3.3.5 \(\int \frac {c+d x^2}{x^2 (a+b x^2)} \, dx\) [205]

Optimal. Leaf size=43 \[ -\frac {c}{a x}-\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {464, 211} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)}{a^{3/2} \sqrt {b}}-\frac {c}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/(x^2*(a + b*x^2)),x]

[Out]

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

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 464

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] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 42, normalized size = 0.98 \begin {gather*} -\frac {c}{a x}+\frac {(-b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 37, normalized size = 0.86

method	result	size
default	\(\frac {\left (a d -b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \sqrt {a b}}-\frac {c}{a x}\)	\(37\)
risch	\(-\frac {c}{a x}-\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) d}{2 \sqrt {-a b}}+\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) b c}{2 \sqrt {-a b}\, a}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) d}{2 \sqrt {-a b}}-\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) b c}{2 \sqrt {-a b}\, a}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 37, normalized size = 0.86 \begin {gather*} -\frac {{\left (b c - a d\right )} \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((d*x^2+c)/x^2/(b*x^2+a),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 1.27, size = 105, normalized size = 2.44 \begin {gather*} \left [\frac {\sqrt {-a b} {\left (b c - a d\right )} x \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, a b c}{2 \, a^{2} b x}, -\frac {\sqrt {a b} {\left (b c - a d\right )} x \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + a b c}{a^{2} b x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (36) = 72\).
time = 0.18, size = 82, normalized size = 1.91 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{3} b}} \left (a d - b c\right ) \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a^{3} b}} \left (a d - b c\right ) \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{2} - \frac {c}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.37, size = 37, normalized size = 0.86 \begin {gather*} -\frac {{\left (b c - a d\right )} \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((d*x^2+c)/x^2/(b*x^2+a),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.06, size = 34, normalized size = 0.79 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )}{a^{3/2}\,\sqrt {b}}-\frac {c}{a\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)/(x^2*(a + b*x^2)),x)

[Out]

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

________________________________________________________________________________________