3.1344 \(\int \frac{A+B x}{(d+e x) (a+c x^2)^2} \, dx\)

Optimal. Leaf size=195 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a B e \left (c d^2-a e^2\right )-A c d \left (3 a e^2+c d^2\right )\right )}{2 a^{3/2} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{a (B d-A e)-x (a B e+A c d)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{e^2 \log \left (a+c x^2\right ) (B d-A e)}{2 \left (a e^2+c d^2\right )^2}-\frac{e^2 (B d-A e) \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.280877, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {823, 801, 635, 205, 260} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a B e \left (c d^2-a e^2\right )-A c d \left (3 a e^2+c d^2\right )\right )}{2 a^{3/2} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{a (B d-A e)-x (a B e+A c d)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{e^2 \log \left (a+c x^2\right ) (B d-A e)}{2 \left (a e^2+c d^2\right )^2}-\frac{e^2 (B d-A e) \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)*(a + c*x^2)^2),x]

[Out]

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

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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]

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

Mathematica [A]  time = 0.184453, size = 158, normalized size = 0.81 \[ \frac{\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a B e \left (a e^2-c d^2\right )\right )}{a^{3/2} \sqrt{c}}+\frac{\left (a e^2+c d^2\right ) (a (A e-B d+B e x)+A c d x)}{a \left (a+c x^2\right )}+e^2 \log \left (a+c x^2\right ) (B d-A e)+2 e^2 (A e-B d) \log (d+e x)}{2 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.024, size = 495, normalized size = 2.5 \begin{align*}{\frac{{e}^{3}\ln \left ( ex+d \right ) A}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{{e}^{2}\ln \left ( ex+d \right ) Bd}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{Acxd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{xA{d}^{3}{c}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) a}}+{\frac{aBx{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{Bcx{d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{aA{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{Ac{d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{aBd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{Bc{d}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{\ln \left ( c{x}^{2}+a \right ) A{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) Bd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{3\,Acd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{A{d}^{3}{c}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{B{e}^{3}a}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{Bc{d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{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((B*x+A)/(e*x+d)/(c*x^2+a)^2,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 110.401, size = 1600, normalized size = 8.21 \begin{align*} \left [-\frac{2 \, B a^{2} c^{2} d^{3} - 2 \, A a^{2} c^{2} d^{2} e + 2 \, B a^{3} c d e^{2} - 2 \, A a^{3} c e^{3} +{\left (A a c^{2} d^{3} - B a^{2} c d^{2} e + 3 \, A a^{2} c d e^{2} + B a^{3} e^{3} +{\left (A c^{3} d^{3} - B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} + B a^{2} c e^{3}\right )} x^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) - 2 \,{\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x - 2 \,{\left (B a^{3} c d e^{2} - A a^{3} c e^{3} +{\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) + 4 \,{\left (B a^{3} c d e^{2} - A a^{3} c e^{3} +{\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )}{4 \,{\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4} +{\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{2}\right )}}, -\frac{B a^{2} c^{2} d^{3} - A a^{2} c^{2} d^{2} e + B a^{3} c d e^{2} - A a^{3} c e^{3} -{\left (A a c^{2} d^{3} - B a^{2} c d^{2} e + 3 \, A a^{2} c d e^{2} + B a^{3} e^{3} +{\left (A c^{3} d^{3} - B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} + B a^{2} c e^{3}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x -{\left (B a^{3} c d e^{2} - A a^{3} c e^{3} +{\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) + 2 \,{\left (B a^{3} c d e^{2} - A a^{3} c e^{3} +{\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4} +{\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*B*a^2*c^2*d^3 - 2*A*a^2*c^2*d^2*e + 2*B*a^3*c*d*e^2 - 2*A*a^3*c*e^3 + (A*a*c^2*d^3 - B*a^2*c*d^2*e +
3*A*a^2*c*d*e^2 + B*a^3*e^3 + (A*c^3*d^3 - B*a*c^2*d^2*e + 3*A*a*c^2*d*e^2 + B*a^2*c*e^3)*x^2)*sqrt(-a*c)*log(
(c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) - 2*(A*a*c^3*d^3 + B*a^2*c^2*d^2*e + A*a^2*c^2*d*e^2 + B*a^3*c*e^3)*
x - 2*(B*a^3*c*d*e^2 - A*a^3*c*e^3 + (B*a^2*c^2*d*e^2 - A*a^2*c^2*e^3)*x^2)*log(c*x^2 + a) + 4*(B*a^3*c*d*e^2
- A*a^3*c*e^3 + (B*a^2*c^2*d*e^2 - A*a^2*c^2*e^3)*x^2)*log(e*x + d))/(a^3*c^3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*
e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^2), -1/2*(B*a^2*c^2*d^3 - A*a^2*c^2*d^2*e + B*a^3*c*d*
e^2 - A*a^3*c*e^3 - (A*a*c^2*d^3 - B*a^2*c*d^2*e + 3*A*a^2*c*d*e^2 + B*a^3*e^3 + (A*c^3*d^3 - B*a*c^2*d^2*e +
3*A*a*c^2*d*e^2 + B*a^2*c*e^3)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) - (A*a*c^3*d^3 + B*a^2*c^2*d^2*e + A*a^2*c
^2*d*e^2 + B*a^3*c*e^3)*x - (B*a^3*c*d*e^2 - A*a^3*c*e^3 + (B*a^2*c^2*d*e^2 - A*a^2*c^2*e^3)*x^2)*log(c*x^2 +
a) + 2*(B*a^3*c*d*e^2 - A*a^3*c*e^3 + (B*a^2*c^2*d*e^2 - A*a^2*c^2*e^3)*x^2)*log(e*x + d))/(a^3*c^3*d^4 + 2*a^
4*c^2*d^2*e^2 + a^5*c*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.18533, size = 362, normalized size = 1.86 \begin{align*} \frac{{\left (B d e^{2} - A e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac{{\left (B d e^{3} - A e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}} + \frac{{\left (A c^{2} d^{3} - B a c d^{2} e + 3 \, A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{a c}} - \frac{B a c d^{3} - A a c d^{2} e + B a^{2} d e^{2} - A a^{2} e^{3} -{\left (A c^{2} d^{3} + B a c d^{2} e + A a c d e^{2} + B a^{2} e^{3}\right )} x}{2 \,{\left (c d^{2} + a e^{2}\right )}^{2}{\left (c x^{2} + a\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(B*d*e^2 - A*e^3)*log(c*x^2 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) - (B*d*e^3 - A*e^4)*log(abs(x*e + d))
/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) + 1/2*(A*c^2*d^3 - B*a*c*d^2*e + 3*A*a*c*d*e^2 + B*a^2*e^3)*arctan(c*x/
sqrt(a*c))/((a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4)*sqrt(a*c)) - 1/2*(B*a*c*d^3 - A*a*c*d^2*e + B*a^2*d*e^2 -
A*a^2*e^3 - (A*c^2*d^3 + B*a*c*d^2*e + A*a*c*d*e^2 + B*a^2*e^3)*x)/((c*d^2 + a*e^2)^2*(c*x^2 + a)*a)