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

Optimal. Leaf size=59 \[ -\frac {\sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {e \log \left (a+c x^2\right )}{2 a}-\frac {d}{a x}+\frac {e \log (x)}{a} \]

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Rubi [A]  time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {801, 635, 205, 260} \begin {gather*} -\frac {\sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {e \log \left (a+c x^2\right )}{2 a}-\frac {d}{a x}+\frac {e \log (x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^2*(a + c*x^2)),x]

[Out]

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

Rule 205

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

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 59, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {e \log \left (a+c x^2\right )}{2 a}-\frac {d}{a x}+\frac {e \log (x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 124, normalized size = 2.10 \begin {gather*} \left [\frac {d x \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} - 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) - e x \log \left (c x^{2} + a\right ) + 2 \, e x \log \relax (x) - 2 \, d}{2 \, a x}, -\frac {2 \, d x \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + e x \log \left (c x^{2} + a\right ) - 2 \, e x \log \relax (x) + 2 \, d}{2 \, a x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 55, normalized size = 0.93 \begin {gather*} -\frac {c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} a} - \frac {e \log \left (c x^{2} + a\right )}{2 \, a} + \frac {e \log \left ({\left | x \right |}\right )}{a} - \frac {d}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 53, normalized size = 0.90 \begin {gather*} -\frac {c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, a}+\frac {e \ln \relax (x )}{a}-\frac {e \ln \left (c \,x^{2}+a \right )}{2 a}-\frac {d}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.26, size = 52, normalized size = 0.88 \begin {gather*} -\frac {c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} a} - \frac {e \log \left (c x^{2} + a\right )}{2 \, a} + \frac {e \log \relax (x)}{a} - \frac {d}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.21, size = 131, normalized size = 2.22 \begin {gather*} \frac {e\,\ln \relax (x)}{a}-\frac {d}{a\,x}-\frac {\ln \left (3\,a^2\,e+d\,\sqrt {-a^3\,c}-3\,e\,x\,\sqrt {-a^3\,c}+a\,c\,d\,x\right )\,\left (a^2\,e+d\,\sqrt {-a^3\,c}\right )}{2\,a^3}-\frac {\ln \left (3\,a^2\,e-d\,\sqrt {-a^3\,c}+3\,e\,x\,\sqrt {-a^3\,c}+a\,c\,d\,x\right )\,\left (a^2\,e-d\,\sqrt {-a^3\,c}\right )}{2\,a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(x^2*(a + c*x^2)),x)

[Out]

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

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sympy [B]  time = 1.62, size = 326, normalized size = 5.53 \begin {gather*} \left (- \frac {e}{2 a} - \frac {d \sqrt {- a^{3} c}}{2 a^{3}}\right ) \log {\left (x + \frac {12 a^{4} e \left (- \frac {e}{2 a} - \frac {d \sqrt {- a^{3} c}}{2 a^{3}}\right )^{2} - 6 a^{3} e^{2} \left (- \frac {e}{2 a} - \frac {d \sqrt {- a^{3} c}}{2 a^{3}}\right ) - 2 a^{2} c d^{2} \left (- \frac {e}{2 a} - \frac {d \sqrt {- a^{3} c}}{2 a^{3}}\right ) - 6 a^{2} e^{3} + 2 a c d^{2} e}{9 a c d e^{2} + c^{2} d^{3}} \right )} + \left (- \frac {e}{2 a} + \frac {d \sqrt {- a^{3} c}}{2 a^{3}}\right ) \log {\left (x + \frac {12 a^{4} e \left (- \frac {e}{2 a} + \frac {d \sqrt {- a^{3} c}}{2 a^{3}}\right )^{2} - 6 a^{3} e^{2} \left (- \frac {e}{2 a} + \frac {d \sqrt {- a^{3} c}}{2 a^{3}}\right ) - 2 a^{2} c d^{2} \left (- \frac {e}{2 a} + \frac {d \sqrt {- a^{3} c}}{2 a^{3}}\right ) - 6 a^{2} e^{3} + 2 a c d^{2} e}{9 a c d e^{2} + c^{2} d^{3}} \right )} - \frac {d}{a x} + \frac {e \log {\relax (x )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-e/(2*a) - d*sqrt(-a**3*c)/(2*a**3))*log(x + (12*a**4*e*(-e/(2*a) - d*sqrt(-a**3*c)/(2*a**3))**2 - 6*a**3*e**
2*(-e/(2*a) - d*sqrt(-a**3*c)/(2*a**3)) - 2*a**2*c*d**2*(-e/(2*a) - d*sqrt(-a**3*c)/(2*a**3)) - 6*a**2*e**3 +
2*a*c*d**2*e)/(9*a*c*d*e**2 + c**2*d**3)) + (-e/(2*a) + d*sqrt(-a**3*c)/(2*a**3))*log(x + (12*a**4*e*(-e/(2*a)
+ d*sqrt(-a**3*c)/(2*a**3))**2 - 6*a**3*e**2*(-e/(2*a) + d*sqrt(-a**3*c)/(2*a**3)) - 2*a**2*c*d**2*(-e/(2*a)
+ d*sqrt(-a**3*c)/(2*a**3)) - 6*a**2*e**3 + 2*a*c*d**2*e)/(9*a*c*d*e**2 + c**2*d**3)) - d/(a*x) + e*log(x)/a

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