\(\int \frac {1}{(d+e x) (a+c x^2)^3} \, dx\) [517]

Optimal result

Integrand size = 17, antiderivative size = 212 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^3} \, dx=\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (c d^2+a e^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac {e^5 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3} \]

[Out]

1/4*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(c*x^2+a)^2+1/8*(4*a^2*e^3+c*d*(7*a*e^2+3*c*d^2)*x)/a^2/(a*e^2+c*d^2)^2/(c*x^2
+a)+e^5*ln(e*x+d)/(a*e^2+c*d^2)^3-1/2*e^5*ln(c*x^2+a)/(a*e^2+c*d^2)^3+1/8*d*(15*a^2*e^4+10*a*c*d^2*e^2+3*c^2*d
^4)*arctan(x*c^(1/2)/a^(1/2))*c^(1/2)/a^(5/2)/(a*e^2+c*d^2)^3

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {755, 837, 815, 649, 211, 266} \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^3} \, dx=\frac {4 a^2 e^3+c d x \left (7 a e^2+3 c d^2\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}+\frac {\sqrt {c} d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^3}+\frac {a e+c d x}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}-\frac {e^5 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]

[In]

Int[1/((d + e*x)*(a + c*x^2)^3),x]

[Out]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 266

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 649

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 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 815

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 837

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^3} \, dx=\frac {\frac {2 \left (c d^2+a e^2\right )^2 (a e+c d x)}{a \left (a+c x^2\right )^2}+\frac {\left (c d^2+a e^2\right ) \left (4 a^2 e^3+3 c^2 d^3 x+7 a c d e^2 x\right )}{a^2 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}+8 e^5 \log (d+e x)-4 e^5 \log \left (a+c x^2\right )}{8 \left (c d^2+a e^2\right )^3} \]

[In]

Integrate[1/((d + e*x)*(a + c*x^2)^3),x]

[Out]

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

Maple [A] (verified)

Time = 2.34 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.15

method	result	size
default	\(\frac {c \left (\frac {\frac {c d \left (7 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right ) x^{3}}{8 a^{2}}+\left (\frac {1}{2} a \,e^{5}+\frac {1}{2} d^{2} e^{3} c \right ) x^{2}+\frac {d \left (9 a^{2} e^{4}+14 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x}{8 a}+\frac {e \left (3 a^{2} e^{4}+4 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{4 c}}{\left (c \,x^{2}+a \right )^{2}}+\frac {-\frac {4 a^{2} e^{5} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (15 a^{2} d \,e^{4}+10 a c \,d^{3} e^{2}+3 c^{2} d^{5}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{8 a^{2}}\right )}{\left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {e^{5} \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{3}}\)	\(243\)
risch	\(\frac {\frac {c^{2} d \left (7 e^{2} a +3 c \,d^{2}\right ) x^{3}}{8 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) a^{2}}+\frac {e^{3} c \,x^{2}}{2 a^{2} e^{4}+4 a c \,d^{2} e^{2}+2 c^{2} d^{4}}+\frac {d c \left (9 e^{2} a +5 c \,d^{2}\right ) x}{8 a \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}+\frac {e \left (3 e^{2} a +c \,d^{2}\right )}{4 a^{2} e^{4}+8 a c \,d^{2} e^{2}+4 c^{2} d^{4}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {e^{5} \ln \left (e x +d \right )}{e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{8} e^{6}+3 a^{7} c \,d^{2} e^{4}+3 a^{6} c^{2} d^{4} e^{2}+a^{5} c^{3} d^{6}\right ) \textit {\_Z}^{2}+16 a^{5} e^{5} \textit {\_Z} +64 a^{2} e^{4}+33 a c \,d^{2} e^{2}+9 c^{2} d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a^{9} e^{10}+11 a^{8} c \,d^{2} e^{8}+14 a^{7} c^{2} d^{4} e^{6}+6 a^{6} c^{3} d^{6} e^{4}-a^{5} c^{4} d^{8} e^{2}-a^{4} c^{5} d^{10}\right ) \textit {\_R}^{2}+\left (24 a^{6} e^{9}+62 a^{5} c \,d^{2} e^{7}+58 a^{4} c^{2} d^{4} e^{5}+26 a^{3} c^{3} d^{6} e^{3}+6 a^{2} c^{4} d^{8} e \right ) \textit {\_R} +98 a^{2} c \,d^{2} e^{6}+84 a \,c^{2} d^{4} e^{4}+18 c^{3} d^{6} e^{2}\right ) x +\left (4 a^{9} d \,e^{9}+16 a^{8} c \,d^{3} e^{7}+24 a^{7} c^{2} d^{5} e^{5}+16 a^{6} c^{3} d^{7} e^{3}+4 a^{5} c^{4} d^{9} e \right ) \textit {\_R}^{2}+\left (a^{6} d \,e^{8}+6 a^{5} c \,d^{3} e^{6}+12 a^{4} c^{2} d^{5} e^{4}+10 a^{3} c^{3} d^{7} e^{2}+3 a^{2} c^{4} d^{9}\right ) \textit {\_R} +112 a^{3} d \,e^{7}+146 a^{2} c \,d^{3} e^{5}+84 a \,c^{2} d^{5} e^{3}+18 c^{3} d^{7} e \right )\right )}{16}\)	\(685\)

