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

Optimal. Leaf size=161 \[ \frac {\left (-2 \sqrt {a} \sqrt {b}-c\right ) \tan ^{-1}\left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3-b x}}{-2 \sqrt {a} \sqrt {b} x+a x^2-b}\right )}{4 a^{3/4} b^{3/4}}+\frac {\left (c-2 \sqrt {a} \sqrt {b}\right ) \tanh ^{-1}\left (\frac {2 \sqrt {a} \sqrt {b} x+a x^2-b}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3-b x}}\right )}{4 a^{3/4} b^{3/4}} \]

________________________________________________________________________________________

Rubi [C]  time = 1.70, antiderivative size = 257, normalized size of antiderivative = 1.60, 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, 6725, 329, 224, 221, 933, 168, 537} \begin {gather*} -\frac {\sqrt {x} \left (2 a \sqrt {b}+\sqrt {-a} c\right ) \sqrt {1-\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {-a}}{\sqrt {a}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{a^{5/4} \sqrt [4]{b} \sqrt {a x^3-b x}}-\frac {\sqrt {x} \left (\frac {c}{\sqrt {-a}}+2 \sqrt {b}\right ) \sqrt {1-\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {a}}{\sqrt {-a}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt [4]{b} \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 + c*x + a*x^2)/((b + 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*a*Sqrt[b] + Sqrt[-a]*c)*Sqrt[x]*Sqrt[1 - (a*x^2)/b]*EllipticPi[Sqrt[-a]/Sqrt[a], ArcSin[(a^(1/
4)*Sqrt[x])/b^(1/4)], -1])/(a^(5/4)*b^(1/4)*Sqrt[-(b*x) + a*x^3]) - ((2*Sqrt[b] + c/Sqrt[-a])*Sqrt[x]*Sqrt[1 -
(a*x^2)/b]*EllipticPi[Sqrt[a]/Sqrt[-a], ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(a^(1/4)*b^(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 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 {-b+c x+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {-b+c x+a x^2}{\sqrt {x} \sqrt {-b+a x^2} \left (b+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+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+a x^2\right )} \, dx}{\sqrt {-b x+a x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \left (\frac {2 b^{3/2}-\frac {b c}{\sqrt {-a}}}{2 b \sqrt {x} \left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt {-b+a x^2}}+\frac {2 b^{3/2}+\frac {b c}{\sqrt {-a}}}{2 b \sqrt {x} \left (\sqrt {b}+\sqrt {-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 (2 \sqrt {b}+\frac {c}{\sqrt {-a}}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}+\sqrt {-a} x\right ) \sqrt {-b+a x^2}} \, dx}{2 \sqrt {-b x+a x^3}}-\frac {\left (\left (2 b^{3/2}-\frac {b c}{\sqrt {-a}}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt {-b+a x^2}} \, dx}{2 b \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 (2 \sqrt {b}+\frac {c}{\sqrt {-a}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}+\sqrt {-a} x\right ) \sqrt {1-\frac {\sqrt {a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {a} x}{\sqrt {b}}}} \, dx}{2 \sqrt {-b x+a x^3}}-\frac {\left (\left (2 b^{3/2}-\frac {b c}{\sqrt {-a}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt {1-\frac {\sqrt {a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {a} x}{\sqrt {b}}}} \, dx}{2 b \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 (2 \sqrt {b}+\frac {c}{\sqrt {-a}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {b}-\sqrt {-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 (\left (2 b^{3/2}-\frac {b c}{\sqrt {-a}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {b}+\sqrt {-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 )}{b \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 a \sqrt {b}+\sqrt {-a} c\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {-a}}{\sqrt {a}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{a^{5/4} \sqrt [4]{b} \sqrt {-b x+a x^3}}-\frac {\left (2 \sqrt {b}+\frac {c}{\sqrt {-a}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {a}}{\sqrt {-a}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.60, size = 136, normalized size = 0.84 \begin {gather*} -\frac {2 x \sqrt {1-\frac {a x^2}{b}} \left (15 b F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {a x^2}{b},-\frac {a x^2}{b}\right )-x \left (5 c F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\frac {a x^2}{b},-\frac {a x^2}{b}\right )+3 a x F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {a x^2}{b},-\frac {a x^2}{b}\right )\right )\right )}{15 b \sqrt {a x^3-b x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*x*Sqrt[1 - (a*x^2)/b]*(15*b*AppellF1[1/4, 1/2, 1, 5/4, (a*x^2)/b, -((a*x^2)/b)] - x*(5*c*AppellF1[3/4, 1/2
, 1, 7/4, (a*x^2)/b, -((a*x^2)/b)] + 3*a*x*AppellF1[5/4, 1/2, 1, 9/4, (a*x^2)/b, -((a*x^2)/b)])))/(15*b*Sqrt[-
(b*x) + a*x^3])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.53, size = 159, normalized size = 0.99 \begin {gather*} \frac {\left (2 \sqrt {a} \sqrt {b}+c\right ) \tan ^{-1}\left (\frac {-b-2 \sqrt {a} \sqrt {b} x+a x^2}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}\right )}{4 a^{3/4} b^{3/4}}+\frac {\left (-2 \sqrt {a} \sqrt {b}+c\right ) \tanh ^{-1}\left (\frac {-b+2 \sqrt {a} \sqrt {b} x+a x^2}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}\right )}{4 a^{3/4} b^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*Sqrt[a]*Sqrt[b] + c)*ArcTan[(-b - 2*Sqrt[a]*Sqrt[b]*x + a*x^2)/(2*a^(1/4)*b^(1/4)*Sqrt[-(b*x) + a*x^3])])/
(4*a^(3/4)*b^(3/4)) + ((-2*Sqrt[a]*Sqrt[b] + c)*ArcTanh[(-b + 2*Sqrt[a]*Sqrt[b]*x + a*x^2)/(2*a^(1/4)*b^(1/4)*
Sqrt[-(b*x) + a*x^3])])/(4*a^(3/4)*b^(3/4))

