3.516 \(\int \frac{d+e x}{(a+c x^2)^3} \, dx\)

Optimal. Leaf size=75 \[ \frac{3 d x}{8 a^2 \left (a+c x^2\right )}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}-\frac{a e-c d x}{4 a c \left (a+c x^2\right )^2} \]

[Out]

-(a*e - c*d*x)/(4*a*c*(a + c*x^2)^2) + (3*d*x)/(8*a^2*(a + c*x^2)) + (3*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5
/2)*Sqrt[c])

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Rubi [A]  time = 0.0197236, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {639, 199, 205} \[ \frac{3 d x}{8 a^2 \left (a+c x^2\right )}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}-\frac{a e-c d x}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a + c*x^2)^3,x]

[Out]

-(a*e - c*d*x)/(4*a*c*(a + c*x^2)^2) + (3*d*x)/(8*a^2*(a + c*x^2)) + (3*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5
/2)*Sqrt[c])

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]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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]

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (a+c x^2\right )^3} \, dx &=-\frac{a e-c d x}{4 a c \left (a+c x^2\right )^2}+\frac{(3 d) \int \frac{1}{\left (a+c x^2\right )^2} \, dx}{4 a}\\ &=-\frac{a e-c d x}{4 a c \left (a+c x^2\right )^2}+\frac{3 d x}{8 a^2 \left (a+c x^2\right )}+\frac{(3 d) \int \frac{1}{a+c x^2} \, dx}{8 a^2}\\ &=-\frac{a e-c d x}{4 a c \left (a+c x^2\right )^2}+\frac{3 d x}{8 a^2 \left (a+c x^2\right )}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.0486735, size = 71, normalized size = 0.95 \[ \frac{\frac{\sqrt{a} \left (-2 a^2 e+5 a c d x+3 c^2 d x^3\right )}{\left (a+c x^2\right )^2}+3 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(a + c*x^2)^3,x]

[Out]

((Sqrt[a]*(-2*a^2*e + 5*a*c*d*x + 3*c^2*d*x^3))/(a + c*x^2)^2 + 3*Sqrt[c]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^
(5/2)*c)

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Maple [A]  time = 0.044, size = 65, normalized size = 0.9 \begin{align*}{\frac{2\,cdx-2\,ae}{8\,ac \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{3\,dx}{8\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{3\,d}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(c*x^2+a)^3,x)

[Out]

1/8*(2*c*d*x-2*a*e)/a/c/(c*x^2+a)^2+3/8*d*x/a^2/(c*x^2+a)+3/8*d/a^2/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))

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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((e*x+d)/(c*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.94332, size = 455, normalized size = 6.07 \begin{align*} \left [\frac{6 \, a c^{2} d x^{3} + 10 \, a^{2} c d x - 4 \, a^{3} e - 3 \,{\left (c^{2} d x^{4} + 2 \, a c d x^{2} + a^{2} d\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{16 \,{\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}, \frac{3 \, a c^{2} d x^{3} + 5 \, a^{2} c d x - 2 \, a^{3} e + 3 \,{\left (c^{2} d x^{4} + 2 \, a c d x^{2} + a^{2} d\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{8 \,{\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/16*(6*a*c^2*d*x^3 + 10*a^2*c*d*x - 4*a^3*e - 3*(c^2*d*x^4 + 2*a*c*d*x^2 + a^2*d)*sqrt(-a*c)*log((c*x^2 - 2*
sqrt(-a*c)*x - a)/(c*x^2 + a)))/(a^3*c^3*x^4 + 2*a^4*c^2*x^2 + a^5*c), 1/8*(3*a*c^2*d*x^3 + 5*a^2*c*d*x - 2*a^
3*e + 3*(c^2*d*x^4 + 2*a*c*d*x^2 + a^2*d)*sqrt(a*c)*arctan(sqrt(a*c)*x/a))/(a^3*c^3*x^4 + 2*a^4*c^2*x^2 + a^5*
c)]

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Sympy [A]  time = 0.658126, size = 124, normalized size = 1.65 \begin{align*} d \left (- \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16}\right ) + \frac{- 2 a^{2} e + 5 a c d x + 3 c^{2} d x^{3}}{8 a^{4} c + 16 a^{3} c^{2} x^{2} + 8 a^{2} c^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x**2+a)**3,x)

[Out]

d*(-3*sqrt(-1/(a**5*c))*log(-a**3*sqrt(-1/(a**5*c)) + x)/16 + 3*sqrt(-1/(a**5*c))*log(a**3*sqrt(-1/(a**5*c)) +
x)/16) + (-2*a**2*e + 5*a*c*d*x + 3*c**2*d*x**3)/(8*a**4*c + 16*a**3*c**2*x**2 + 8*a**2*c**3*x**4)

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Giac [A]  time = 1.31913, size = 82, normalized size = 1.09 \begin{align*} \frac{3 \, d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2}} + \frac{3 \, c^{2} d x^{3} + 5 \, a c d x - 2 \, a^{2} e}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+a)^3,x, algorithm="giac")

[Out]

3/8*d*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2) + 1/8*(3*c^2*d*x^3 + 5*a*c*d*x - 2*a^2*e)/((c*x^2 + a)^2*a^2*c)