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

Optimal. Leaf size=39 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x^3-b x}}{a x^2-b}\right )}{\sqrt {c}} \]

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Rubi [C]  time = 1.99, antiderivative size = 254, normalized size of antiderivative = 6.51, number of steps used = 14, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2056, 6728, 329, 224, 221, 933, 168, 537} \begin {gather*} -\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {2 \sqrt {a} \sqrt {b}}{c-\sqrt {c^2+4 a b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a x^3-b x}}-\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {2 \sqrt {a} \sqrt {b}}{c+\sqrt {c^2+4 a b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a x^3-b x}}+\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a x^3-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^(1/4)*Sqrt[x]*Sqrt[1 - (a*x^2)/b]*EllipticF[ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(a^(1/4)*Sqrt[-(b*x)
+ a*x^3]) - (2*b^(1/4)*Sqrt[x]*Sqrt[1 - (a*x^2)/b]*EllipticPi[(-2*Sqrt[a]*Sqrt[b])/(c - Sqrt[4*a*b + c^2]), Ar
cSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(a^(1/4)*Sqrt[-(b*x) + a*x^3]) - (2*b^(1/4)*Sqrt[x]*Sqrt[1 - (a*x^2)/b]*
EllipticPi[(-2*Sqrt[a]*Sqrt[b])/(c + Sqrt[4*a*b + c^2]), ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(a^(1/4)*Sqrt
[-(b*x) + a*x^3])

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)
/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 933

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]
), x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] &&  !GtQ[a, 0]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {b+a x^2}{\left (-b+c x+a x^2\right ) \sqrt {-b x+a x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {b+a x^2}{\sqrt {x} \sqrt {-b+a x^2} \left (-b+c x+a x^2\right )} \, dx}{\sqrt {-b x+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {-b+a x^2}}+\frac {2 b-c x}{\sqrt {x} \sqrt {-b+a x^2} \left (-b+c x+a x^2\right )}\right ) \, dx}{\sqrt {-b x+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-b+a x^2}} \, dx}{\sqrt {-b x+a x^3}}+\frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {2 b-c x}{\sqrt {x} \sqrt {-b+a x^2} \left (-b+c x+a x^2\right )} \, dx}{\sqrt {-b x+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \left (\frac {-c+\sqrt {4 a b+c^2}}{\sqrt {x} \left (c-\sqrt {4 a b+c^2}+2 a x\right ) \sqrt {-b+a x^2}}+\frac {-c-\sqrt {4 a b+c^2}}{\sqrt {x} \left (c+\sqrt {4 a b+c^2}+2 a x\right ) \sqrt {-b+a x^2}}\right ) \, dx}{\sqrt {-b x+a x^3}}+\frac {\left (2 \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}\\ &=\frac {\left (\left (-c-\sqrt {4 a b+c^2}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (c+\sqrt {4 a b+c^2}+2 a x\right ) \sqrt {-b+a x^2}} \, dx}{\sqrt {-b x+a x^3}}+\frac {\left (\left (-c+\sqrt {4 a b+c^2}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (c-\sqrt {4 a b+c^2}+2 a x\right ) \sqrt {-b+a x^2}} \, dx}{\sqrt {-b x+a x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a x^4}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}\\ &=\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-b x+a x^3}}+\frac {\left (\left (-c-\sqrt {4 a b+c^2}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (c+\sqrt {4 a b+c^2}+2 a x\right ) \sqrt {1-\frac {\sqrt {a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {a} x}{\sqrt {b}}}} \, dx}{\sqrt {-b x+a x^3}}+\frac {\left (\left (-c+\sqrt {4 a b+c^2}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (c-\sqrt {4 a b+c^2}+2 a x\right ) \sqrt {1-\frac {\sqrt {a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {a} x}{\sqrt {b}}}} \, dx}{\sqrt {-b x+a x^3}}\\ &=\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-b x+a x^3}}-\frac {\left (2 \left (-c-\sqrt {4 a b+c^2}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-c-\sqrt {4 a b+c^2}-2 a x^2\right ) \sqrt {1-\frac {\sqrt {a} x^2}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {a} x^2}{\sqrt {b}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}-\frac {\left (2 \left (-c+\sqrt {4 a b+c^2}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-c+\sqrt {4 a b+c^2}-2 a x^2\right ) \sqrt {1-\frac {\sqrt {a} x^2}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {a} x^2}{\sqrt {b}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}\\ &=\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-b x+a x^3}}-\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {2 \sqrt {a} \sqrt {b}}{c-\sqrt {4 a b+c^2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-b x+a x^3}}-\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {2 \sqrt {a} \sqrt {b}}{c+\sqrt {4 a b+c^2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-b x+a x^3}}\\ \end {align*}