[In]

int(1/(e*x+d)/(c*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

c/(a*e^2+c*d^2)^3*((1/8*c*d*(7*a^2*e^4+10*a*c*d^2*e^2+3*c^2*d^4)/a^2*x^3+(1/2*a*e^5+1/2*d^2*e^3*c)*x^2+1/8*d*(
9*a^2*e^4+14*a*c*d^2*e^2+5*c^2*d^4)/a*x+1/4*e*(3*a^2*e^4+4*a*c*d^2*e^2+c^2*d^4)/c)/(c*x^2+a)^2+1/8/a^2*(-4*a^2
*e^5/c*ln(c*x^2+a)+(15*a^2*d*e^4+10*a*c*d^3*e^2+3*c^2*d^5)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))))+e^5*ln(e*x+d)
/(a*e^2+c*d^2)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (196) = 392\).

Time = 2.27 (sec) , antiderivative size = 1020, normalized size of antiderivative = 4.81 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^3} \, dx=\left [\frac {4 \, a^{2} c^{2} d^{4} e + 16 \, a^{3} c d^{2} e^{3} + 12 \, a^{4} e^{5} + 2 \, {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 7 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 8 \, {\left (a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{5} + 10 \, a^{3} c d^{3} e^{2} + 15 \, a^{4} d e^{4} + {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 15 \, a^{2} c^{2} d e^{4}\right )} x^{4} + 2 \, {\left (3 \, a c^{3} d^{5} + 10 \, a^{2} c^{2} d^{3} e^{2} + 15 \, a^{3} c d e^{4}\right )} x^{2}\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} + 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) + 2 \, {\left (5 \, a c^{3} d^{5} + 14 \, a^{2} c^{2} d^{3} e^{2} + 9 \, a^{3} c d e^{4}\right )} x - 8 \, {\left (a^{2} c^{2} e^{5} x^{4} + 2 \, a^{3} c e^{5} x^{2} + a^{4} e^{5}\right )} \log \left (c x^{2} + a\right ) + 16 \, {\left (a^{2} c^{2} e^{5} x^{4} + 2 \, a^{3} c e^{5} x^{2} + a^{4} e^{5}\right )} \log \left (e x + d\right )}{16 \, {\left (a^{4} c^{3} d^{6} + 3 \, a^{5} c^{2} d^{4} e^{2} + 3 \, a^{6} c d^{2} e^{4} + a^{7} e^{6} + {\left (a^{2} c^{5} d^{6} + 3 \, a^{3} c^{4} d^{4} e^{2} + 3 \, a^{4} c^{3} d^{2} e^{4} + a^{5} c^{2} e^{6}\right )} x^{4} + 2 \, {\left (a^{3} c^{4} d^{6} + 3 \, a^{4} c^{3} d^{4} e^{2} + 3 \, a^{5} c^{2} d^{2} e^{4} + a^{6} c e^{6}\right )} x^{2}\right )}}, \frac {2 \, a^{2} c^{2} d^{4} e + 8 \, a^{3} c d^{2} e^{3} + 6 \, a^{4} e^{5} + {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 7 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 4 \, {\left (a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{5} + 10 \, a^{3} c d^{3} e^{2} + 15 \, a^{4} d e^{4} + {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 15 \, a^{2} c^{2} d e^{4}\right )} x^{4} + 2 \, {\left (3 \, a c^{3} d^{5} + 10 \, a^{2} c^{2} d^{3} e^{2} + 15 \, a^{3} c d e^{4}\right )} x^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + {\left (5 \, a c^{3} d^{5} + 14 \, a^{2} c^{2} d^{3} e^{2} + 9 \, a^{3} c d e^{4}\right )} x - 4 \, {\left (a^{2} c^{2} e^{5} x^{4} + 2 \, a^{3} c e^{5} x^{2} + a^{4} e^{5}\right )} \log \left (c x^{2} + a\right ) + 8 \, {\left (a^{2} c^{2} e^{5} x^{4} + 2 \, a^{3} c e^{5} x^{2} + a^{4} e^{5}\right )} \log \left (e x + d\right )}{8 \, {\left (a^{4} c^{3} d^{6} + 3 \, a^{5} c^{2} d^{4} e^{2} + 3 \, a^{6} c d^{2} e^{4} + a^{7} e^{6} + {\left (a^{2} c^{5} d^{6} + 3 \, a^{3} c^{4} d^{4} e^{2} + 3 \, a^{4} c^{3} d^{2} e^{4} + a^{5} c^{2} e^{6}\right )} x^{4} + 2 \, {\left (a^{3} c^{4} d^{6} + 3 \, a^{4} c^{3} d^{4} e^{2} + 3 \, a^{5} c^{2} d^{2} e^{4} + a^{6} c e^{6}\right )} x^{2}\right )}}\right ] \]

