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

Optimal. Leaf size=113 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.24, antiderivative size = 159, normalized size of antiderivative = 1.41, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2056, 466, 404, 212, 206, 203} \begin {gather*} -\frac {\sqrt {x} \sqrt {a x^2+b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}-\frac {\sqrt {x} \sqrt {a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 404

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[a/c, Subst[Int[1/(1 - 4*a*b*x^4), x], x
, x/Sqrt[a + b*x^4]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && PosQ[a*b]

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.05, size = 60, normalized size = 0.53 \begin {gather*} -\frac {2 \sqrt {x \left (a x^2+b\right )} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};-\frac {a x^2}{b},\frac {a x^2}{b}\right )}{b \sqrt {\frac {a x^2}{b}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[x*(b + a*x^2)]*AppellF1[1/4, -1/2, 1, 5/4, -((a*x^2)/b), (a*x^2)/b])/(b*Sqrt[1 + (a*x^2)/b])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.36, size = 113, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.73, size = 326, normalized size = 2.88 \begin {gather*} -\left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \arctan \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {a x^{3} + b x} a b \left (\frac {1}{a b}\right )^{\frac {3}{4}}}{a x^{2} + b}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} + 8 \, \sqrt {a x^{3} + b x} {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a b}\right )^{\frac {3}{4}}\right )} + 4 \, {\left (a^{2} b x^{3} + a b^{2} x\right )} \sqrt {\frac {1}{a b}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} - 8 \, \sqrt {a x^{3} + b x} {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a b}\right )^{\frac {3}{4}}\right )} + 4 \, {\left (a^{2} b x^{3} + a b^{2} x\right )} \sqrt {\frac {1}{a b}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1/4)^(1/4)*(1/(a*b))^(1/4)*arctan(4*(1/4)^(3/4)*sqrt(a*x^3 + b*x)*a*b*(1/(a*b))^(3/4)/(a*x^2 + b)) - 1/4*(1/
4)^(1/4)*(1/(a*b))^(1/4)*log((a^2*x^4 + 6*a*b*x^2 + b^2 + 8*sqrt(a*x^3 + b*x)*((1/4)^(1/4)*a*b*x*(1/(a*b))^(1/
4) + (1/4)^(3/4)*(a^2*b*x^2 + a*b^2)*(1/(a*b))^(3/4)) + 4*(a^2*b*x^3 + a*b^2*x)*sqrt(1/(a*b)))/(a^2*x^4 - 2*a*
b*x^2 + b^2)) + 1/4*(1/4)^(1/4)*(1/(a*b))^(1/4)*log((a^2*x^4 + 6*a*b*x^2 + b^2 - 8*sqrt(a*x^3 + b*x)*((1/4)^(1
/4)*a*b*x*(1/(a*b))^(1/4) + (1/4)^(3/4)*(a^2*b*x^2 + a*b^2)*(1/(a*b))^(3/4)) + 4*(a^2*b*x^3 + a*b^2*x)*sqrt(1/
(a*b)))/(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} + 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+b)/(a*x^2-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 - b)), x)

________________________________________________________________________________________

maple [C]  time = 0.14, size = 395, normalized size = 3.50

method	result	size
elliptic	\(\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}+\frac {b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\)	\(395\)
default	\(\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}+2 b \left (\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\right )\)	\(417\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2+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))+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))-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} + 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+b)/(a*x^2-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 - 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(-(b + a*x^2)/((b*x + a*x^3)^(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}{\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+b)/(a*x**2-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)), x)

________________________________________________________________________________________