3.1904 \(\int \frac {(d+e x)^2}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\)

Optimal. Leaf size=173 \[ -\frac {3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac {c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}-\frac {e^3}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac {4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \]

[Out]

-1/3*c*d/(-a*e^2+c*d^2)^2/(c*d*x+a*e)^3+c*d*e/(-a*e^2+c*d^2)^3/(c*d*x+a*e)^2-3*c*d*e^2/(-a*e^2+c*d^2)^4/(c*d*x
+a*e)-e^3/(-a*e^2+c*d^2)^4/(e*x+d)-4*c*d*e^3*ln(c*d*x+a*e)/(-a*e^2+c*d^2)^5+4*c*d*e^3*ln(e*x+d)/(-a*e^2+c*d^2)
^5

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Rubi [A]  time = 0.15, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 44} \[ -\frac {e^3}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac {3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac {c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}-\frac {4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-(c*d)/(3*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^3) + (c*d*e)/((c*d^2 - a*e^2)^3*(a*e + c*d*x)^2) - (3*c*d*e^2)/((c*d
^2 - a*e^2)^4*(a*e + c*d*x)) - e^3/((c*d^2 - a*e^2)^4*(d + e*x)) - (4*c*d*e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2
)^5 + (4*c*d*e^3*Log[d + e*x])/(c*d^2 - a*e^2)^5

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac {1}{(a e+c d x)^4 (d+e x)^2} \, dx\\ &=\int \left (\frac {c^2 d^2}{\left (c d^2-a e^2\right )^2 (a e+c d x)^4}-\frac {2 c^2 d^2 e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^3}+\frac {3 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac {4 c^2 d^2 e^3}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {e^4}{\left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {4 c d e^4}{\left (c d^2-a e^2\right )^5 (d+e x)}\right ) \, dx\\ &=-\frac {c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}+\frac {c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {e^3}{\left (c d^2-a e^2\right )^4 (d+e x)}-\frac {4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 157, normalized size = 0.91 \[ \frac {\frac {9 c d e^2 \left (c d^2-a e^2\right )}{a e+c d x}-\frac {3 c d e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {c d \left (c d^2-a e^2\right )^3}{(a e+c d x)^3}+\frac {3 c d^2 e^3-3 a e^5}{d+e x}+12 c d e^3 \log (a e+c d x)-12 c d e^3 \log (d+e x)}{3 \left (a e^2-c d^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

((c*d*(c*d^2 - a*e^2)^3)/(a*e + c*d*x)^3 - (3*c*d*e*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (9*c*d*e^2*(c*d^2 - a
*e^2))/(a*e + c*d*x) + (3*c*d^2*e^3 - 3*a*e^5)/(d + e*x) + 12*c*d*e^3*Log[a*e + c*d*x] - 12*c*d*e^3*Log[d + e*
x])/(3*(-(c*d^2) + a*e^2)^5)

