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

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

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Rubi [A]  time = 0.72, antiderivative size = 443, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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 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]

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 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 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])

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*}

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Mathematica [A]  time = 0.49, size = 378, normalized size = 0.85 \begin {gather*} \frac {\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 {\left (a e^2+c d^2\right ) \left (4 a^3 B e^4+a^2 c e^2 (A e (16 d-7 e x)-2 B d (6 d-7 e x))-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 {\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 (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\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} \end {gather*}

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)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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giac [A]  time = 0.22, size = 836, normalized size = 1.89 \begin {gather*} \frac {{\left (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}\right )} \arctan \left (\frac {{\left (c d - \frac {c d^{2}}{x e + d} - \frac {a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{8 \, {\left (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}\right )} \sqrt {a c}} + \frac {{\left (5 \, B c d^{2} e^{4} - 6 \, A c d e^{5} - B a e^{6}\right )} \log \left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \, {\left (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}\right )}} + \frac {\frac {B d e^{10}}{x e + d} - \frac {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}} + \frac {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} - \frac {{\left (9 \, A c^{5} d^{6} e^{2} - 6 \, B a c^{4} d^{5} e^{3} + 41 \, 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}\right )} e^{\left (-1\right )}}{x e + d} + \frac {{\left (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}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {{\left (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}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{8 \, {\left (c d^{2} + a e^{2}\right )}^{4} a^{2} {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}^{2}} \end {gather*}

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)

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maple [B]  time = 0.12, size = 1410, normalized size = 3.18

result too large to display

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]

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

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maxima [B]  time = 1.39, size = 997, normalized size = 2.25 \begin {gather*} \frac {{\left (5 \, B c d^{2} e^{4} - 6 \, A c d e^{5} - B a e^{6}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (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}\right )}} - \frac {{\left (5 \, B c d^{2} e^{4} - 6 \, A c d e^{5} - B a e^{6}\right )} \log \left (e x + d\right )}{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}} + \frac {{\left (3 \, A c^{4} d^{6} - 2 \, B a c^{3} d^{5} e + 15 \, A a c^{3} d^{4} e^{2} - 20 \, B a^{2} c^{2} d^{3} e^{3} + 45 \, A a^{2} c^{2} d^{2} e^{4} + 30 \, B a^{3} c d e^{5} - 15 \, A a^{3} c e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, {\left (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}\right )} \sqrt {a c}} - \frac {2 \, B a^{2} c^{2} d^{5} - 4 \, A a^{2} c^{2} d^{4} e + 12 \, B a^{3} c d^{3} e^{2} - 20 \, A a^{3} c d^{2} e^{3} - 14 \, B a^{4} d e^{4} + 8 \, A a^{4} e^{5} - {\left (3 \, A c^{4} d^{4} e - 2 \, B a c^{3} d^{3} e^{2} + 12 \, A a c^{3} d^{2} e^{3} + 22 \, B a^{2} c^{2} d e^{4} - 15 \, A a^{2} c^{2} e^{5}\right )} x^{4} - {\left (3 \, A c^{4} d^{5} - 2 \, B a c^{3} d^{4} e + 12 \, A a c^{3} d^{3} e^{2} + 2 \, B a^{2} c^{2} d^{2} e^{3} + 9 \, A a^{2} c^{2} d e^{4} + 4 \, B a^{3} c e^{5}\right )} x^{3} - {\left (5 \, A a c^{3} d^{4} e - 10 \, B a^{2} c^{2} d^{3} e^{2} + 28 \, A a^{2} c^{2} d^{2} e^{3} + 38 \, B a^{3} c d e^{4} - 25 \, A a^{3} c e^{5}\right )} x^{2} - {\left (5 \, A a c^{3} d^{5} + 16 \, A a^{2} c^{2} d^{3} e^{2} + 6 \, B a^{3} c d^{2} e^{3} + 11 \, A a^{3} c d e^{4} + 6 \, B a^{4} e^{5}\right )} x}{8 \, {\left (a^{4} c^{3} d^{7} + 3 \, a^{5} c^{2} d^{5} e^{2} + 3 \, a^{6} c d^{3} e^{4} + a^{7} d e^{6} + {\left (a^{2} c^{5} d^{6} e + 3 \, a^{3} c^{4} d^{4} e^{3} + 3 \, a^{4} c^{3} d^{2} e^{5} + a^{5} c^{2} e^{7}\right )} x^{5} + {\left (a^{2} c^{5} d^{7} + 3 \, a^{3} c^{4} d^{5} e^{2} + 3 \, a^{4} c^{3} d^{3} e^{4} + a^{5} c^{2} d e^{6}\right )} x^{4} + 2 \, {\left (a^{3} c^{4} d^{6} e + 3 \, a^{4} c^{3} d^{4} e^{3} + 3 \, a^{5} c^{2} d^{2} e^{5} + a^{6} c e^{7}\right )} x^{3} + 2 \, {\left (a^{3} c^{4} d^{7} + 3 \, a^{4} c^{3} d^{5} e^{2} + 3 \, a^{5} c^{2} d^{3} e^{4} + a^{6} c d e^{6}\right )} x^{2} + {\left (a^{4} c^{3} d^{6} e + 3 \, a^{5} c^{2} d^{4} e^{3} + 3 \, a^{6} c d^{2} e^{5} + a^{7} e^{7}\right )} x\right )}} \end {gather*}

