3.3.88 \(\int \frac {d+e x}{a-c x^2} \, dx\)

Optimal. Leaf size=43 \[ \frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-\frac {e \log \left (a-c x^2\right )}{2 c} \]

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {635, 208, 260} \begin {gather*} \frac {d \tanh ^{-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]

Int[(d + e*x)/(a - c*x^2),x]

[Out]

(d*ArcTanh[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) - (e*Log[a - c*x^2])/(2*c)

Rule 208

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

Rule 260

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 635

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 \tanh ^{-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 = 43, normalized size = 1.00 \begin {gather*} \frac {d \tanh ^{-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*ArcTanh[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) - (e*Log[a - c*x^2])/(2*c)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{a-c x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)/(a - c*x^2),x]

[Out]

IntegrateAlgebraic[(d + e*x)/(a - c*x^2), x]

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fricas [A]  time = 0.42, size = 102, normalized size = 2.37 \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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giac [A]  time = 0.15, size = 37, normalized size = 0.86 \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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maple [A]  time = 0.04, size = 34, normalized size = 0.79 \begin {gather*} \frac {d \arctanh \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}-\frac {e \ln \left (c \,x^{2}-a \right )}{2 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(-c*x^2+a),x)

[Out]

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

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maxima [A]  time = 1.28, size = 49, normalized size = 1.14 \begin {gather*} -\frac {d \log \left (\frac {c x - \sqrt {a c}}{c x + \sqrt {a c}}\right )}{2 \, \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]

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

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mupad [B]  time = 0.21, size = 103, normalized size = 2.40 \begin {gather*} \frac {d\,\ln \left (a\,c+x\,\sqrt {a\,c^3}\right )\,\sqrt {a\,c^3}}{2\,a\,c^2}-\frac {e\,\ln \left (x\,\sqrt {a\,c^3}-a\,c\right )}{2\,c}-\frac {e\,\ln \left (a\,c+x\,\sqrt {a\,c^3}\right )}{2\,c}-\frac {d\,\ln \left (x\,\sqrt {a\,c^3}-a\,c\right )\,\sqrt {a\,c^3}}{2\,a\,c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a - c*x^2),x)

[Out]

(d*log(a*c + x*(a*c^3)^(1/2))*(a*c^3)^(1/2))/(2*a*c^2) - (e*log(x*(a*c^3)^(1/2) - a*c))/(2*c) - (e*log(a*c + x
*(a*c^3)^(1/2)))/(2*c) - (d*log(x*(a*c^3)^(1/2) - a*c)*(a*c^3)^(1/2))/(2*a*c^2)

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sympy [B]  time = 0.28, size = 119, normalized size = 2.77 \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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