### 3.502 $$\int \frac{1}{(d+e x) (a+c x^2)} \, dx$$

Optimal. Leaf size=86 $-\frac{e \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac{e \log (d+e x)}{a e^2+c d^2}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^2+c d^2\right )}$

[Out]

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

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Rubi [A]  time = 0.0380176, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.294, Rules used = {706, 31, 635, 205, 260} $-\frac{e \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac{e \log (d+e x)}{a e^2+c d^2}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^2+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)*(a + c*x^2)),x]

[Out]

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

Rule 706

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 + a*e^2), Int[1/(d + e*x), x],
x] + Dist[1/(c*d^2 + a*e^2), Int[(c*d - c*e*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a
*e^2, 0]

Rule 31

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

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0354208, size = 63, normalized size = 0.73 $\frac{\frac{2 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a}}-e \log \left (a+c x^2\right )+2 e \log (d+e x)}{2 a e^2+2 c d^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)*(a + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.049, size = 77, normalized size = 0.9 \begin{align*} -{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,a{e}^{2}+2\,c{d}^{2}}}+{\frac{cd}{a{e}^{2}+c{d}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{e\ln \left ( ex+d \right ) }{a{e}^{2}+c{d}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(c*x^2+a),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.14763, size = 298, normalized size = 3.47 \begin{align*} \left [\frac{d \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} + 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) - e \log \left (c x^{2} + a\right ) + 2 \, e \log \left (e x + d\right )}{2 \,{\left (c d^{2} + a e^{2}\right )}}, \frac{2 \, d \sqrt{\frac{c}{a}} \arctan \left (x \sqrt{\frac{c}{a}}\right ) - e \log \left (c x^{2} + a\right ) + 2 \, e \log \left (e x + d\right )}{2 \,{\left (c d^{2} + a e^{2}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.80207, size = 1134, normalized size = 13.19 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x**2+a),x)

[Out]

e*log(x + (-12*a**4*e**8/(a*e**2 + c*d**2)**2 - 20*a**3*c*d**2*e**6/(a*e**2 + c*d**2)**2 + 6*a**3*e**6/(a*e**2
+ c*d**2) - 4*a**2*c**2*d**4*e**4/(a*e**2 + c*d**2)**2 + 12*a**2*c*d**2*e**4/(a*e**2 + c*d**2) + 6*a**2*e**4
+ 4*a*c**3*d**6*e**2/(a*e**2 + c*d**2)**2 + 6*a*c**2*d**4*e**2/(a*e**2 + c*d**2) - a*c*d**2*e**2 + c**2*d**4)/
(9*a*c*d*e**3 + c**2*d**3*e))/(a*e**2 + c*d**2) + (-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d*
*2)))*log(x + (-12*a**4*e**6*(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))**2 - 20*a**3*c*
d**2*e**4*(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))**2 + 6*a**3*e**5*(-e/(2*(a*e**2 +
c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2))) - 4*a**2*c**2*d**4*e**2*(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a
*c)/(2*a*(a*e**2 + c*d**2)))**2 + 12*a**2*c*d**2*e**3*(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 +
c*d**2))) + 6*a**2*e**4 + 4*a*c**3*d**6*(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))**2 +
6*a*c**2*d**4*e*(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2))) - a*c*d**2*e**2 + c**2*d**4
)/(9*a*c*d*e**3 + c**2*d**3*e)) + (-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))*log(x + (-
12*a**4*e**6*(-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))**2 - 20*a**3*c*d**2*e**4*(-e/(2
*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))**2 + 6*a**3*e**5*(-e/(2*(a*e**2 + c*d**2)) + d*sqr
t(-a*c)/(2*a*(a*e**2 + c*d**2))) - 4*a**2*c**2*d**4*e**2*(-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2
+ c*d**2)))**2 + 12*a**2*c*d**2*e**3*(-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2))) + 6*a*
*2*e**4 + 4*a*c**3*d**6*(-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))**2 + 6*a*c**2*d**4*e
*(-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2))) - a*c*d**2*e**2 + c**2*d**4)/(9*a*c*d*e**3
+ c**2*d**3*e))

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Giac [A]  time = 1.1845, size = 107, normalized size = 1.24 \begin{align*} \frac{c d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt{a c}} - \frac{e \log \left (c x^{2} + a\right )}{2 \,{\left (c d^{2} + a e^{2}\right )}} + \frac{e^{2} \log \left ({\left | x e + d \right |}\right )}{c d^{2} e + a e^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+a),x, algorithm="giac")

[Out]

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