3.23.100 \(\int \frac {\sqrt {b^2+a^2 x^3} (2 b^2+c x^3+a^2 x^6)}{x (-b^2+a^2 x^6)} \, dx\)

Optimal. Leaf size=176 \[ \frac {\left (c \sqrt {b-a}-3 a b \sqrt {b-a}\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {b-a}}\right )}{3 a \sqrt {b}}+\frac {\left (-c \sqrt {a+b}-3 a b \sqrt {a+b}\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )}{3 a \sqrt {b}}+\frac {2}{3} \sqrt {a^2 x^3+b^2}+\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right ) \]

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Rubi [A]  time = 0.78, antiderivative size = 223, normalized size of antiderivative = 1.27, number of steps used = 14, number of rules used = 7, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6725, 266, 50, 63, 208, 444, 205} \begin {gather*} \frac {(3 a b-c) \sqrt {a^2 x^3+b^2}}{3 a b}+\frac {(3 a b+c) \sqrt {a^2 x^3+b^2}}{3 a b}-\frac {\sqrt {a-b} (3 a b-c) \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )}{3 a \sqrt {b}}-\frac {\sqrt {a+b} (3 a b+c) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )}{3 a \sqrt {b}}-\frac {4}{3} \sqrt {a^2 x^3+b^2}+\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x*(-b^2 + a^2*x^6)),x]

[Out]

(-4*Sqrt[b^2 + a^2*x^3])/3 + ((3*a*b - c)*Sqrt[b^2 + a^2*x^3])/(3*a*b) + ((3*a*b + c)*Sqrt[b^2 + a^2*x^3])/(3*
a*b) - (Sqrt[a - b]*(3*a*b - c)*ArcTan[Sqrt[b^2 + a^2*x^3]/(Sqrt[a - b]*Sqrt[b])])/(3*a*Sqrt[b]) + (4*b*ArcTan
h[Sqrt[b^2 + a^2*x^3]/b])/3 - (Sqrt[a + b]*(3*a*b + c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/(Sqrt[b]*Sqrt[a + b])])/(3*
a*Sqrt[b])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x \left (-b^2+a^2 x^6\right )} \, dx &=\int \left (-\frac {2 \sqrt {b^2+a^2 x^3}}{x}-\frac {(3 a b+c) x^2 \sqrt {b^2+a^2 x^3}}{2 b \left (b-a x^3\right )}+\frac {(3 a b-c) x^2 \sqrt {b^2+a^2 x^3}}{2 b \left (b+a x^3\right )}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {b^2+a^2 x^3}}{x} \, dx\right )+\frac {(3 a b-c) \int \frac {x^2 \sqrt {b^2+a^2 x^3}}{b+a x^3} \, dx}{2 b}-\frac {(3 a b+c) \int \frac {x^2 \sqrt {b^2+a^2 x^3}}{b-a x^3} \, dx}{2 b}\\ &=-\left (\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x} \, dx,x,x^3\right )\right )+\frac {(3 a b-c) \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{b+a x} \, dx,x,x^3\right )}{6 b}-\frac {(3 a b+c) \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{b-a x} \, dx,x,x^3\right )}{6 b}\\ &=-\frac {4}{3} \sqrt {b^2+a^2 x^3}+\frac {(3 a b-c) \sqrt {b^2+a^2 x^3}}{3 a b}+\frac {(3 a b+c) \sqrt {b^2+a^2 x^3}}{3 a b}-\frac {1}{3} \left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-\frac {1}{6} ((a-b) (3 a b-c)) \operatorname {Subst}\left (\int \frac {1}{(b+a x) \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-\frac {1}{6} ((a+b) (3 a b+c)) \operatorname {Subst}\left (\int \frac {1}{(b-a x) \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )\\ &=-\frac {4}{3} \sqrt {b^2+a^2 x^3}+\frac {(3 a b-c) \sqrt {b^2+a^2 x^3}}{3 a b}+\frac {(3 a b+c) \sqrt {b^2+a^2 x^3}}{3 a b}-\frac {\left (4 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 a^2}-\frac {((a-b) (3 a b-c)) \operatorname {Subst}\left (\int \frac {1}{b-\frac {b^2}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 a^2}-\frac {((a+b) (3 a b+c)) \operatorname {Subst}\left (\int \frac {1}{b+\frac {b^2}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 a^2}\\ &=-\frac {4}{3} \sqrt {b^2+a^2 x^3}+\frac {(3 a b-c) \sqrt {b^2+a^2 x^3}}{3 a b}+\frac {(3 a b+c) \sqrt {b^2+a^2 x^3}}{3 a b}-\frac {\sqrt {a-b} (3 a b-c) \tan ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )}{3 a \sqrt {b}}+\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )-\frac {\sqrt {a+b} (3 a b+c) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 a \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 194, normalized size = 1.10 \begin {gather*} \frac {c \sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )-\sqrt {a+b} (3 a b+c) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )+2 a \sqrt {b} \sqrt {a^2 x^3+b^2}-3 a b \sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )+4 a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 a \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x*(-b^2 + a^2*x^6)),x]

