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

Optimal. Leaf size=147 \[ -\frac {\left (2 \sqrt {a} \sqrt {b}-c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (2 \sqrt {a} \sqrt {b}+c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}} \]

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Rubi [C]  time = 1.50, antiderivative size = 284, normalized size of antiderivative = 1.93, number of steps used = 13, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 6725, 329, 220, 933, 168, 537} \begin {gather*} -\frac {\sqrt {x} \left (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {\frac {a x^2}{b}+1} \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 {\sqrt {x} \left (\frac {c}{\sqrt {a}}+2 \sqrt {b}\right ) \sqrt {\frac {a x^2}{b}+1} \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 {\sqrt {x} \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[x]*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/
b^(1/4)], 1/2])/(a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3]) - ((2*Sqrt[b] - c/Sqrt[a])*Sqrt[x]*Sqrt[1 + (a*x^2)/b]*Ell
ipticPi[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]) -
((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 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

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} \left (-b+a x^2\right ) \sqrt {b+a x^2}} \, 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} \left (-b+a x^2\right ) \sqrt {b+a x^2}}\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} \left (-b+a x^2\right ) \sqrt {b+a x^2}} \, 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 {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \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 \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 {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \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 \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 {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \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 \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 {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \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}}-\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*}

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Mathematica [C]  time = 0.43, size = 133, normalized size = 0.90 \begin {gather*} -\frac {2 x \sqrt {\frac {a x^2}{b}+1} \left (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 )+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 )\right )}{15 b \sqrt {x \left (a x^2+b\right )}} \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[x
*(b + a*x^2)])

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IntegrateAlgebraic [A]  time = 0.41, size = 147, normalized size = 1.00 \begin {gather*} -\frac {\left (2 \sqrt {a} \sqrt {b}-c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (2 \sqrt {a} \sqrt {b}+c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} 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]

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

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fricas [B]  time = 0.92, size = 1557, normalized size = 10.59

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)*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))

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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)

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maple [C]  time = 0.14, size = 691, normalized size = 4.70

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 )}\)	\(691\)
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 )}\)	\(710\)

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))*Ellipti
cPi(((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))

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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)

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

[Out]

\text{Hanged}

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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)

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