[In]

integrate(1/(e*x+d)/(c*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/16*(4*a^2*c^2*d^4*e + 16*a^3*c*d^2*e^3 + 12*a^4*e^5 + 2*(3*c^4*d^5 + 10*a*c^3*d^3*e^2 + 7*a^2*c^2*d*e^4)*x^
3 + 8*(a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^2 + (3*a^2*c^2*d^5 + 10*a^3*c*d^3*e^2 + 15*a^4*d*e^4 + (3*c^4*d^5 + 10*a
*c^3*d^3*e^2 + 15*a^2*c^2*d*e^4)*x^4 + 2*(3*a*c^3*d^5 + 10*a^2*c^2*d^3*e^2 + 15*a^3*c*d*e^4)*x^2)*sqrt(-c/a)*l
og((c*x^2 + 2*a*x*sqrt(-c/a) - a)/(c*x^2 + a)) + 2*(5*a*c^3*d^5 + 14*a^2*c^2*d^3*e^2 + 9*a^3*c*d*e^4)*x - 8*(a
^2*c^2*e^5*x^4 + 2*a^3*c*e^5*x^2 + a^4*e^5)*log(c*x^2 + a) + 16*(a^2*c^2*e^5*x^4 + 2*a^3*c*e^5*x^2 + a^4*e^5)*
log(e*x + d))/(a^4*c^3*d^6 + 3*a^5*c^2*d^4*e^2 + 3*a^6*c*d^2*e^4 + a^7*e^6 + (a^2*c^5*d^6 + 3*a^3*c^4*d^4*e^2
+ 3*a^4*c^3*d^2*e^4 + a^5*c^2*e^6)*x^4 + 2*(a^3*c^4*d^6 + 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 + a^6*c*e^6)*x
^2), 1/8*(2*a^2*c^2*d^4*e + 8*a^3*c*d^2*e^3 + 6*a^4*e^5 + (3*c^4*d^5 + 10*a*c^3*d^3*e^2 + 7*a^2*c^2*d*e^4)*x^3
+ 4*(a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^2 + (3*a^2*c^2*d^5 + 10*a^3*c*d^3*e^2 + 15*a^4*d*e^4 + (3*c^4*d^5 + 10*a*
c^3*d^3*e^2 + 15*a^2*c^2*d*e^4)*x^4 + 2*(3*a*c^3*d^5 + 10*a^2*c^2*d^3*e^2 + 15*a^3*c*d*e^4)*x^2)*sqrt(c/a)*arc
tan(x*sqrt(c/a)) + (5*a*c^3*d^5 + 14*a^2*c^2*d^3*e^2 + 9*a^3*c*d*e^4)*x - 4*(a^2*c^2*e^5*x^4 + 2*a^3*c*e^5*x^2
+ a^4*e^5)*log(c*x^2 + a) + 8*(a^2*c^2*e^5*x^4 + 2*a^3*c*e^5*x^2 + a^4*e^5)*log(e*x + d))/(a^4*c^3*d^6 + 3*a^
5*c^2*d^4*e^2 + 3*a^6*c*d^2*e^4 + a^7*e^6 + (a^2*c^5*d^6 + 3*a^3*c^4*d^4*e^2 + 3*a^4*c^3*d^2*e^4 + a^5*c^2*e^6
)*x^4 + 2*(a^3*c^4*d^6 + 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 + a^6*c*e^6)*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(c*x**2+a)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.82 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^3} \, dx=-\frac {e^{5} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac {e^{5} \log \left (e x + d\right )}{c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}} + \frac {{\left (3 \, c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, {\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} \sqrt {a c}} + \frac {4 \, a^{2} c e^{3} x^{2} + 2 \, a^{2} c d^{2} e + 6 \, a^{3} e^{3} + {\left (3 \, c^{3} d^{3} + 7 \, a c^{2} d e^{2}\right )} x^{3} + {\left (5 \, a c^{2} d^{3} + 9 \, a^{2} c d e^{2}\right )} x}{8 \, {\left (a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4} + {\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4}\right )} x^{2}\right )}} \]