[Out]

(2*a*Sqrt[b]*Sqrt[b^2 + a^2*x^3] - 3*a*Sqrt[a - b]*b*ArcTan[Sqrt[b^2 + a^2*x^3]/(Sqrt[a - b]*Sqrt[b])] + Sqrt[
a - b]*c*ArcTan[Sqrt[b^2 + a^2*x^3]/(Sqrt[a - b]*Sqrt[b])] + 4*a*b^(3/2)*ArcTanh[Sqrt[b^2 + a^2*x^3]/b] - Sqrt
[a + b]*(3*a*b + c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/(Sqrt[b]*Sqrt[a + b])])/(3*a*Sqrt[b])

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IntegrateAlgebraic [A]  time = 0.24, size = 176, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt {b^2+a^2 x^3}+\frac {\left (-3 a \sqrt {a-b} b+\sqrt {a-b} c\right ) \tan ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )}{3 a \sqrt {b}}+\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )+\frac {\left (-3 a b \sqrt {a+b}-\sqrt {a+b} c\right ) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 a \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x*(-b^2 + a^2*x^6)),x]

[Out]

(2*Sqrt[b^2 + a^2*x^3])/3 + ((-3*a*Sqrt[a - b]*b + Sqrt[a - b]*c)*ArcTan[Sqrt[b^2 + a^2*x^3]/(Sqrt[a - b]*Sqrt
[b])])/(3*a*Sqrt[b]) + (4*b*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/3 + ((-3*a*b*Sqrt[a + b] - Sqrt[a + b]*c)*ArcTanh[
Sqrt[b^2 + a^2*x^3]/(Sqrt[b]*Sqrt[a + b])])/(3*a*Sqrt[b])

