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

Optimal. Leaf size=26 \[ \frac {2 \sqrt {a x^3-b x}}{b-a x^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 17, normalized size of antiderivative = 0.65, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2056, 449} \begin {gather*} -\frac {2 x}{\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]

(-2*x)/Sqrt[-(b*x) + a*x^3]

Rule 449

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

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 {b+a x^2}{\sqrt {x} \left (-b+a x^2\right )^{3/2}} \, dx}{\sqrt {-b x+a x^3}}\\ &=-\frac {2 x}{\sqrt {-b x+a x^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 17, normalized size = 0.65 \begin {gather*} -\frac {2 x}{\sqrt {a x^3-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/Sqrt[-(b*x) + a*x^3]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.17, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-b x+a x^3}}{b-a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-(b*x) + a*x^3])/(b - a*x^2)

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 25, normalized size = 0.96 \begin {gather*} -\frac {2 \, \sqrt {a x^{3} - b x}}{a x^{2} - b} \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]

-2*sqrt(a*x^3 - b*x)/(a*x^2 - b)

________________________________________________________________________________________

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 [A]  time = 0.12, size = 16, normalized size = 0.62

method	result	size
gosper	\(-\frac {2 x}{\sqrt {a \,x^{3}-b x}}\)	\(16\)
elliptic	\(-\frac {2 x}{\sqrt {\left (x^{2}-\frac {b}{a}\right ) a x}}\)	\(19\)
trager	\(-\frac {2 \sqrt {a \,x^{3}-b x}}{a \,x^{2}-b}\)	\(26\)
default	\(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}+2 b \left (-\frac {x}{b \sqrt {\left (x^{2}-\frac {b}{a}\right ) a x}}-\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{2 b a \sqrt {a \,x^{3}-b x}}\right )\)	\(231\)

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]

-2*x/(a*x^3-b*x)^(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 [B]  time = 0.24, size = 24, normalized size = 0.92 \begin {gather*} \frac {2\,\sqrt {a\,x^3-b\,x}}{b-a\,x^2} \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]

(2*(a*x^3 - b*x)^(1/2))/(b - a*x^2)

________________________________________________________________________________________

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)

________________________________________________________________________________________