### 3.522 $$\int \frac{d+e x}{(a+c x^2)^4} \, dx$$

Optimal. Leaf size=93 $\frac{5 d x}{16 a^3 \left (a+c x^2\right )}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \sqrt{c}}-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3}$

[Out]

-(a*e - c*d*x)/(6*a*c*(a + c*x^2)^3) + (5*d*x)/(24*a^2*(a + c*x^2)^2) + (5*d*x)/(16*a^3*(a + c*x^2)) + (5*d*Ar
cTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*Sqrt[c])

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Rubi [A]  time = 0.0280046, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {639, 199, 205} $\frac{5 d x}{16 a^3 \left (a+c x^2\right )}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \sqrt{c}}-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/(a + c*x^2)^4,x]

[Out]

-(a*e - c*d*x)/(6*a*c*(a + c*x^2)^3) + (5*d*x)/(24*a^2*(a + c*x^2)^2) + (5*d*x)/(16*a^3*(a + c*x^2)) + (5*d*Ar
cTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*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 )^4} \, dx &=-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac{(5 d) \int \frac{1}{\left (a+c x^2\right )^3} \, dx}{6 a}\\ &=-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac{(5 d) \int \frac{1}{\left (a+c x^2\right )^2} \, dx}{8 a^2}\\ &=-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac{5 d x}{16 a^3 \left (a+c x^2\right )}+\frac{(5 d) \int \frac{1}{a+c x^2} \, dx}{16 a^3}\\ &=-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac{5 d x}{16 a^3 \left (a+c x^2\right )}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.0463111, size = 83, normalized size = 0.89 $\frac{\frac{\sqrt{a} \left (33 a^2 c d x-8 a^3 e+40 a c^2 d x^3+15 c^3 d x^5\right )}{\left (a+c x^2\right )^3}+15 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{48 a^{7/2} c}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)/(a + c*x^2)^4,x]

[Out]

((Sqrt[a]*(-8*a^3*e + 33*a^2*c*d*x + 40*a*c^2*d*x^3 + 15*c^3*d*x^5))/(a + c*x^2)^3 + 15*Sqrt[c]*d*ArcTan[(Sqrt
[c]*x)/Sqrt[a]])/(48*a^(7/2)*c)

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Maple [A]  time = 0.044, size = 81, normalized size = 0.9 \begin{align*}{\frac{2\,cdx-2\,ae}{12\,ac \left ( c{x}^{2}+a \right ) ^{3}}}+{\frac{5\,dx}{24\,{a}^{2} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{5\,dx}{16\,{a}^{3} \left ( c{x}^{2}+a \right ) }}+{\frac{5\,d}{16\,{a}^{3}}\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)^4,x)

[Out]

1/12*(2*c*d*x-2*a*e)/a/c/(c*x^2+a)^3+5/24*d*x/a^2/(c*x^2+a)^2+5/16*d*x/a^3/(c*x^2+a)+5/16*d/a^3/(a*c)^(1/2)*ar
ctan(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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.09654, size = 608, normalized size = 6.54 \begin{align*} \left [\frac{30 \, a c^{3} d x^{5} + 80 \, a^{2} c^{2} d x^{3} + 66 \, a^{3} c d x - 16 \, a^{4} e - 15 \,{\left (c^{3} d x^{6} + 3 \, a c^{2} d x^{4} + 3 \, a^{2} c d x^{2} + a^{3} d\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{96 \,{\left (a^{4} c^{4} x^{6} + 3 \, a^{5} c^{3} x^{4} + 3 \, a^{6} c^{2} x^{2} + a^{7} c\right )}}, \frac{15 \, a c^{3} d x^{5} + 40 \, a^{2} c^{2} d x^{3} + 33 \, a^{3} c d x - 8 \, a^{4} e + 15 \,{\left (c^{3} d x^{6} + 3 \, a c^{2} d x^{4} + 3 \, a^{2} c d x^{2} + a^{3} d\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{48 \,{\left (a^{4} c^{4} x^{6} + 3 \, a^{5} c^{3} x^{4} + 3 \, a^{6} c^{2} x^{2} + a^{7} c\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+a)^4,x, algorithm="fricas")

[Out]

[1/96*(30*a*c^3*d*x^5 + 80*a^2*c^2*d*x^3 + 66*a^3*c*d*x - 16*a^4*e - 15*(c^3*d*x^6 + 3*a*c^2*d*x^4 + 3*a^2*c*d
*x^2 + a^3*d)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)))/(a^4*c^4*x^6 + 3*a^5*c^3*x^4 + 3*a^6*c
^2*x^2 + a^7*c), 1/48*(15*a*c^3*d*x^5 + 40*a^2*c^2*d*x^3 + 33*a^3*c*d*x - 8*a^4*e + 15*(c^3*d*x^6 + 3*a*c^2*d*
x^4 + 3*a^2*c*d*x^2 + a^3*d)*sqrt(a*c)*arctan(sqrt(a*c)*x/a))/(a^4*c^4*x^6 + 3*a^5*c^3*x^4 + 3*a^6*c^2*x^2 + a
^7*c)]

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Sympy [A]  time = 0.838742, size = 150, normalized size = 1.61 \begin{align*} d \left (- \frac{5 \sqrt{- \frac{1}{a^{7} c}} \log{\left (- a^{4} \sqrt{- \frac{1}{a^{7} c}} + x \right )}}{32} + \frac{5 \sqrt{- \frac{1}{a^{7} c}} \log{\left (a^{4} \sqrt{- \frac{1}{a^{7} c}} + x \right )}}{32}\right ) + \frac{- 8 a^{3} e + 33 a^{2} c d x + 40 a c^{2} d x^{3} + 15 c^{3} d x^{5}}{48 a^{6} c + 144 a^{5} c^{2} x^{2} + 144 a^{4} c^{3} x^{4} + 48 a^{3} c^{4} x^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x**2+a)**4,x)

[Out]

d*(-5*sqrt(-1/(a**7*c))*log(-a**4*sqrt(-1/(a**7*c)) + x)/32 + 5*sqrt(-1/(a**7*c))*log(a**4*sqrt(-1/(a**7*c)) +
x)/32) + (-8*a**3*e + 33*a**2*c*d*x + 40*a*c**2*d*x**3 + 15*c**3*d*x**5)/(48*a**6*c + 144*a**5*c**2*x**2 + 14
4*a**4*c**3*x**4 + 48*a**3*c**4*x**6)

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Giac [A]  time = 1.3043, size = 99, normalized size = 1.06 \begin{align*} \frac{5 \, d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3}} + \frac{15 \, c^{3} d x^{5} + 40 \, a c^{2} d x^{3} + 33 \, a^{2} c d x - 8 \, a^{3} e}{48 \,{\left (c x^{2} + a\right )}^{3} a^{3} c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+a)^4,x, algorithm="giac")

[Out]

5/16*d*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^3) + 1/48*(15*c^3*d*x^5 + 40*a*c^2*d*x^3 + 33*a^2*c*d*x - 8*a^3*e)/(
(c*x^2 + a)^3*a^3*c)