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fricas [A]  time = 0.74, size = 735, normalized size = 4.18 \begin {gather*} \left [\frac {4 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 4 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - {\left (3 \, a b - c\right )} \sqrt {-\frac {a - b}{b}} \log \left (\frac {a^{2} x^{3} - a b + 2 \, b^{2} + 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {-\frac {a - b}{b}}}{a x^{3} + b}\right ) + {\left (3 \, a b + c\right )} \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} x^{3} + a b + 2 \, b^{2} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {\frac {a + b}{b}}}{a x^{3} - b}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2}} a}{6 \, a}, \frac {4 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 4 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, {\left (3 \, a b - c\right )} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + {\left (3 \, a b + c\right )} \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} x^{3} + a b + 2 \, b^{2} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {\frac {a + b}{b}}}{a x^{3} - b}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2}} a}{6 \, a}, \frac {4 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 4 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, {\left (3 \, a b + c\right )} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - {\left (3 \, a b - c\right )} \sqrt {-\frac {a - b}{b}} \log \left (\frac {a^{2} x^{3} - a b + 2 \, b^{2} + 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {-\frac {a - b}{b}}}{a x^{3} + b}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2}} a}{6 \, a}, \frac {2 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 2 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (3 \, a b + c\right )} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + {\left (3 \, a b - c\right )} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} a}{3 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x/(a^2*x^6-b^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.30, size = 183, normalized size = 1.04 \begin {gather*} \frac {2}{3} \, b \log \left ({\left | b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right ) - \frac {2}{3} \, b \log \left ({\left | -b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right ) + \frac {2}{3} \, \sqrt {a^{2} x^{3} + b^{2}} - \frac {{\left (3 \, a^{2} b - 3 \, a b^{2} - a c + b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {a b - b^{2}}}\right )}{3 \, \sqrt {a b - b^{2}} a} + \frac {{\left (3 \, a^{2} b + 3 \, a b^{2} + a c + b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {-a b - b^{2}}}\right )}{3 \, \sqrt {-a b - b^{2}} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x/(a^2*x^6-b^2),x, algorithm="giac")

[Out]

2/3*b*log(abs(b + sqrt(a^2*x^3 + b^2))) - 2/3*b*log(abs(-b + sqrt(a^2*x^3 + b^2))) + 2/3*sqrt(a^2*x^3 + b^2) -
1/3*(3*a^2*b - 3*a*b^2 - a*c + b*c)*arctan(sqrt(a^2*x^3 + b^2)/sqrt(a*b - b^2))/(sqrt(a*b - b^2)*a) + 1/3*(3*
a^2*b + 3*a*b^2 + a*c + b*c)*arctan(sqrt(a^2*x^3 + b^2)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*a)

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maple [C]  time = 0.79, size = 943, normalized size = 5.36

method	result	size
default	\(\frac {\left (3 a b +c \right ) \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3 a}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i a \left (2 x +\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, a -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a -\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 b^{2}}{2 b \left (a +b \right )}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{2}}\right )}{2 b}+\frac {\left (3 a b -c \right ) \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3 a}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i a \left (2 x +\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, a -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a -\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 b^{2}}{2 b \left (a -b \right )}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{2}}\right )}{2 b}-\frac {4 \sqrt {a^{2} x^{3}+b^{2}}}{3}+\frac {4 b^{2} \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\)	\(943\)
elliptic	\(\text {Expression too large to display}\)	\(3175\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x/(a^2*x^6-b^2),x,method=_RETURNVERBOSE)

[Out]