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fricas [B]  time = 1.04, size = 837, normalized size = 4.84 \[ -\frac {c^{4} d^{8} - 6 \, a c^{3} d^{6} e^{2} + 18 \, a^{2} c^{2} d^{4} e^{4} - 10 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 12 \, {\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (c^{4} d^{6} e^{2} + 4 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 2 \, {\left (c^{4} d^{7} e - 9 \, a c^{3} d^{5} e^{3} - 3 \, a^{2} c^{2} d^{3} e^{5} + 11 \, a^{3} c d e^{7}\right )} x + 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a^{3} c d^{2} e^{6} + {\left (c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right ) - 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a^{3} c d^{2} e^{6} + {\left (c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (a^{3} c^{5} d^{11} e^{3} - 5 \, a^{4} c^{4} d^{9} e^{5} + 10 \, a^{5} c^{3} d^{7} e^{7} - 10 \, a^{6} c^{2} d^{5} e^{9} + 5 \, a^{7} c d^{3} e^{11} - a^{8} d e^{13} + {\left (c^{8} d^{13} e - 5 \, a c^{7} d^{11} e^{3} + 10 \, a^{2} c^{6} d^{9} e^{5} - 10 \, a^{3} c^{5} d^{7} e^{7} + 5 \, a^{4} c^{4} d^{5} e^{9} - a^{5} c^{3} d^{3} e^{11}\right )} x^{4} + {\left (c^{8} d^{14} - 2 \, a c^{7} d^{12} e^{2} - 5 \, a^{2} c^{6} d^{10} e^{4} + 20 \, a^{3} c^{5} d^{8} e^{6} - 25 \, a^{4} c^{4} d^{6} e^{8} + 14 \, a^{5} c^{3} d^{4} e^{10} - 3 \, a^{6} c^{2} d^{2} e^{12}\right )} x^{3} + 3 \, {\left (a c^{7} d^{13} e - 4 \, a^{2} c^{6} d^{11} e^{3} + 5 \, a^{3} c^{5} d^{9} e^{5} - 5 \, a^{5} c^{3} d^{5} e^{9} + 4 \, a^{6} c^{2} d^{3} e^{11} - a^{7} c d e^{13}\right )} x^{2} + {\left (3 \, a^{2} c^{6} d^{12} e^{2} - 14 \, a^{3} c^{5} d^{10} e^{4} + 25 \, a^{4} c^{4} d^{8} e^{6} - 20 \, a^{5} c^{3} d^{6} e^{8} + 5 \, a^{6} c^{2} d^{4} e^{10} + 2 \, a^{7} c d^{2} e^{12} - a^{8} e^{14}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(c^4*d^8 - 6*a*c^3*d^6*e^2 + 18*a^2*c^2*d^4*e^4 - 10*a^3*c*d^2*e^6 - 3*a^4*e^8 + 12*(c^4*d^5*e^3 - a*c^3*
d^3*e^5)*x^3 + 6*(c^4*d^6*e^2 + 4*a*c^3*d^4*e^4 - 5*a^2*c^2*d^2*e^6)*x^2 - 2*(c^4*d^7*e - 9*a*c^3*d^5*e^3 - 3*
a^2*c^2*d^3*e^5 + 11*a^3*c*d*e^7)*x + 12*(c^4*d^4*e^4*x^4 + a^3*c*d^2*e^6 + (c^4*d^5*e^3 + 3*a*c^3*d^3*e^5)*x^
3 + 3*(a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + (3*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*log(c*d*x + a*e) - 12*(c^4*
d^4*e^4*x^4 + a^3*c*d^2*e^6 + (c^4*d^5*e^3 + 3*a*c^3*d^3*e^5)*x^3 + 3*(a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 +
(3*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*log(e*x + d))/(a^3*c^5*d^11*e^3 - 5*a^4*c^4*d^9*e^5 + 10*a^5*c^3*d^7*e^7
- 10*a^6*c^2*d^5*e^9 + 5*a^7*c*d^3*e^11 - a^8*d*e^13 + (c^8*d^13*e - 5*a*c^7*d^11*e^3 + 10*a^2*c^6*d^9*e^5 - 1
0*a^3*c^5*d^7*e^7 + 5*a^4*c^4*d^5*e^9 - a^5*c^3*d^3*e^11)*x^4 + (c^8*d^14 - 2*a*c^7*d^12*e^2 - 5*a^2*c^6*d^10*
e^4 + 20*a^3*c^5*d^8*e^6 - 25*a^4*c^4*d^6*e^8 + 14*a^5*c^3*d^4*e^10 - 3*a^6*c^2*d^2*e^12)*x^3 + 3*(a*c^7*d^13*
e - 4*a^2*c^6*d^11*e^3 + 5*a^3*c^5*d^9*e^5 - 5*a^5*c^3*d^5*e^9 + 4*a^6*c^2*d^3*e^11 - a^7*c*d*e^13)*x^2 + (3*a
^2*c^6*d^12*e^2 - 14*a^3*c^5*d^10*e^4 + 25*a^4*c^4*d^8*e^6 - 20*a^5*c^3*d^6*e^8 + 5*a^6*c^2*d^4*e^10 + 2*a^7*c
*d^2*e^12 - a^8*e^14)*x)