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]

1/2*(5*B*c*d^2*e^4 - 6*A*c*d*e^5 - B*a*e^6)*log(c*x^2 + a)/(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) - (5*B*c*d^2*e^4 - 6*A*c*d*e^5 - B*a*e^6)*log(e*x + d)/(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) + 1/8*(3*A*c^4*d^6 - 2*B*a*c^3*d^5*e + 15*A*a*c^3*d^4*e^2 - 20*B
*a^2*c^2*d^3*e^3 + 45*A*a^2*c^2*d^2*e^4 + 30*B*a^3*c*d*e^5 - 15*A*a^3*c*e^6)*arctan(c*x/sqrt(a*c))/((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)) - 1/8*(2*B*a^2*c^2*d^5 - 4*
A*a^2*c^2*d^4*e + 12*B*a^3*c*d^3*e^2 - 20*A*a^3*c*d^2*e^3 - 14*B*a^4*d*e^4 + 8*A*a^4*e^5 - (3*A*c^4*d^4*e - 2*
B*a*c^3*d^3*e^2 + 12*A*a*c^3*d^2*e^3 + 22*B*a^2*c^2*d*e^4 - 15*A*a^2*c^2*e^5)*x^4 - (3*A*c^4*d^5 - 2*B*a*c^3*d
^4*e + 12*A*a*c^3*d^3*e^2 + 2*B*a^2*c^2*d^2*e^3 + 9*A*a^2*c^2*d*e^4 + 4*B*a^3*c*e^5)*x^3 - (5*A*a*c^3*d^4*e -
10*B*a^2*c^2*d^3*e^2 + 28*A*a^2*c^2*d^2*e^3 + 38*B*a^3*c*d*e^4 - 25*A*a^3*c*e^5)*x^2 - (5*A*a*c^3*d^5 + 16*A*a
^2*c^2*d^3*e^2 + 6*B*a^3*c*d^2*e^3 + 11*A*a^3*c*d*e^4 + 6*B*a^4*e^5)*x)/(a^4*c^3*d^7 + 3*a^5*c^2*d^5*e^2 + 3*a
^6*c*d^3*e^4 + a^7*d*e^6 + (a^2*c^5*d^6*e + 3*a^3*c^4*d^4*e^3 + 3*a^4*c^3*d^2*e^5 + a^5*c^2*e^7)*x^5 + (a^2*c^
5*d^7 + 3*a^3*c^4*d^5*e^2 + 3*a^4*c^3*d^3*e^4 + a^5*c^2*d*e^6)*x^4 + 2*(a^3*c^4*d^6*e + 3*a^4*c^3*d^4*e^3 + 3*
a^5*c^2*d^2*e^5 + a^6*c*e^7)*x^3 + 2*(a^3*c^4*d^7 + 3*a^4*c^3*d^5*e^2 + 3*a^5*c^2*d^3*e^4 + a^6*c*d*e^6)*x^2 +
(a^4*c^3*d^6*e + 3*a^5*c^2*d^4*e^3 + 3*a^6*c*d^2*e^5 + a^7*e^7)*x)

