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

Optimal. Leaf size=19 \[ \frac {2 \sqrt {a x^3-b x}}{x} \]

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Rubi [A]  time = 0.10, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2036} \begin {gather*} \frac {2 \sqrt {a x^3-b x}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-(b*x) + a*x^3])/x

Rule 2036

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[(c*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(a*(m + j*p + 1)), x] /; FreeQ[{a, b, c, d, e,
j, m, n, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && EqQ[a*d*(m + j*p + 1) - b*c*(m + n
+ p*(j + n) + 1), 0] && (GtQ[e, 0] || IntegersQ[j]) && NeQ[m + j*p + 1, 0]

Rubi steps

\begin {align*} \int \frac {b+a x^2}{x \sqrt {-b x+a x^3}} \, dx &=\frac {2 \sqrt {-b x+a x^3}}{x}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a x^3-b x}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-(b*x) + a*x^3])/x

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IntegrateAlgebraic [A]  time = 0.20, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-b x+a x^3}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-(b*x) + a*x^3])/x

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fricas [A]  time = 0.47, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 \, \sqrt {a x^{3} - b x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a*x^3 - b*x)/x

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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} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.13, size = 18, normalized size = 0.95

method	result	size
trager	\(\frac {2 \sqrt {a \,x^{3}-b x}}{x}\)	\(18\)
gosper	\(\frac {2 a \,x^{2}-2 b}{\sqrt {a \,x^{3}-b x}}\)	\(24\)
risch	\(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}\)	\(25\)
elliptic	\(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}\)	\(25\)
default	\(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{\sqrt {a \,x^{3}-b x}}+b \left (\frac {2 a \,x^{2}-2 b}{b \sqrt {x \left (a \,x^{2}-b \right )}}-\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{b \sqrt {a \,x^{3}-b x}}\right )\)	\(322\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.21, size = 17, normalized size = 0.89 \begin {gather*} \frac {2\,\sqrt {a\,x^3-b\,x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a*x^3 - b*x)^(1/2))/x

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} + b}{x \sqrt {x \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)/x/(a*x**3-b*x)**(1/2),x)

[Out]

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

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