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giac [B]  time = 0.32, size = 672, normalized size = 3.88 \[ \frac {8 \, {\left (c^{3} d^{5} e^{3} - 2 \, a c^{2} d^{3} e^{5} + a^{2} c d e^{7}\right )} \arctan \left (-\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} - \frac {12 \, c^{5} d^{7} x^{5} e^{5} + 30 \, c^{5} d^{8} x^{4} e^{4} + 22 \, c^{5} d^{9} x^{3} e^{3} + 3 \, c^{5} d^{10} x^{2} e^{2} + c^{5} d^{12} - 24 \, a c^{4} d^{5} x^{5} e^{7} - 30 \, a c^{4} d^{6} x^{4} e^{6} + 32 \, a c^{4} d^{7} x^{3} e^{5} + 51 \, a c^{4} d^{8} x^{2} e^{4} + 6 \, a c^{4} d^{9} x e^{3} - 7 \, a c^{4} d^{10} e^{2} + 12 \, a^{2} c^{3} d^{3} x^{5} e^{9} - 30 \, a^{2} c^{3} d^{4} x^{4} e^{8} - 108 \, a^{2} c^{3} d^{5} x^{3} e^{7} - 54 \, a^{2} c^{3} d^{6} x^{2} e^{6} + 36 \, a^{2} c^{3} d^{7} x e^{5} + 24 \, a^{2} c^{3} d^{8} e^{4} + 30 \, a^{3} c^{2} d^{2} x^{4} e^{10} + 32 \, a^{3} c^{2} d^{3} x^{3} e^{9} - 54 \, a^{3} c^{2} d^{4} x^{2} e^{8} - 84 \, a^{3} c^{2} d^{5} x e^{7} - 28 \, a^{3} c^{2} d^{6} e^{6} + 22 \, a^{4} c d x^{3} e^{11} + 51 \, a^{4} c d^{2} x^{2} e^{10} + 36 \, a^{4} c d^{3} x e^{9} + 7 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} x^{2} e^{12} + 6 \, a^{5} d x e^{11} + 3 \, a^{5} d^{2} e^{10}}{3 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*(c^3*d^5*e^3 - 2*a*c^2*d^3*e^5 + a^2*c*d*e^7)*arctan(-(2*c*d*x*e + c*d^2 + a*e^2)/sqrt(-c^2*d^4 + 2*a*c*d^2*
e^2 - a^2*e^4))/((c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 -
6*a^5*c*d^2*e^10 + a^6*e^12)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) - 1/3*(12*c^5*d^7*x^5*e^5 + 30*c^5*d^8
*x^4*e^4 + 22*c^5*d^9*x^3*e^3 + 3*c^5*d^10*x^2*e^2 + c^5*d^12 - 24*a*c^4*d^5*x^5*e^7 - 30*a*c^4*d^6*x^4*e^6 +
32*a*c^4*d^7*x^3*e^5 + 51*a*c^4*d^8*x^2*e^4 + 6*a*c^4*d^9*x*e^3 - 7*a*c^4*d^10*e^2 + 12*a^2*c^3*d^3*x^5*e^9 -
30*a^2*c^3*d^4*x^4*e^8 - 108*a^2*c^3*d^5*x^3*e^7 - 54*a^2*c^3*d^6*x^2*e^6 + 36*a^2*c^3*d^7*x*e^5 + 24*a^2*c^3*
d^8*e^4 + 30*a^3*c^2*d^2*x^4*e^10 + 32*a^3*c^2*d^3*x^3*e^9 - 54*a^3*c^2*d^4*x^2*e^8 - 84*a^3*c^2*d^5*x*e^7 - 2
8*a^3*c^2*d^6*e^6 + 22*a^4*c*d*x^3*e^11 + 51*a^4*c*d^2*x^2*e^10 + 36*a^4*c*d^3*x*e^9 + 7*a^4*c*d^4*e^8 + 3*a^5
*x^2*e^12 + 6*a^5*d*x*e^11 + 3*a^5*d^2*e^10)/((c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d
^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^3)