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mupad [B]  time = 5.51, size = 3015, normalized size = 6.81

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(5*A*c^2*d^3 + 6*B*a^2*e^3 + 11*A*a*c*d*e^2))/(8*a*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) - (4*A*a^2*e^5 + B
*c^2*d^5 - 7*B*a^2*d*e^4 - 2*A*c^2*d^4*e - 10*A*a*c*d^2*e^3 + 6*B*a*c*d^3*e^2)/(4*(a*e^2 + c*d^2)*(a^2*e^4 + c
^2*d^4 + 2*a*c*d^2*e^2)) + (x^3*(3*A*c^3*d^3 + 4*B*a^2*c*e^3 + 9*A*a*c^2*d*e^2 - 2*B*a*c^2*d^2*e))/(8*a^2*(a^2
*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (x^4*(3*A*c^4*d^4*e - 15*A*a^2*c^2*e^5 + 12*A*a*c^3*d^2*e^3 - 2*B*a*c^3*d^3
*e^2 + 22*B*a^2*c^2*d*e^4))/(8*a^2*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)) + (x^2*(5*A*c^3*d^
4*e - 25*A*a^2*c*e^5 + 28*A*a*c^2*d^2*e^3 - 10*B*a*c^2*d^3*e^2 + 38*B*a^2*c*d*e^4))/(8*a*(a*e^2 + c*d^2)*(a^2*
e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))/(a^2*d + c^2*d*x^4 + c^2*e*x^5 + a^2*e*x + 2*a*c*d*x^2 + 2*a*c*e*x^3) - (log(
d + e*x)*(c*(5*B*d^2*e^4 - 6*A*d*e^5) - B*a*e^6))/(a^4*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a
^2*c^2*d^4*e^4) + (log(576*B^2*a^14*e^16*(-a^5*c)^(1/2) - 9*A^2*c^8*d^16*(-a^5*c)^(3/2) - 225*A^2*a^8*e^16*(-a
^5*c)^(3/2) + 19836*A^2*a^2*d^2*e^14*(-a^5*c)^(5/2) + 4056*B^2*a^2*d^4*e^12*(-a^5*c)^(5/2) + 3708*B^2*a^8*d^2*
e^14*(-a^5*c)^(3/2) + 23796*A^2*c^2*d^6*e^10*(-a^5*c)^(5/2) + 13840*B^2*c^2*d^8*e^8*(-a^5*c)^(5/2) + 576*B^2*a
^16*c*e^16*x + 9*A^2*a^7*c^10*d^16*x + 225*A^2*a^15*c^2*e^16*x - 19236*A*B*a^2*d^3*e^13*(-a^5*c)^(5/2) - 33540
*A*B*c^2*d^7*e^9*(-a^5*c)^(5/2) + 40572*A^2*a*c*d^4*e^12*(-a^5*c)^(5/2) + 21820*B^2*a*c*d^6*e^10*(-a^5*c)^(5/2
) + 108*A^2*a^8*c^9*d^14*e^2*x + 684*A^2*a^9*c^8*d^12*e^4*x + 2340*A^2*a^10*c^7*d^10*e^6*x + 4590*A^2*a^11*c^6