________________________________________________________________________________________

fricas [B]  time = 0.92, size = 1593, normalized size = 9.89

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(1/2)*sqrt(-(a*b*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^3)) + 4*c)/(a*b))*log(-(16*a^2*b^4 - b^2
*c^4 + (16*a^4*b^2 - a^2*c^4)*x^4 - 6*(16*a^3*b^3 - a*b*c^4)*x^2 + 4*sqrt(1/2)*(4*a^2*b^3*c - a*b^2*c^3 - (4*a
^3*b^2*c - a^2*b*c^3)*x^2 - 4*(4*a^3*b^3 - a^2*b^2*c^2)*x + 2*(a^4*b^3*x^2 - a^3*b^3*c*x - a^3*b^4)*sqrt(-(16*
a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^3)))*sqrt(a*x^3 - b*x)*sqrt(-(a*b*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*
b^3)) + 4*c)/(a*b)) + 4*((4*a^4*b^3 + a^3*b^2*c^2)*x^3 - (4*a^3*b^4 + a^2*b^3*c^2)*x)*sqrt(-(16*a^2*b^2 - 8*a*
b*c^2 + c^4)/(a^3*b^3)))/(a^2*x^4 + 2*a*b*x^2 + b^2)) + 1/8*sqrt(1/2)*sqrt(-(a*b*sqrt(-(16*a^2*b^2 - 8*a*b*c^2
+ c^4)/(a^3*b^3)) + 4*c)/(a*b))*log(-(16*a^2*b^4 - b^2*c^4 + (16*a^4*b^2 - a^2*c^4)*x^4 - 6*(16*a^3*b^3 - a*b
*c^4)*x^2 - 4*sqrt(1/2)*(4*a^2*b^3*c - a*b^2*c^3 - (4*a^3*b^2*c - a^2*b*c^3)*x^2 - 4*(4*a^3*b^3 - a^2*b^2*c^2)
*x + 2*(a^4*b^3*x^2 - a^3*b^3*c*x - a^3*b^4)*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^3)))*sqrt(a*x^3 - b*x
)*sqrt(-(a*b*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^3)) + 4*c)/(a*b)) + 4*((4*a^4*b^3 + a^3*b^2*c^2)*x^3
- (4*a^3*b^4 + a^2*b^3*c^2)*x)*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^3)))/(a^2*x^4 + 2*a*b*x^2 + b^2)) -
1/8*sqrt(1/2)*sqrt((a*b*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^3)) - 4*c)/(a*b))*log(-(16*a^2*b^4 - b^2*
c^4 + (16*a^4*b^2 - a^2*c^4)*x^4 - 6*(16*a^3*b^3 - a*b*c^4)*x^2 + 4*sqrt(1/2)*(4*a^2*b^3*c - a*b^2*c^3 - (4*a^
3*b^2*c - a^2*b*c^3)*x^2 - 4*(4*a^3*b^3 - a^2*b^2*c^2)*x - 2*(a^4*b^3*x^2 - a^3*b^3*c*x - a^3*b^4)*sqrt(-(16*a
^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^3)))*sqrt(a*x^3 - b*x)*sqrt((a*b*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^
3)) - 4*c)/(a*b)) - 4*((4*a^4*b^3 + a^3*b^2*c^2)*x^3 - (4*a^3*b^4 + a^2*b^3*c^2)*x)*sqrt(-(16*a^2*b^2 - 8*a*b*
c^2 + c^4)/(a^3*b^3)))/(a^2*x^4 + 2*a*b*x^2 + b^2)) + 1/8*sqrt(1/2)*sqrt((a*b*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 +
c^4)/(a^3*b^3)) - 4*c)/(a*b))*log(-(16*a^2*b^4 - b^2*c^4 + (16*a^4*b^2 - a^2*c^4)*x^4 - 6*(16*a^3*b^3 - a*b*c^
4)*x^2 - 4*sqrt(1/2)*(4*a^2*b^3*c - a*b^2*c^3 - (4*a^3*b^2*c - a^2*b*c^3)*x^2 - 4*(4*a^3*b^3 - a^2*b^2*c^2)*x
- 2*(a^4*b^3*x^2 - a^3*b^3*c*x - a^3*b^4)*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^3)))*sqrt(a*x^3 - b*x)*s
qrt((a*b*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^3)) - 4*c)/(a*b)) - 4*((4*a^4*b^3 + a^3*b^2*c^2)*x^3 - (4
*a^3*b^4 + a^2*b^3*c^2)*x)*sqrt(-(16*a^2*b^2 - 8*a*b*c^2 + c^4)/(a^3*b^3)))/(a^2*x^4 + 2*a*b*x^2 + b^2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.15, size = 664, normalized size = 4.12

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 {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {b \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 {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{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 {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}+\frac {b \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 {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\)	\(664\)
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 {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {b \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 {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{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 {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}+\frac {b \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 {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\)	\(681\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________