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

Optimal. Leaf size=60 \[ -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b x}}{a^2 x^2+b}\right )}{\sqrt {a} \sqrt {b}} \]

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Rubi [C]  time = 2.12, antiderivative size = 290, normalized size of antiderivative = 4.83, number of steps used = 13, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {2056, 6728, 329, 220, 933, 168, 537} \begin {gather*} \frac {\sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt [4]{b} \sqrt {a^2 x^3+b x}}-\frac {2 \sqrt [4]{-b} \sqrt {x} \sqrt {\frac {a^2 x^2}{b}+1} \Pi \left (\frac {\sqrt {-b}}{\left (\sqrt {b-1}-\sqrt {b}\right ) \sqrt {b}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{-b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {a^2 x^3+b x}}-\frac {2 \sqrt [4]{-b} \sqrt {x} \sqrt {\frac {a^2 x^2}{b}+1} \Pi \left (-\frac {\sqrt {-b}}{\left (\sqrt {b-1}+\sqrt {b}\right ) \sqrt {b}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{-b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {a^2 x^3+b x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Mathematica [C]  time = 2.31, size = 204, normalized size = 3.40 \begin {gather*} -\frac {2 i x^{3/2} \sqrt {\frac {b}{a^2 x^2}+1} \left (-\Pi \left (-\frac {i a \sqrt {b}}{a b-\sqrt {a^2 (b-1) b}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {i a \sqrt {b}}{a b+\sqrt {a^2 (b-1) b}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{a}}}{\sqrt {x}}\right )\right |-1\right )+F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {\frac {i \sqrt {b}}{a}} \sqrt {x \left (a^2 x^2+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.34, size = 60, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b x+a^2 x^3}}{b+a^2 x^2}\right )}{\sqrt {a} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.50, size = 219, normalized size = 3.65 \begin {gather*} \left [\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} - 12 \, a^{3} b x^{3} - 12 \, a b^{2} x + 2 \, {\left (2 \, a^{2} b^{2} + a^{2} b\right )} x^{2} + 4 \, \sqrt {2} {\left (a^{3} b x^{2} - 2 \, a^{2} b^{2} x + a b^{2}\right )} \sqrt {a^{2} x^{3} + b x} \sqrt {-\frac {1}{a b}} + b^{2}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 4 \, a b^{2} x + 2 \, {\left (2 \, a^{2} b^{2} + a^{2} b\right )} x^{2} + b^{2}}\right ), \frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a b}} \arctan \left (\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a b x + b\right )} \sqrt {\frac {1}{a b}}}{4 \, \sqrt {a^{2} x^{3} + b x}}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(2)*sqrt(-1/(a*b))*log((a^4*x^4 - 12*a^3*b*x^3 - 12*a*b^2*x + 2*(2*a^2*b^2 + a^2*b)*x^2 + 4*sqrt(2)*(
a^3*b*x^2 - 2*a^2*b^2*x + a*b^2)*sqrt(a^2*x^3 + b*x)*sqrt(-1/(a*b)) + b^2)/(a^4*x^4 + 4*a^3*b*x^3 + 4*a*b^2*x
+ 2*(2*a^2*b^2 + a^2*b)*x^2 + b^2)), 1/2*sqrt(2)*sqrt(1/(a*b))*arctan(1/4*sqrt(2)*(a^2*x^2 - 2*a*b*x + b)*sqrt
(1/(a*b))/sqrt(a^2*x^3 + b*x))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.28, size = 1081, normalized size = 18.02

method	result	size
elliptic	\(\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b x}}+\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b +\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) b^{2}}{a^{2} \sqrt {b^{2}-b}\, \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}-\frac {\sqrt {b^{2}-b}}{a}\right )}-\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b +\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{a^{2} \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}-\frac {\sqrt {b^{2}-b}}{a}\right )}-\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b +\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{a^{2} \sqrt {b^{2}-b}\, \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}-\frac {\sqrt {b^{2}-b}}{a}\right )}-\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b -\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) b^{2}}{a^{2} \sqrt {b^{2}-b}\, \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}+\frac {\sqrt {b^{2}-b}}{a}\right )}-\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b -\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{a^{2} \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}+\frac {\sqrt {b^{2}-b}}{a}\right )}+\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b -\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{a^{2} \sqrt {b^{2}-b}\, \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}+\frac {\sqrt {b^{2}-b}}{a}\right )}\)	\(1081\)
default	\(\frac {\sqrt {-b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b x}}-2 b \left (-\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b +\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{2 a^{2} \sqrt {b^{2}-b}\, \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}-\frac {\sqrt {b^{2}-b}}{a}\right )}+\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b +\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}-\frac {\sqrt {b^{2}-b}}{a}\right )}+\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b +\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {b^{2}-b}\, \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}-\frac {\sqrt {b^{2}-b}}{a}\right )}+\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b -\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{2 a^{2} \sqrt {b^{2}-b}\, \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}+\frac {\sqrt {b^{2}-b}}{a}\right )}+\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b -\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}+\frac {\sqrt {b^{2}-b}}{a}\right )}-\frac {\sqrt {-b}\, \sqrt {\frac {x a}{\sqrt {-b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\sqrt {-b}}{a \left (-\frac {\sqrt {-b}}{a}-\frac {-b -\sqrt {b^{2}-b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {b^{2}-b}\, \sqrt {a^{2} x^{3}+b x}\, \left (-\frac {\sqrt {-b}}{a}+\frac {b}{a}+\frac {\sqrt {b^{2}-b}}{a}\right )}\right )\)	\(1096\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-1>0)', see `assume?` for mor
e details)Is b-1 positive, negative or zero?

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mupad [B]  time = 3.28, size = 78, normalized size = 1.30 \begin {gather*} \frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,b}{4}+\frac {\sqrt {2}\,a^2\,x^2}{4}-\frac {\sqrt {2}\,a\,b\,x}{2}+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3+b\,x}\,1{}\mathrm {i}}{a^2\,x^2+2\,b\,a\,x+b}\right )\,1{}\mathrm {i}}{2\,\sqrt {a}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*log(((2^(1/2)*b)/4 + (2^(1/2)*a^2*x^2)/4 + a^(1/2)*b^(1/2)*(b*x + a^2*x^3)^(1/2)*1i - (2^(1/2)*a*b*x)
/2)/(b + a^2*x^2 + 2*a*b*x))*1i)/(2*a^(1/2)*b^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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