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Mathematica [C]  time = 1.65, size = 204, normalized size = 5.23 \begin {gather*} -\frac {2 i x^{3/2} \sqrt {1-\frac {b}{a x^2}} \left (-\Pi \left (\frac {2 \sqrt {a} \sqrt {b}}{c-\sqrt {c^2+4 a b}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {2 \sqrt {a} \sqrt {b}}{c+\sqrt {c^2+4 a b}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )+F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \sqrt {a x^3-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*Sqrt[1 - b/(a*x^2)]*x^(3/2)*(EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]/Sqrt[x]], -1] - EllipticPi[(
2*Sqrt[a]*Sqrt[b])/(c - Sqrt[4*a*b + c^2]), I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]/Sqrt[x]], -1] - EllipticPi[(2*S
qrt[a]*Sqrt[b])/(c + Sqrt[4*a*b + c^2]), I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]/Sqrt[x]], -1]))/(Sqrt[-(Sqrt[b]/Sq
rt[a])]*Sqrt[-(b*x) + a*x^3])

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IntegrateAlgebraic [A]  time = 0.31, size = 39, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {-b x+a x^3}}{-b+a x^2}\right )}{\sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 172, normalized size = 4.41 \begin {gather*} \left [-\frac {\sqrt {-c} \log \left (\frac {a^{2} x^{4} - 6 \, a c x^{3} + 6 \, b c x - {\left (2 \, a b - c^{2}\right )} x^{2} + b^{2} - 4 \, \sqrt {a x^{3} - b x} {\left (a x^{2} - c x - b\right )} \sqrt {-c}}{a^{2} x^{4} + 2 \, a c x^{3} - 2 \, b c x - {\left (2 \, a b - c^{2}\right )} x^{2} + b^{2}}\right )}{2 \, c}, \frac {\arctan \left (\frac {\sqrt {a x^{3} - b x} {\left (a x^{2} - c x - b\right )} \sqrt {c}}{2 \, {\left (a c x^{3} - b c x\right )}}\right )}{\sqrt {c}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-c)*log((a^2*x^4 - 6*a*c*x^3 + 6*b*c*x - (2*a*b - c^2)*x^2 + b^2 - 4*sqrt(a*x^3 - b*x)*(a*x^2 - c*x
- b)*sqrt(-c))/(a^2*x^4 + 2*a*c*x^3 - 2*b*c*x - (2*a*b - c^2)*x^2 + b^2))/c, arctan(1/2*sqrt(a*x^3 - b*x)*(a*
x^2 - c*x - b)*sqrt(c)/(a*c*x^3 - 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^{2} + b}{\sqrt {a x^{3} - b x} {\left (a x^{2} + c x - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.25, size = 1089, normalized size = 27.92

