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

Optimal. Leaf size=140 \[ -\frac {\tanh ^{-1}\left (\frac {\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}-\frac {b^{3/4}}{2 \sqrt [4]{a}}+\sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^3-b x}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3-b x}}{-2 \sqrt {a} \sqrt {b} x+a x^2-b}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}} \]

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Rubi [A]  time = 0.22, antiderivative size = 176, normalized size of antiderivative = 1.26, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2056, 466, 405} \begin {gather*} -\frac {\sqrt {x} \sqrt {a x^2-b} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right )}{\sqrt [4]{b} \sqrt {a x^2-b}}\right )}{\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 [4]{a} \sqrt {x} \left (\sqrt {a} x+\sqrt {b}\right )}{\sqrt [4]{b} \sqrt {a x^2-b}}\right )}{\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[(a^(1/4)*Sqrt[x]*(Sqrt[b] - Sqrt[a]*x))/(b^(1/4)*Sqrt[-b + a*x^2])])/(a^(1/
4)*b^(1/4)*Sqrt[-(b*x) + a*x^3])) - (Sqrt[x]*Sqrt[-b + a*x^2]*ArcTanh[(a^(1/4)*Sqrt[x]*(Sqrt[b] + Sqrt[a]*x))/
(b^(1/4)*Sqrt[-b + a*x^2])])/(a^(1/4)*b^(1/4)*Sqrt[-(b*x) + a*x^3])

Rule 405

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-(a*b), 4]}, Simp[(a*ArcTan[(q*
x*(a + q^2*x^2))/(a*Sqrt[a + b*x^4])])/(2*c*q), x] + Simp[(a*ArcTanh[(q*x*(a - q^2*x^2))/(a*Sqrt[a + b*x^4])])
/(2*c*q), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[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 {\sqrt {x} \sqrt {-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right )}{\sqrt [4]{b} \sqrt {-b+a x^2}}\right )}{\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 [4]{a} \sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right )}{\sqrt [4]{b} \sqrt {-b+a x^2}}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 62, normalized size = 0.44 \begin {gather*} \frac {2 \sqrt {a x^3-b x} 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 {1-\frac {a x^2}{b}}} \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[-(b*x) + a*x^3]*AppellF1[1/4, -1/2, 1, 5/4, (a*x^2)/b, -((a*x^2)/b)])/(b*Sqrt[1 - (a*x^2)/b])

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IntegrateAlgebraic [A]  time = 0.42, size = 150, normalized size = 1.07 \begin {gather*} \frac {\tan ^{-1}\left (\frac {-\frac {b^{3/4}}{2 \sqrt [4]{a}}-\sqrt [4]{a} \sqrt [4]{b} x+\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}}{\sqrt {-b x+a x^3}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {-\frac {b^{3/4}}{2 \sqrt [4]{a}}+\sqrt [4]{a} \sqrt [4]{b} x+\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}}{\sqrt {-b x+a x^3}}\right )}{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[(-1/2*b^(3/4)/a^(1/4) - a^(1/4)*b^(1/4)*x + (a^(3/4)*x^2)/(2*b^(1/4)))/Sqrt[-(b*x) + a*x^3]]/(2*a^(1/4)
*b^(1/4)) - ArcTanh[(-1/2*b^(3/4)/a^(1/4) + a^(1/4)*b^(1/4)*x + (a^(3/4)*x^2)/(2*b^(1/4)))/Sqrt[-(b*x) + a*x^3
]]/(2*a^(1/4)*b^(1/4))

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fricas [B]  time = 0.62, size = 344, normalized size = 2.46 \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))

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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} + 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)

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maple [C]  time = 0.14, size = 380, normalized size = 2.71

method	result	size
elliptic	\(\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}-\frac {b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\)	\(380\)
default	\(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}-2 b \left (\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\right )\)	\(400\)

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))

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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} + 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)

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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 - a*x^2)/((a*x^3 - b*x)^(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}{\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)

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