3.1353 \(\int \frac{A+B x}{(d+e x)^2 (a+c x^2)^3} \, dx\)

Optimal. Leaf size=443 \[ -\frac{e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (-5 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (2 a B d e \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )-3 A \left (15 a^2 c d^2 e^4-5 a^3 e^6+5 a c^2 d^4 e^2+c^3 d^6\right )\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}-\frac{x \left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (3 a e^2+c d^2\right )\right )+a e \left (-5 a A e^2+6 a B d e+A c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}-\frac{a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}+\frac{e^4 \log \left (a+c x^2\right ) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{2 \left (a e^2+c d^2\right )^4}-\frac{e^4 \log (d+e x) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{\left (a e^2+c d^2\right )^4} \]

[Out]

-(e*(2*a*B*d*e*(c*d^2 - 11*a*e^2) - 3*A*(c^2*d^4 + 4*a*c*d^2*e^2 - 5*a^2*e^4)))/(8*a^2*(c*d^2 + a*e^2)^3*(d +
e*x)) - (a*(B*d - A*e) - (A*c*d + a*B*e)*x)/(4*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^2) - (a*e*(A*c*d^2 + 6*
a*B*d*e - 5*a*A*e^2) + (2*a*B*e*(c*d^2 - 2*a*e^2) - 3*A*c*d*(c*d^2 + 3*a*e^2))*x)/(8*a^2*(c*d^2 + a*e^2)^2*(d
+ e*x)*(a + c*x^2)) - (Sqrt[c]*(2*a*B*d*e*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4) - 3*A*(c^3*d^6 + 5*a*c^2*d^4
*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*(c*d^2 + a*e^2)^4) - (e^4*(5*B*c
*d^2 - 6*A*c*d*e - a*B*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^4 + (e^4*(5*B*c*d^2 - 6*A*c*d*e - a*B*e^2)*Log[a + c
*x^2])/(2*(c*d^2 + a*e^2)^4)

________________________________________________________________________________________

Rubi [A]  time = 0.724492, antiderivative size = 443, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (-5 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (2 a B d e \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )-3 A \left (15 a^2 c d^2 e^4-5 a^3 e^6+5 a c^2 d^4 e^2+c^3 d^6\right )\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}-\frac{x \left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (3 a e^2+c d^2\right )\right )+a e \left (-5 a A e^2+6 a B d e+A c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}-\frac{a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}+\frac{e^4 \log \left (a+c x^2\right ) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{2 \left (a e^2+c d^2\right )^4}-\frac{e^4 \log (d+e x) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{\left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^2*(a + c*x^2)^3),x]

[Out]

-(e*(2*a*B*d*e*(c*d^2 - 11*a*e^2) - 3*A*(c^2*d^4 + 4*a*c*d^2*e^2 - 5*a^2*e^4)))/(8*a^2*(c*d^2 + a*e^2)^3*(d +
e*x)) - (a*(B*d - A*e) - (A*c*d + a*B*e)*x)/(4*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^2) - (a*e*(A*c*d^2 + 6*
a*B*d*e - 5*a*A*e^2) + (2*a*B*e*(c*d^2 - 2*a*e^2) - 3*A*c*d*(c*d^2 + 3*a*e^2))*x)/(8*a^2*(c*d^2 + a*e^2)^2*(d
+ e*x)*(a + c*x^2)) - (Sqrt[c]*(2*a*B*d*e*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4) - 3*A*(c^3*d^6 + 5*a*c^2*d^4
*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*(c*d^2 + a*e^2)^4) - (e^4*(5*B*c
*d^2 - 6*A*c*d*e - a*B*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^4 + (e^4*(5*B*c*d^2 - 6*A*c*d*e - a*B*e^2)*Log[a + c
*x^2])/(2*(c*d^2 + a*e^2)^4)