method	result	size
elliptic	\(\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}+\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}-\frac {-c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c^{2}}{2 \sqrt {4 a b +c^{2}}\, a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}-\frac {-c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}+\frac {2 \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}-\frac {-c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{\sqrt {4 a b +c^{2}}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c^{2}}{2 \sqrt {4 a b +c^{2}}\, a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}-\frac {2 \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{\sqrt {4 a b +c^{2}}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}\)	\(1089\)
default	\(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}+\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}-\frac {-c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c^{2}}{2 \sqrt {4 a b +c^{2}}\, a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}-\frac {-c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}+\frac {2 \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}-\frac {-c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{\sqrt {4 a b +c^{2}}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}-\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c^{2}}{2 \sqrt {4 a b +c^{2}}\, a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}-\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}-\frac {2 \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {c +\sqrt {4 a b +c^{2}}}{2 a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{\sqrt {4 a b +c^{2}}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {c}{2 a}+\frac {\sqrt {4 a b +c^{2}}}{2 a}\right )}\)	\(1106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(a*b)^(1/2)*(x*a/(a*b)^(1/2)+1)^(1/2)*(-2*x*a/(a*b)^(1/2)+2)^(1/2)*(-x*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1
/2)*EllipticF(((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/2/(4*a*b+c^2)^(1/2)/a^2*(a*b)^(1/2)*(x*
a/(a*b)^(1/2)+1)^(1/2)*(-2*x*a/(a*b)^(1/2)+2)^(1/2)*(-x*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1/2)/(-1/a*(a*b)^(1/
2)+1/2*c/a-1/2/a*(4*a*b+c^2)^(1/2))*EllipticPi(((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),-1/a*(a*b)^(1/2)/(-1/
a*(a*b)^(1/2)-1/2/a*(-c+(4*a*b+c^2)^(1/2))),1/2*2^(1/2))*c^2-1/2/a^2*(a*b)^(1/2)*(x*a/(a*b)^(1/2)+1)^(1/2)*(-2
*x*a/(a*b)^(1/2)+2)^(1/2)*(-x*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1/2)/(-1/a*(a*b)^(1/2)+1/2*c/a-1/2/a*(4*a*b+c^
2)^(1/2))*EllipticPi(((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),-1/a*(a*b)^(1/2)/(-1/a*(a*b)^(1/2)-1/2/a*(-c+(4
*a*b+c^2)^(1/2))),1/2*2^(1/2))*c+2/(4*a*b+c^2)^(1/2)/a*(a*b)^(1/2)*(x*a/(a*b)^(1/2)+1)^(1/2)*(-2*x*a/(a*b)^(1/
2)+2)^(1/2)*(-x*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1/2)/(-1/a*(a*b)^(1/2)+1/2*c/a-1/2/a*(4*a*b+c^2)^(1/2))*Elli
pticPi(((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),-1/a*(a*b)^(1/2)/(-1/a*(a*b)^(1/2)-1/2/a*(-c+(4*a*b+c^2)^(1/2
))),1/2*2^(1/2))*b-1/2/(4*a*b+c^2)^(1/2)/a^2*(a*b)^(1/2)*(x*a/(a*b)^(1/2)+1)^(1/2)*(-2*x*a/(a*b)^(1/2)+2)^(1/2
)*(-x*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1/2)/(-1/a*(a*b)^(1/2)+1/2*c/a+1/2/a*(4*a*b+c^2)^(1/2))*EllipticPi(((x
+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),-1/a*(a*b)^(1/2)/(-1/a*(a*b)^(1/2)+1/2*(c+(4*a*b+c^2)^(1/2))/a),1/2*2^(
1/2))*c^2-1/2/a^2*(a*b)^(1/2)*(x*a/(a*b)^(1/2)+1)^(1/2)*(-2*x*a/(a*b)^(1/2)+2)^(1/2)*(-x*a/(a*b)^(1/2))^(1/2)/
(a*x^3-b*x)^(1/2)/(-1/a*(a*b)^(1/2)+1/2*c/a+1/2/a*(4*a*b+c^2)^(1/2))*EllipticPi(((x+1/a*(a*b)^(1/2))*a/(a*b)^(
1/2))^(1/2),-1/a*(a*b)^(1/2)/(-1/a*(a*b)^(1/2)+1/2*(c+(4*a*b+c^2)^(1/2))/a),1/2*2^(1/2))*c-2/(4*a*b+c^2)^(1/2)
/a*(a*b)^(1/2)*(x*a/(a*b)^(1/2)+1)^(1/2)*(-2*x*a/(a*b)^(1/2)+2)^(1/2)*(-x*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1/
2)/(-1/a*(a*b)^(1/2)+1/2*c/a+1/2/a*(4*a*b+c^2)^(1/2))*EllipticPi(((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),-1/
a*(a*b)^(1/2)/(-1/a*(a*b)^(1/2)+1/2*(c+(4*a*b+c^2)^(1/2))/a),1/2*2^(1/2))*b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.19, size = 51, normalized size = 1.31 \begin {gather*} \frac {\ln \left (\frac {b+c\,x-a\,x^2-\sqrt {c}\,\sqrt {a\,x^3-b\,x}\,2{}\mathrm {i}}{a\,x^2+c\,x-b}\right )\,1{}\mathrm {i}}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((b + c*x - a*x^2 - c^(1/2)*(a*x^3 - b*x)^(1/2)*2i)/(c*x - b + a*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^{2} + b}{\sqrt {x \left (a x^{2} - b\right )} \left (a x^{2} - b + c x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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