[In]

integrate(1/(e*x+d)/(c*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/2*e^5*log(c*x^2 + a)/(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6) + e^5*log(e*x + d)/(c^3*d^6 +
3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6) + 1/8*(3*c^3*d^5 + 10*a*c^2*d^3*e^2 + 15*a^2*c*d*e^4)*arctan(c*x/
sqrt(a*c))/((a^2*c^3*d^6 + 3*a^3*c^2*d^4*e^2 + 3*a^4*c*d^2*e^4 + a^5*e^6)*sqrt(a*c)) + 1/8*(4*a^2*c*e^3*x^2 +
2*a^2*c*d^2*e + 6*a^3*e^3 + (3*c^3*d^3 + 7*a*c^2*d*e^2)*x^3 + (5*a*c^2*d^3 + 9*a^2*c*d*e^2)*x)/(a^4*c^2*d^4 +
2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 + 2*(a^3*c^3*d^4 + 2*a^4*c^2*d
^2*e^2 + a^5*c*e^4)*x^2)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.70 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^3} \, dx=\frac {e^{6} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} - \frac {e^{5} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac {{\left (3 \, c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, {\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} \sqrt {a c}} + \frac {2 \, a^{2} c^{2} d^{4} e + 8 \, a^{3} c d^{2} e^{3} + 6 \, a^{4} e^{5} + {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 7 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 4 \, {\left (a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{2} + {\left (5 \, a c^{3} d^{5} + 14 \, a^{2} c^{2} d^{3} e^{2} + 9 \, a^{3} c d e^{4}\right )} x}{8 \, {\left (c d^{2} + a e^{2}\right )}^{3} {\left (c x^{2} + a\right )}^{2} a^{2}} \]

[In]

integrate(1/(e*x+d)/(c*x^2+a)^3,x, algorithm="giac")

[Out]

e^6*log(abs(e*x + d))/(c^3*d^6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e^7) - 1/2*e^5*log(c*x^2 + a)/(c^3*
d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6) + 1/8*(3*c^3*d^5 + 10*a*c^2*d^3*e^2 + 15*a^2*c*d*e^4)*arcta
n(c*x/sqrt(a*c))/((a^2*c^3*d^6 + 3*a^3*c^2*d^4*e^2 + 3*a^4*c*d^2*e^4 + a^5*e^6)*sqrt(a*c)) + 1/8*(2*a^2*c^2*d^
4*e + 8*a^3*c*d^2*e^3 + 6*a^4*e^5 + (3*c^4*d^5 + 10*a*c^3*d^3*e^2 + 7*a^2*c^2*d*e^4)*x^3 + 4*(a^2*c^2*d^2*e^3
+ a^3*c*e^5)*x^2 + (5*a*c^3*d^5 + 14*a^2*c^2*d^3*e^2 + 9*a^3*c*d*e^4)*x)/((c*d^2 + a*e^2)^3*(c*x^2 + a)^2*a^2)