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)^2 \left (a+c x^2\right )^3} \, dx &=-\frac{a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac{\int \frac{-c \left (3 A c d^2-2 a B d e+5 a A e^2\right )-4 c e (A c d+a B e) x}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx}{4 a c \left (c d^2+a e^2\right )}\\ &=-\frac{a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (A c d^2+6 a B d e-5 a A e^2\right )+\left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac{\int \frac{-c^2 \left (2 a B d e \left (c d^2+7 a e^2\right )-3 A \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )\right )-2 c^2 e \left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{(d+e x)^2 \left (a+c x^2\right )} \, dx}{8 a^2 c^2 \left (c d^2+a e^2\right )^2}\\ &=-\frac{a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (A c d^2+6 a B d e-5 a A e^2\right )+\left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac{\int \left (\frac{c^2 e^2 \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (c^2 d^4+4 a c d^2 e^2-5 a^2 e^4\right )\right )}{\left (c d^2+a e^2\right ) (d+e x)^2}+\frac{8 a^2 c^2 e^5 \left (-5 B c d^2+6 A c d e+a B e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac{c^3 \left (-2 a B d e \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )+3 A \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )+8 a^2 e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{8 a^2 c^2 \left (c d^2+a e^2\right )^2}\\ &=-\frac{e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (c^2 d^4+4 a c d^2 e^2-5 a^2 e^4\right )\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac{a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (A c d^2+6 a B d e-5 a A e^2\right )+\left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}-\frac{e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac{c \int \frac{-2 a B d e \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )+3 A \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )+8 a^2 e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) x}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^4}\\ &=-\frac{e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (c^2 d^4+4 a c d^2 e^2-5 a^2 e^4\right )\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac{a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (A c d^2+6 a B d e-5 a A e^2\right )+\left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}-\frac{e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac{\left (c e^4 \left (5 B c d^2-6 A c d e-a B e^2\right )\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^4}-\frac{\left (c \left (2 a B d e \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-3 A \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )\right )\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^4}\\ &=-\frac{e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (c^2 d^4+4 a c d^2 e^2-5 a^2 e^4\right )\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac{a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (A c d^2+6 a B d e-5 a A e^2\right )+\left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}-\frac{\sqrt{c} \left (2 a B d e \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-3 A \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (c d^2+a e^2\right )^4}-\frac{e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac{e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4}\\ \end{align*}

Mathematica [A]  time = 0.5434, size = 378, normalized size = 0.85 \[ \frac{\frac{\left (a e^2+c d^2\right ) \left (a^2 c e^2 (A e (16 d-7 e x)-2 B d (6 d-7 e x))+4 a^3 B e^4-2 a c^2 d^2 e x (B d-6 A e)+3 A c^3 d^4 x\right )}{a^2 \left (a+c x^2\right )}+\frac{2 \left (a e^2+c d^2\right )^2 \left (a^2 B e^2-a c (A e (e x-2 d)+B d (d-2 e x))+A c^2 d^2 x\right )}{a \left (a+c x^2\right )^2}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (3 A \left (15 a^2 c d^2 e^4-5 a^3 e^6+5 a c^2 d^4 e^2+c^3 d^6\right )+2 a B d e \left (15 a^2 e^4-10 a c d^2 e^2-c^2 d^4\right )\right )}{a^{5/2}}-4 e^4 \log \left (a+c x^2\right ) \left (a B e^2+6 A c d e-5 B c d^2\right )-\frac{8 e^4 \left (a e^2+c d^2\right ) (A e-B d)}{d+e x}+8 e^4 \log (d+e x) \left (a B e^2+6 A c d e-5 B c d^2\right )}{8 \left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^2*(a + c*x^2)^3),x]

[Out]

((-8*e^4*(-(B*d) + A*e)*(c*d^2 + a*e^2))/(d + e*x) + ((c*d^2 + a*e^2)*(4*a^3*B*e^4 + 3*A*c^3*d^4*x - 2*a*c^2*d
^2*e*(B*d - 6*A*e)*x + a^2*c*e^2*(-2*B*d*(6*d - 7*e*x) + A*e*(16*d - 7*e*x))))/(a^2*(a + c*x^2)) + (2*(c*d^2 +
a*e^2)^2*(a^2*B*e^2 + A*c^2*d^2*x - a*c*(B*d*(d - 2*e*x) + A*e*(-2*d + e*x))))/(a*(a + c*x^2)^2) + (Sqrt[c]*(
2*a*B*d*e*(-(c^2*d^4) - 10*a*c*d^2*e^2 + 15*a^2*e^4) + 3*A*(c^3*d^6 + 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a
^3*e^6))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 8*e^4*(-5*B*c*d^2 + 6*A*c*d*e + a*B*e^2)*Log[d + e*x] - 4*e^4*
(-5*B*c*d^2 + 6*A*c*d*e + a*B*e^2)*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^4)

