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

Optimal. Leaf size=57 \[ \frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}+\frac {c d x-a e}{2 a c \left (a+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {639, 205} \[ \frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}-\frac {a e-c d x}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a + c*x^2)^2,x]

[Out]

-(a*e - c*d*x)/(2*a*c*(a + c*x^2)) + (d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[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 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 57, normalized size = 1.00 \[ \frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}+\frac {c d x-a e}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(a + c*x^2)^2,x]

[Out]

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

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fricas [A]  time = 0.99, size = 140, normalized size = 2.46 \[ \left [\frac {2 \, a c d x - 2 \, a^{2} e - {\left (c d x^{2} + a d\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{4 \, {\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}, \frac {a c d x - a^{2} e + {\left (c d x^{2} + a d\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{2 \, {\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*a*c*d*x - 2*a^2*e - (c*d*x^2 + a*d)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)))/(a^2*c^2
*x^2 + a^3*c), 1/2*(a*c*d*x - a^2*e + (c*d*x^2 + a*d)*sqrt(a*c)*arctan(sqrt(a*c)*x/a))/(a^2*c^2*x^2 + a^3*c)]

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giac [A]  time = 0.15, size = 48, normalized size = 0.84 \[ \frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a} + \frac {c d x - a e}{2 \, {\left (c x^{2} + a\right )} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 49, normalized size = 0.86 \[ \frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a}+\frac {2 c d x -2 a e}{4 \left (c \,x^{2}+a \right ) a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*c*d*x-2*a*e)/a/c/(c*x^2+a)+1/2*d/a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)

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maxima [A]  time = 2.99, size = 48, normalized size = 0.84 \[ \frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a} + \frac {c d x - a e}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.31, size = 44, normalized size = 0.77 \[ \frac {d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {c}}-\frac {\frac {e}{2\,c}-\frac {d\,x}{2\,a}}{c\,x^2+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a + c*x^2)^2,x)

[Out]

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

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sympy [A]  time = 0.30, size = 90, normalized size = 1.58 \[ d \left (- \frac {\sqrt {- \frac {1}{a^{3} c}} \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} c}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} c}} \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} c}} + x \right )}}{4}\right ) + \frac {- a e + c d x}{2 a^{2} c + 2 a c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*(-sqrt(-1/(a**3*c))*log(-a**2*sqrt(-1/(a**3*c)) + x)/4 + sqrt(-1/(a**3*c))*log(a**2*sqrt(-1/(a**3*c)) + x)/4
) + (-a*e + c*d*x)/(2*a**2*c + 2*a*c**2*x**2)

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