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maple [A]  time = 0.06, size = 173, normalized size = 1.00 \[ -\frac {4 c d \,e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{5}}+\frac {4 c d \,e^{3} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{5}}-\frac {3 c d \,e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right )}-\frac {c d e}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c d x +a e \right )^{2}}-\frac {e^{3}}{\left (a \,e^{2}-c \,d^{2}\right )^{4} \left (e x +d \right )}-\frac {c d}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (c d x +a e \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*c*d/(a*e^2-c*d^2)^2/(c*d*x+a*e)^3+4*c*d/(a*e^2-c*d^2)^5*e^3*ln(c*d*x+a*e)-3*c*d/(a*e^2-c*d^2)^4*e^2/(c*d*
x+a*e)-c*d/(a*e^2-c*d^2)^3*e/(c*d*x+a*e)^2-e^3/(a*e^2-c*d^2)^4/(e*x+d)-4*c*d/(a*e^2-c*d^2)^5*e^3*ln(e*x+d)

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maxima [B]  time = 1.44, size = 656, normalized size = 3.79 \[ -\frac {4 \, c d e^{3} \log \left (c d x + a e\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} + \frac {4 \, c d e^{3} \log \left (e x + d\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} - \frac {12 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} - 5 \, a c^{2} d^{4} e^{2} + 13 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} + 5 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} - 11 \, a^{2} c d e^{5}\right )} x}{3 \, {\left (a^{3} c^{4} d^{9} e^{3} - 4 \, a^{4} c^{3} d^{7} e^{5} + 6 \, a^{5} c^{2} d^{5} e^{7} - 4 \, a^{6} c d^{3} e^{9} + a^{7} d e^{11} + {\left (c^{7} d^{11} e - 4 \, a c^{6} d^{9} e^{3} + 6 \, a^{2} c^{5} d^{7} e^{5} - 4 \, a^{3} c^{4} d^{5} e^{7} + a^{4} c^{3} d^{3} e^{9}\right )} x^{4} + {\left (c^{7} d^{12} - a c^{6} d^{10} e^{2} - 6 \, a^{2} c^{5} d^{8} e^{4} + 14 \, a^{3} c^{4} d^{6} e^{6} - 11 \, a^{4} c^{3} d^{4} e^{8} + 3 \, a^{5} c^{2} d^{2} e^{10}\right )} x^{3} + 3 \, {\left (a c^{6} d^{11} e - 3 \, a^{2} c^{5} d^{9} e^{3} + 2 \, a^{3} c^{4} d^{7} e^{5} + 2 \, a^{4} c^{3} d^{5} e^{7} - 3 \, a^{5} c^{2} d^{3} e^{9} + a^{6} c d e^{11}\right )} x^{2} + {\left (3 \, a^{2} c^{5} d^{10} e^{2} - 11 \, a^{3} c^{4} d^{8} e^{4} + 14 \, a^{4} c^{3} d^{6} e^{6} - 6 \, a^{5} c^{2} d^{4} e^{8} - a^{6} c d^{2} e^{10} + a^{7} e^{12}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*c*d*e^3*log(c*d*x + a*e)/(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^
2*e^8 - a^5*e^10) + 4*c*d*e^3*log(e*x + d)/(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e
^6 + 5*a^4*c*d^2*e^8 - a^5*e^10) - 1/3*(12*c^3*d^3*e^3*x^3 + c^3*d^6 - 5*a*c^2*d^4*e^2 + 13*a^2*c*d^2*e^4 + 3*
a^3*e^6 + 6*(c^3*d^4*e^2 + 5*a*c^2*d^2*e^4)*x^2 - 2*(c^3*d^5*e - 8*a*c^2*d^3*e^3 - 11*a^2*c*d*e^5)*x)/(a^3*c^4
*d^9*e^3 - 4*a^4*c^3*d^7*e^5 + 6*a^5*c^2*d^5*e^7 - 4*a^6*c*d^3*e^9 + a^7*d*e^11 + (c^7*d^11*e - 4*a*c^6*d^9*e^
3 + 6*a^2*c^5*d^7*e^5 - 4*a^3*c^4*d^5*e^7 + a^4*c^3*d^3*e^9)*x^4 + (c^7*d^12 - a*c^6*d^10*e^2 - 6*a^2*c^5*d^8*
e^4 + 14*a^3*c^4*d^6*e^6 - 11*a^4*c^3*d^4*e^8 + 3*a^5*c^2*d^2*e^10)*x^3 + 3*(a*c^6*d^11*e - 3*a^2*c^5*d^9*e^3
+ 2*a^3*c^4*d^7*e^5 + 2*a^4*c^3*d^5*e^7 - 3*a^5*c^2*d^3*e^9 + a^6*c*d*e^11)*x^2 + (3*a^2*c^5*d^10*e^2 - 11*a^3
*c^4*d^8*e^4 + 14*a^4*c^3*d^6*e^6 - 6*a^5*c^2*d^4*e^8 - a^6*c*d^2*e^10 + a^7*e^12)*x)

