3.3.83 \(\int \frac {d+e x}{a+c x^2} \, dx\) [283]

Optimal. Leaf size=42 \[ \frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {e \log \left (a+c x^2\right )}{2 c} \]

[Out]

1/2*e*ln(c*x^2+a)/c+d*arctan(x*c^(1/2)/a^(1/2))/a^(1/2)/c^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {649, 211, 266} \begin {gather*} \frac {d \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {e \log \left (a+c x^2\right )}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a + c*x^2),x]

[Out]

(d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) + (e*Log[a + c*x^2])/(2*c)

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 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 42, normalized size = 1.00 \begin {gather*} \frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {e \log \left (a+c x^2\right )}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(a + c*x^2),x]

[Out]

(d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) + (e*Log[a + c*x^2])/(2*c)

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Maple [A]
time = 0.54, size = 32, normalized size = 0.76

method	result	size
default	\(\frac {e \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}\)	\(32\)
risch	\(\frac {\ln \left (-\sqrt {-a c}\, x +a \right ) d \sqrt {-a c}}{2 a c}+\frac {\ln \left (-\sqrt {-a c}\, x +a \right ) e}{2 c}-\frac {\ln \left (\sqrt {-a c}\, x +a \right ) d \sqrt {-a c}}{2 a c}+\frac {\ln \left (\sqrt {-a c}\, x +a \right ) e}{2 c}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e*ln(c*x^2+a)/c+d/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))

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Maxima [A]
time = 0.53, size = 32, normalized size = 0.76 \begin {gather*} \frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} + \frac {e \log \left (c x^{2} + a\right )}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*arctan(c*x/sqrt(a*c))/sqrt(a*c) + 1/2*e*log(c*x^2 + a)/c

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Fricas [A]
time = 5.00, size = 100, normalized size = 2.38 \begin {gather*} \left [\frac {a e \log \left (c x^{2} + a\right ) - \sqrt {-a c} d \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{2 \, a c}, \frac {a e \log \left (c x^{2} + a\right ) + 2 \, \sqrt {a c} d \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{2 \, a c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (37) = 74\).
time = 0.12, size = 124, normalized size = 2.95 \begin {gather*} \left (\frac {e}{2 c} - \frac {d \sqrt {- a c^{3}}}{2 a c^{2}}\right ) \log {\left (x + \frac {2 a c \left (\frac {e}{2 c} - \frac {d \sqrt {- a c^{3}}}{2 a c^{2}}\right ) - a e}{c d} \right )} + \left (\frac {e}{2 c} + \frac {d \sqrt {- a c^{3}}}{2 a c^{2}}\right ) \log {\left (x + \frac {2 a c \left (\frac {e}{2 c} + \frac {d \sqrt {- a c^{3}}}{2 a c^{2}}\right ) - a e}{c d} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e/(2*c) - d*sqrt(-a*c**3)/(2*a*c**2))*log(x + (2*a*c*(e/(2*c) - d*sqrt(-a*c**3)/(2*a*c**2)) - a*e)/(c*d)) + (
e/(2*c) + d*sqrt(-a*c**3)/(2*a*c**2))*log(x + (2*a*c*(e/(2*c) + d*sqrt(-a*c**3)/(2*a*c**2)) - a*e)/(c*d))

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Giac [A]
time = 0.94, size = 32, normalized size = 0.76 \begin {gather*} \frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} + \frac {e \log \left (c x^{2} + a\right )}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*arctan(c*x/sqrt(a*c))/sqrt(a*c) + 1/2*e*log(c*x^2 + a)/c

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Mupad [B]
time = 0.05, size = 32, normalized size = 0.76 \begin {gather*} \frac {e\,\ln \left (c\,x^2+a\right )}{2\,c}+\frac {d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a + c*x^2),x)

[Out]

(e*log(a + c*x^2))/(2*c) + (d*atan((c^(1/2)*x)/a^(1/2)))/(a^(1/2)*c^(1/2))

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