2}{3}} c +2 a^{2} b \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 \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha c +i \sqrt {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 ^{2} a -2 i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b +i \sqrt {3}\, a b c -3 \left (-a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha c -3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b -3 a b c}{2 a b c}, \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 c \right ) \sqrt {a \,x^{3}+b}}\right )}{a^{2} b c}\)	\(842\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3-2*b)/(a*x^3+b)^(1/2)/(a*x^3+c*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/c*2^(1/2)*sum((_alpha
^2*c+3*b)/_alpha/(3*_alpha*a+2*c)*(-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*c+I*(-a^2*b)^(2/3)*3^(
1/2)*c+(-a^2*b)^(1/3)*_alpha^2*a^2+_alpha*(-a^2*b)^(2/3)*a+(-a^2*b)^(1/3)*_alpha*a*c+(-a^2*b)^(2/3)*c+2*a^2*b)
*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*(-a^2*b)^(2/3)*3^(1/2)*_alpha*c+I*3^(1/2)*_alpha*a^2*b-3*(
-a^2*b)^(2/3)*_alpha^2*a-2*I*(-a^2*b)^(1/3)*3^(1/2)*a*b+I*3^(1/2)*a*b*c-3*(-a^2*b)^(2/3)*_alpha*c-3*_alpha*a^2
*b-3*a*b*c)/b/c,(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)),_al
pha=RootOf(_Z^3*a+_Z^2*c+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 (a x^{3} + c x^{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)/(a*x^3+c*x^2+b),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.38, size = 51, normalized size = 1.96 \begin {gather*} \frac {\ln \left (\frac {b+a\,x^3-c\,x^2+\sqrt {c}\,x\,\sqrt {a\,x^3+b}\,2{}\mathrm {i}}{a\,x^3+c\,x^2+b}\right )\,1{}\mathrm {i}}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral((a*x**3 - 2*b)/(sqrt(a*x**3 + b)*(a*x**3 + b + c*x**2)), x)

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