1/2*(3*a*b+c)/b*(2/3/a*(a^2*x^3+b^2)^(1/2)+1/3*I/a^2*2^(1/2)*sum((-a*b^2)^(1/3)*(1/2*I*a*(2*x+1/a*(-I*3^(1/2)*
(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(a*(x-1/a*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(
-a*b^2)^(1/3)))^(1/2)*(-1/2*I*a*(2*x+1/a*(I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(a^2
*x^3+b^2)^(1/2)*(I*(-a*b^2)^(1/3)*_alpha*3^(1/2)*a-I*3^(1/2)*(-a*b^2)^(2/3)+2*_alpha^2*a^2-(-a*b^2)^(1/3)*_alp
ha*a-(-a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/a*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/a*(-a*b^2)^(1/3))*3^(1/2)
*a/(-a*b^2)^(1/3))^(1/2),-1/2*(2*I*(-a*b^2)^(1/3)*3^(1/2)*_alpha^2*a-I*(-a*b^2)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)
*b^2-3*(-a*b^2)^(2/3)*_alpha-3*b^2)/b/(a+b),(I*3^(1/2)/a*(-a*b^2)^(1/3)/(-3/2/a*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/a
*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*a-b)))+1/2*(3*a*b-c)/b*(2/3/a*(a^2*x^3+b^2)^(1/2)+1/3*I/a^2*2^(1/2
)*sum((-a*b^2)^(1/3)*(1/2*I*a*(2*x+1/a*(-I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(a*(x
-1/a*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*a*(2*x+1/a*(I*3^(1/2)*(-a*b^2
)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(a^2*x^3+b^2)^(1/2)*(I*(-a*b^2)^(1/3)*_alpha*3^(1/2)*a-I*3^(1/2
)*(-a*b^2)^(2/3)+2*_alpha^2*a^2-(-a*b^2)^(1/3)*_alpha*a-(-a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/a*(-a
*b^2)^(1/3)-1/2*I*3^(1/2)/a*(-a*b^2)^(1/3))*3^(1/2)*a/(-a*b^2)^(1/3))^(1/2),1/2*(2*I*(-a*b^2)^(1/3)*3^(1/2)*_a
lpha^2*a-I*(-a*b^2)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)*b^2-3*(-a*b^2)^(2/3)*_alpha-3*b^2)/b/(a-b),(I*3^(1/2)/a*(-a
*b^2)^(1/3)/(-3/2/a*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/a*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*a+b)))-4/3*(a^2*
x^3+b^2)^(1/2)+4/3*b^2*arctanh((a^2*x^3+b^2)^(1/2)/(b^2)^(1/2))/(b^2)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} x^{6} + c x^{3} + 2 \, b^{2}\right )} \sqrt {a^{2} x^{3} + b^{2}}}{{\left (a^{2} x^{6} - b^{2}\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x/(a^2*x^6-b^2),x, algorithm="maxima")

[Out]

integrate((a^2*x^6 + c*x^3 + 2*b^2)*sqrt(a^2*x^3 + b^2)/((a^2*x^6 - b^2)*x), x)

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mupad [B]  time = 9.52, size = 204, normalized size = 1.16 \begin {gather*} \frac {2\,\sqrt {a^2\,x^3+b^2}}{3}+\frac {2\,b\,\ln \left (\frac {{\left (b+\sqrt {a^2\,x^3+b^2}\right )}^3\,\left (b-\sqrt {a^2\,x^3+b^2}\right )}{x^6}\right )}{3}+\frac {\ln \left (\frac {a\,b+2\,b^2+a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a+b}}{b-a\,x^3}\right )\,\sqrt {a+b}\,\left (c+3\,a\,b\right )}{6\,a\,\sqrt {b}}+\frac {\ln \left (\frac {2\,b^2-a\,b+a^2\,x^3+\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a-b}\,2{}\mathrm {i}}{a\,x^3+b}\right )\,\sqrt {a-b}\,\left (c-3\,a\,b\right )\,1{}\mathrm {i}}{6\,a\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b^2 + a^2*x^3)^(1/2)*(c*x^3 + 2*b^2 + a^2*x^6))/(x*(b^2 - a^2*x^6)),x)

[Out]

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

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sympy [A]  time = 114.32, size = 162, normalized size = 0.92 \begin {gather*} - \frac {2 b \log {\left (b - \sqrt {a^{2} x^{3} + b^{2}} \right )}}{3} + \frac {2 b \log {\left (b + \sqrt {a^{2} x^{3} + b^{2}} \right )}}{3} + \frac {2 \sqrt {a^{2} x^{3} + b^{2}}}{3} - \frac {\left (a - b\right ) \left (3 a b - c\right ) \operatorname {atan}{\left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {a b - b^{2}}} \right )}}{3 a \sqrt {a b - b^{2}}} + \frac {\left (a + b\right ) \left (3 a b + c\right ) \operatorname {atan}{\left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {- a b - b^{2}}} \right )}}{3 a \sqrt {- a b - b^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*x**3+b**2)**(1/2)*(a**2*x**6+c*x**3+2*b**2)/x/(a**2*x**6-b**2),x)

[Out]

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

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