Mupad [B] (verification not implemented)

Time = 10.57 (sec) , antiderivative size = 987, normalized size of antiderivative = 4.66 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^3} \, dx=\frac {\frac {c\,d^2\,e+3\,a\,e^3}{4\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {x^3\,\left (3\,c^3\,d^3+7\,a\,c^2\,d\,e^2\right )}{8\,a^2\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {x\,\left (5\,c^2\,d^3+9\,a\,c\,d\,e^2\right )}{8\,a\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {c\,e^3\,x^2}{2\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}}{a^2+2\,a\,c\,x^2+c^2\,x^4}+\frac {e^5\,\ln \left (d+e\,x\right )}{{\left (c\,d^2+a\,e^2\right )}^3}-\frac {\ln \left (9\,c^6\,d^{14}\,{\left (-a^5\,c\right )}^{3/2}-576\,a^{12}\,e^{14}\,\sqrt {-a^5\,c}-1326\,d^4\,e^{10}\,{\left (-a^5\,c\right )}^{5/2}+9\,a^7\,c^8\,d^{14}\,x+1377\,a^6\,d^2\,e^{12}\,{\left (-a^5\,c\right )}^{3/2}+576\,a^{14}\,c\,e^{14}\,x+319\,a^2\,c^4\,d^{10}\,e^4\,{\left (-a^5\,c\right )}^{3/2}+740\,a^3\,c^3\,d^8\,e^6\,{\left (-a^5\,c\right )}^{3/2}+1015\,a^4\,c^2\,d^6\,e^8\,{\left (-a^5\,c\right )}^{3/2}+78\,a^8\,c^7\,d^{12}\,e^2\,x+319\,a^9\,c^6\,d^{10}\,e^4\,x+740\,a^{10}\,c^5\,d^8\,e^6\,x+1015\,a^{11}\,c^4\,d^6\,e^8\,x+1326\,a^{12}\,c^3\,d^4\,e^{10}\,x+1377\,a^{13}\,c^2\,d^2\,e^{12}\,x+78\,a\,c^5\,d^{12}\,e^2\,{\left (-a^5\,c\right )}^{3/2}\right )\,\left (8\,a^5\,e^5+3\,c^2\,d^5\,\sqrt {-a^5\,c}+15\,a^2\,d\,e^4\,\sqrt {-a^5\,c}+10\,a\,c\,d^3\,e^2\,\sqrt {-a^5\,c}\right )}{16\,\left (a^8\,e^6+3\,a^7\,c\,d^2\,e^4+3\,a^6\,c^2\,d^4\,e^2+a^5\,c^3\,d^6\right )}+\frac {\ln \left (576\,a^{10}\,e^{14}\,\sqrt {-a^5\,c}+9\,a^5\,c^8\,d^{14}\,x+9\,a^3\,c^7\,d^{14}\,\sqrt {-a^5\,c}-1377\,a^4\,d^2\,e^{12}\,{\left (-a^5\,c\right )}^{3/2}-319\,c^4\,d^{10}\,e^4\,{\left (-a^5\,c\right )}^{3/2}+576\,a^{12}\,c\,e^{14}\,x-1015\,a^2\,c^2\,d^6\,e^8\,{\left (-a^5\,c\right )}^{3/2}+78\,a^4\,c^6\,d^{12}\,e^2\,\sqrt {-a^5\,c}+78\,a^6\,c^7\,d^{12}\,e^2\,x+319\,a^7\,c^6\,d^{10}\,e^4\,x+740\,a^8\,c^5\,d^8\,e^6\,x+1015\,a^9\,c^4\,d^6\,e^8\,x+1326\,a^{10}\,c^3\,d^4\,e^{10}\,x+1377\,a^{11}\,c^2\,d^2\,e^{12}\,x-740\,a\,c^3\,d^8\,e^6\,{\left (-a^5\,c\right )}^{3/2}-1326\,a^3\,c\,d^4\,e^{10}\,{\left (-a^5\,c\right )}^{3/2}\right )\,\left (3\,c^2\,d^5\,\sqrt {-a^5\,c}-8\,a^5\,e^5+15\,a^2\,d\,e^4\,\sqrt {-a^5\,c}+10\,a\,c\,d^3\,e^2\,\sqrt {-a^5\,c}\right )}{16\,\left (a^8\,e^6+3\,a^7\,c\,d^2\,e^4+3\,a^6\,c^2\,d^4\,e^2+a^5\,c^3\,d^6\right )} \]

