3.2.32 \(\int \frac {a c+b c x^2}{x^2 (a+b x^2)^2} \, dx\) [132]

Optimal. Leaf size=36 \[ -\frac {c}{a x}-\frac {\sqrt {b} c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {21, 331, 211} \begin {gather*} -\frac {\sqrt {b} c \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {c}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*c + b*c*x^2)/(x^2*(a + b*x^2)^2),x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 211

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

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx &=c \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx\\ &=-\frac {c}{a x}-\frac {(b c) \int \frac {1}{a+b x^2} \, dx}{a}\\ &=-\frac {c}{a x}-\frac {\sqrt {b} c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 36, normalized size = 1.00 \begin {gather*} c \left (-\frac {1}{a x}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*c + b*c*x^2)/(x^2*(a + b*x^2)^2),x]

[Out]

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

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Maple [A]
time = 0.07, size = 32, normalized size = 0.89

method	result	size
default	\(c \left (-\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \sqrt {a b}}-\frac {1}{a x}\right )\)	\(32\)
risch	\(-\frac {c}{a x}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{3} \textit {\_Z}^{2}+b \,c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{3}+2 b \,c^{2}\right ) x +a^{2} c \textit {\_R} \right )\right )}{2}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*c*x^2+a*c)/x^2/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

c*(-b/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))-1/a/x)

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Maxima [A]
time = 0.57, size = 31, normalized size = 0.86 \begin {gather*} -\frac {b c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {c}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*c*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - c/(a*x)

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Fricas [A]
time = 1.04, size = 86, normalized size = 2.39 \begin {gather*} \left [\frac {c x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2 \, c}{2 \, a x}, -\frac {c x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + c}{a x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(c*x*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 2*c)/(a*x), -(c*x*sqrt(b/a)*arctan(x*sq
rt(b/a)) + c)/(a*x)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (31) = 62\).
time = 0.07, size = 66, normalized size = 1.83 \begin {gather*} c \left (\frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (- \frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (\frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {1}{a x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c*x**2+a*c)/x**2/(b*x**2+a)**2,x)

[Out]

c*(sqrt(-b/a**3)*log(-a**2*sqrt(-b/a**3)/b + x)/2 - sqrt(-b/a**3)*log(a**2*sqrt(-b/a**3)/b + x)/2 - 1/(a*x))

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Giac [A]
time = 0.95, size = 31, normalized size = 0.86 \begin {gather*} -\frac {b c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {c}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*c*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - c/(a*x)

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Mupad [B]
time = 0.05, size = 28, normalized size = 0.78 \begin {gather*} -\frac {c}{a\,x}-\frac {\sqrt {b}\,c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c + b*c*x^2)/(x^2*(a + b*x^2)^2),x)

[Out]

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

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