*d^8*e^8*x + 23796*A^2*a^12*c^5*d^6*e^10*x + 40572*A^2*a^13*c^4*d^4*e^12*x + 19836*A^2*a^14*c^3*d^2*e^14*x + 4
*B^2*a^9*c^8*d^14*e^2*x + 88*B^2*a^10*c^7*d^12*e^4*x + 444*B^2*a^11*c^6*d^10*e^6*x + 13840*B^2*a^12*c^5*d^8*e^
8*x + 21820*B^2*a^13*c^4*d^6*e^10*x + 4056*B^2*a^14*c^3*d^4*e^12*x - 3708*B^2*a^15*c^2*d^2*e^14*x - 108*A^2*a*
c^7*d^14*e^2*(-a^5*c)^(3/2) - 6012*A*B*a^8*d*e^15*(-a^5*c)^(3/2) - 684*A^2*a^2*c^6*d^12*e^4*(-a^5*c)^(3/2) - 2
340*A^2*a^3*c^5*d^10*e^6*(-a^5*c)^(3/2) - 4590*A^2*a^4*c^4*d^8*e^8*(-a^5*c)^(3/2) - 4*B^2*a^2*c^6*d^14*e^2*(-a
^5*c)^(3/2) - 88*B^2*a^3*c^5*d^12*e^4*(-a^5*c)^(3/2) - 444*B^2*a^4*c^4*d^10*e^6*(-a^5*c)^(3/2) - 12*A*B*a^8*c^
9*d^15*e*x + 6012*A*B*a^15*c^2*d*e^15*x - 57348*A*B*a*c*d^5*e^11*(-a^5*c)^(5/2) + 12*A*B*a*c^7*d^15*e*(-a^5*c)
^(3/2) - 204*A*B*a^9*c^8*d^13*e^3*x - 972*A*B*a^10*c^7*d^11*e^5*x - 2220*A*B*a^11*c^6*d^9*e^7*x - 33540*A*B*a^
12*c^5*d^7*e^9*x - 57348*A*B*a^13*c^4*d^5*e^11*x - 19236*A*B*a^14*c^3*d^3*e^13*x + 204*A*B*a^2*c^6*d^13*e^3*(-
a^5*c)^(3/2) + 972*A*B*a^3*c^5*d^11*e^5*(-a^5*c)^(3/2) + 2220*A*B*a^4*c^4*d^9*e^7*(-a^5*c)^(3/2))*(c*(a^5*((5*
B*d^2*e^4)/2 - 3*A*d*e^5) + a^2*((45*A*d^2*e^4*(-a^5*c)^(1/2))/16 - (5*B*d^3*e^3*(-a^5*c)^(1/2))/4)) - a^3*((1
5*A*e^6*(-a^5*c)^(1/2))/16 - (15*B*d*e^5*(-a^5*c)^(1/2))/8) - (B*a^6*e^6)/2 + a*c^2*((15*A*d^4*e^2*(-a^5*c)^(1
/2))/16 - (B*d^5*e*(-a^5*c)^(1/2))/8) + (3*A*c^3*d^6*(-a^5*c)^(1/2))/16))/(a^9*e^8 + a^5*c^4*d^8 + 4*a^8*c*d^2
*e^6 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4) - (log(225*A^2*a^8*e^16*(-a^5*c)^(3/2) + 9*A^2*c^8*d^16*(-a^5*c)
^(3/2) - 576*B^2*a^14*e^16*(-a^5*c)^(1/2) - 19836*A^2*a^2*d^2*e^14*(-a^5*c)^(5/2) - 4056*B^2*a^2*d^4*e^12*(-a^
5*c)^(5/2) - 3708*B^2*a^8*d^2*e^14*(-a^5*c)^(3/2) - 23796*A^2*c^2*d^6*e^10*(-a^5*c)^(5/2) - 13840*B^2*c^2*d^8*
e^8*(-a^5*c)^(5/2) + 576*B^2*a^16*c*e^16*x + 9*A^2*a^7*c^10*d^16*x + 225*A^2*a^15*c^2*e^16*x + 19236*A*B*a^2*d