[In]

int(1/((a + c*x^2)^3*(d + e*x)),x)

[Out]

((3*a*e^3 + c*d^2*e)/(4*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (x^3*(3*c^3*d^3 + 7*a*c^2*d*e^2))/(8*a^2*(a^2*e
^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (x*(5*c^2*d^3 + 9*a*c*d*e^2))/(8*a*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (c*
e^3*x^2)/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))/(a^2 + c^2*x^4 + 2*a*c*x^2) + (e^5*log(d + e*x))/(a*e^2 + c*
d^2)^3 - (log(9*c^6*d^14*(-a^5*c)^(3/2) - 576*a^12*e^14*(-a^5*c)^(1/2) - 1326*d^4*e^10*(-a^5*c)^(5/2) + 9*a^7*
c^8*d^14*x + 1377*a^6*d^2*e^12*(-a^5*c)^(3/2) + 576*a^14*c*e^14*x + 319*a^2*c^4*d^10*e^4*(-a^5*c)^(3/2) + 740*
a^3*c^3*d^8*e^6*(-a^5*c)^(3/2) + 1015*a^4*c^2*d^6*e^8*(-a^5*c)^(3/2) + 78*a^8*c^7*d^12*e^2*x + 319*a^9*c^6*d^1
0*e^4*x + 740*a^10*c^5*d^8*e^6*x + 1015*a^11*c^4*d^6*e^8*x + 1326*a^12*c^3*d^4*e^10*x + 1377*a^13*c^2*d^2*e^12
*x + 78*a*c^5*d^12*e^2*(-a^5*c)^(3/2))*(8*a^5*e^5 + 3*c^2*d^5*(-a^5*c)^(1/2) + 15*a^2*d*e^4*(-a^5*c)^(1/2) + 1
0*a*c*d^3*e^2*(-a^5*c)^(1/2)))/(16*(a^8*e^6 + a^5*c^3*d^6 + 3*a^7*c*d^2*e^4 + 3*a^6*c^2*d^4*e^2)) + (log(576*a
^10*e^14*(-a^5*c)^(1/2) + 9*a^5*c^8*d^14*x + 9*a^3*c^7*d^14*(-a^5*c)^(1/2) - 1377*a^4*d^2*e^12*(-a^5*c)^(3/2)
- 319*c^4*d^10*e^4*(-a^5*c)^(3/2) + 576*a^12*c*e^14*x - 1015*a^2*c^2*d^6*e^8*(-a^5*c)^(3/2) + 78*a^4*c^6*d^12*
e^2*(-a^5*c)^(1/2) + 78*a^6*c^7*d^12*e^2*x + 319*a^7*c^6*d^10*e^4*x + 740*a^8*c^5*d^8*e^6*x + 1015*a^9*c^4*d^6
*e^8*x + 1326*a^10*c^3*d^4*e^10*x + 1377*a^11*c^2*d^2*e^12*x - 740*a*c^3*d^8*e^6*(-a^5*c)^(3/2) - 1326*a^3*c*d
^4*e^10*(-a^5*c)^(3/2))*(3*c^2*d^5*(-a^5*c)^(1/2) - 8*a^5*e^5 + 15*a^2*d*e^4*(-a^5*c)^(1/2) + 10*a*c*d^3*e^2*(
-a^5*c)^(1/2)))/(16*(a^8*e^6 + a^5*c^3*d^6 + 3*a^7*c*d^2*e^4 + 3*a^6*c^2*d^4*e^2))