________________________________________________________________________________________

Maple [B]  time = 0.025, size = 1410, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*d^3*a*e^3-7/4*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*a*d^4*e^2+5/8*c^3/(a*e^2+c
*d^2)^4/(c*x^2+a)^2*x^3*A*d^2*e^4+3/2*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^3*B*d^3*e^3+45/8*c^2/(a*e^2+c*d^2)^4/(
a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A*d^2*e^4+6*e^5/(a*e^2+c*d^2)^4*ln(e*x+d)*A*c*d+1/2*c^3/(a*e^2+c*d^2)^4/(c*
x^2+a)^2*A*d^5*e-3*c/(a*e^2+c*d^2)^4*ln(c*x^2+a)*A*d*e^5+5/2*c/(a*e^2+c*d^2)^4*ln(c*x^2+a)*B*d^2*e^4-5*e^4/(a*
e^2+c*d^2)^4*ln(e*x+d)*B*c*d^2-e^5/(a*e^2+c*d^2)^3/(e*x+d)*A-1/2/(a*e^2+c*d^2)^4*a*ln(c*x^2+a)*B*e^6+e^6/(a*e^
2+c*d^2)^4*ln(e*x+d)*a*B+e^4/(a*e^2+c*d^2)^3/(e*x+d)*B*d+3/4/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*a^3*e^6-1/4*c^3/(a*
e^2+c*d^2)^4/(c*x^2+a)^2*B*d^6+15/8*c^3/(a*e^2+c*d^2)^4/a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A*d^4*e^2+15/4*c
/(a*e^2+c*d^2)^4*a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*B*d*e^5-1/4*c^3/(a*e^2+c*d^2)^4/a/(a*c)^(1/2)*arctan(x*
c/(a*c)^(1/2))*B*d^5*e-1/4*c^4/(a*e^2+c*d^2)^4/(c*x^2+a)^2/a*x^3*B*d^5*e+2*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^2
*A*a*d*e^5-c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^2*B*a*d^2*e^4+3/8*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a*x*A*d^2*e^4+5
/2*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a*x*B*d^3*e^3+9/4*c/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a^2*x*B*d*e^5+15/8*c^4/(a*e
^2+c*d^2)^4/(c*x^2+a)^2/a*x^3*A*d^4*e^2+7/4*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a*x^3*B*d*e^5-15/8*c/(a*e^2+c*d^2)
^4*a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A*e^6-5/2*c^2/(a*e^2+c*d^2)^4/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*B*d
^3*e^3+17/8*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x*A*d^4*e^2+1/4*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x*B*d^5*e-7/8*c^2/
(a*e^2+c*d^2)^4/(c*x^2+a)^2*a*x^3*A*e^6+3/8*c^5/(a*e^2+c*d^2)^4/(c*x^2+a)^2/a^2*x^3*A*d^6+1/2*c/(a*e^2+c*d^2)^
4/(c*x^2+a)^2*x^2*B*a^2*e^6-9/8*c/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a^2*x*A*e^6+5/2*c/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*
a^2*d*e^5-3/4*c/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*a^2*d^2*e^4+2*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^2*A*d^3*e^3-3/2*
c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^2*B*d^4*e^2+5/8*c^4/(a*e^2+c*d^2)^4/(c*x^2+a)^2/a*x*A*d^6+3/8*c^4/(a*e^2+c*d
^2)^4/a^2/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A*d^6

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [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)^2/(c*x^2+a)^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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)**2/(c*x**2+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.27589, size = 1129, normalized size = 2.55 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*A*c^4*d^6*e^2 - 2*B*a*c^3*d^5*e^3 + 15*A*a*c^3*d^4*e^4 - 20*B*a^2*c^2*d^3*e^5 + 45*A*a^2*c^2*d^2*e^6 +
30*B*a^3*c*d*e^7 - 15*A*a^3*c*e^8)*arctan((c*d - c*d^2/(x*e + d) - a*e^2/(x*e + d))*e^(-1)/sqrt(a*c))*e^(-2)/(
(a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8)*sqrt(a*c)) + 1/2*(5*B*c*d^2*
e^4 - 6*A*c*d*e^5 - B*a*e^6)*log(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)/(c^4*d^8 + 4*a*c
^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8) + (B*d*e^10/(x*e + d) - A*e^11/(x*e + d))/(c^3*d^6
*e^6 + 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 + a^3*e^12) + 1/8*(3*A*c^5*d^5*e - 2*B*a*c^4*d^4*e^2 + 14*A*a*c^4*d^
3*e^3 + 32*B*a^2*c^3*d^2*e^4 - 29*A*a^2*c^3*d*e^5 - 6*B*a^3*c^2*e^6 - (9*A*c^5*d^6*e^2 - 6*B*a*c^4*d^5*e^3 + 4
1*A*a*c^4*d^4*e^4 + 116*B*a^2*c^3*d^3*e^5 - 121*A*a^2*c^3*d^2*e^6 - 38*B*a^3*c^2*d*e^7 + 7*A*a^3*c^2*e^8)*e^(-
1)/(x*e + d) + (9*A*c^5*d^7*e^3 - 6*B*a*c^4*d^6*e^4 + 45*A*a*c^4*d^5*e^5 + 140*B*a^2*c^3*d^4*e^6 - 145*A*a^2*c
^3*d^3*e^7 - 22*B*a^3*c^2*d^2*e^8 - 21*A*a^3*c^2*d*e^9 - 8*B*a^4*c*e^10)*e^(-2)/(x*e + d)^2 - (3*A*c^5*d^8*e^4
- 2*B*a*c^4*d^7*e^5 + 18*A*a*c^4*d^6*e^6 + 58*B*a^2*c^3*d^5*e^7 - 60*A*a^2*c^3*d^4*e^8 + 26*B*a^3*c^2*d^3*e^9
- 66*A*a^3*c^2*d^2*e^10 - 34*B*a^4*c*d*e^11 + 9*A*a^4*c*e^12)*e^(-3)/(x*e + d)^3)/((c*d^2 + a*e^2)^4*a^2*(c -
2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)^2)