^3*e^13*(-a^5*c)^(5/2) + 33540*A*B*c^2*d^7*e^9*(-a^5*c)^(5/2) - 40572*A^2*a*c*d^4*e^12*(-a^5*c)^(5/2) - 21820*
B^2*a*c*d^6*e^10*(-a^5*c)^(5/2) + 108*A^2*a^8*c^9*d^14*e^2*x + 684*A^2*a^9*c^8*d^12*e^4*x + 2340*A^2*a^10*c^7*
d^10*e^6*x + 4590*A^2*a^11*c^6*d^8*e^8*x + 23796*A^2*a^12*c^5*d^6*e^10*x + 40572*A^2*a^13*c^4*d^4*e^12*x + 198
36*A^2*a^14*c^3*d^2*e^14*x + 4*B^2*a^9*c^8*d^14*e^2*x + 88*B^2*a^10*c^7*d^12*e^4*x + 444*B^2*a^11*c^6*d^10*e^6
*x + 13840*B^2*a^12*c^5*d^8*e^8*x + 21820*B^2*a^13*c^4*d^6*e^10*x + 4056*B^2*a^14*c^3*d^4*e^12*x - 3708*B^2*a^
15*c^2*d^2*e^14*x + 108*A^2*a*c^7*d^14*e^2*(-a^5*c)^(3/2) + 6012*A*B*a^8*d*e^15*(-a^5*c)^(3/2) + 684*A^2*a^2*c
^6*d^12*e^4*(-a^5*c)^(3/2) + 2340*A^2*a^3*c^5*d^10*e^6*(-a^5*c)^(3/2) + 4590*A^2*a^4*c^4*d^8*e^8*(-a^5*c)^(3/2
) + 4*B^2*a^2*c^6*d^14*e^2*(-a^5*c)^(3/2) + 88*B^2*a^3*c^5*d^12*e^4*(-a^5*c)^(3/2) + 444*B^2*a^4*c^4*d^10*e^6*
(-a^5*c)^(3/2) - 12*A*B*a^8*c^9*d^15*e*x + 6012*A*B*a^15*c^2*d*e^15*x + 57348*A*B*a*c*d^5*e^11*(-a^5*c)^(5/2)
- 12*A*B*a*c^7*d^15*e*(-a^5*c)^(3/2) - 204*A*B*a^9*c^8*d^13*e^3*x - 972*A*B*a^10*c^7*d^11*e^5*x - 2220*A*B*a^1
1*c^6*d^9*e^7*x - 33540*A*B*a^12*c^5*d^7*e^9*x - 57348*A*B*a^13*c^4*d^5*e^11*x - 19236*A*B*a^14*c^3*d^3*e^13*x
- 204*A*B*a^2*c^6*d^13*e^3*(-a^5*c)^(3/2) - 972*A*B*a^3*c^5*d^11*e^5*(-a^5*c)^(3/2) - 2220*A*B*a^4*c^4*d^9*e^
7*(-a^5*c)^(3/2))*((B*a^6*e^6)/2 - a^3*((15*A*e^6*(-a^5*c)^(1/2))/16 - (15*B*d*e^5*(-a^5*c)^(1/2))/8) - c*(a^5
*((5*B*d^2*e^4)/2 - 3*A*d*e^5) - a^2*((45*A*d^2*e^4*(-a^5*c)^(1/2))/16 - (5*B*d^3*e^3*(-a^5*c)^(1/2))/4)) + a*
c^2*((15*A*d^4*e^2*(-a^5*c)^(1/2))/16 - (B*d^5*e*(-a^5*c)^(1/2))/8) + (3*A*c^3*d^6*(-a^5*c)^(1/2))/16))/(a^9*e
^8 + a^5*c^4*d^8 + 4*a^8*c*d^2*e^6 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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