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mupad [B]  time = 0.98, size = 617, normalized size = 3.57 \[ \frac {8\,c\,d\,e^3\,\mathrm {atanh}\left (\frac {a^5\,e^{10}-3\,a^4\,c\,d^2\,e^8+2\,a^3\,c^2\,d^4\,e^6+2\,a^2\,c^3\,d^6\,e^4-3\,a\,c^4\,d^8\,e^2+c^5\,d^{10}}{{\left (a\,e^2-c\,d^2\right )}^5}+\frac {2\,c\,d\,e\,x\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{{\left (a\,e^2-c\,d^2\right )}^5}\right )}{{\left (a\,e^2-c\,d^2\right )}^5}-\frac {\frac {3\,a^3\,e^6+13\,a^2\,c\,d^2\,e^4-5\,a\,c^2\,d^4\,e^2+c^3\,d^6}{3\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {2\,e\,x\,\left (11\,a^2\,c\,d\,e^4+8\,a\,c^2\,d^3\,e^2-c^3\,d^5\right )}{3\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {2\,e^2\,x^2\,\left (c^3\,d^4+5\,a\,c^2\,d^2\,e^2\right )}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}+\frac {4\,c^3\,d^3\,e^3\,x^3}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}}{x\,\left (a^3\,e^4+3\,c\,a^2\,d^2\,e^2\right )+x^3\,\left (c^3\,d^4+3\,a\,c^2\,d^2\,e^2\right )+x^2\,\left (3\,a^2\,c\,d\,e^3+3\,a\,c^2\,d^3\,e\right )+a^3\,d\,e^3+c^3\,d^3\,e\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*c*d*e^3*atanh((a^5*e^10 + c^5*d^10 - 3*a*c^4*d^8*e^2 - 3*a^4*c*d^2*e^8 + 2*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*
e^6)/(a*e^2 - c*d^2)^5 + (2*c*d*e*x*(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
))/(a*e^2 - c*d^2)^5))/(a*e^2 - c*d^2)^5 - ((3*a^3*e^6 + c^3*d^6 - 5*a*c^2*d^4*e^2 + 13*a^2*c*d^2*e^4)/(3*(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)) + (2*e*x*(8*a*c^2*d^3*e^2 - c^3*d^5 +
11*a^2*c*d*e^4))/(3*(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)) + (2*e^2*x^2
*(c^3*d^4 + 5*a*c^2*d^2*e^2))/(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) + (4
*c^3*d^3*e^3*x^3)/(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))/(x*(a^3*e^4 + 3
*a^2*c*d^2*e^2) + x^3*(c^3*d^4 + 3*a*c^2*d^2*e^2) + x^2*(3*a*c^2*d^3*e + 3*a^2*c*d*e^3) + a^3*d*e^3 + c^3*d^3*
e*x^4)

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sympy [B]  time = 2.96, size = 1006, normalized size = 5.82 \[ - \frac {4 c d e^{3} \log {\left (x + \frac {- \frac {4 a^{6} c d e^{15}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {24 a^{5} c^{2} d^{3} e^{13}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {60 a^{4} c^{3} d^{5} e^{11}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {80 a^{3} c^{4} d^{7} e^{9}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {60 a^{2} c^{5} d^{9} e^{7}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {24 a c^{6} d^{11} e^{5}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 a c d e^{5} - \frac {4 c^{7} d^{13} e^{3}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 c^{2} d^{3} e^{3}}{8 c^{2} d^{2} e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {4 c d e^{3} \log {\left (x + \frac {\frac {4 a^{6} c d e^{15}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {24 a^{5} c^{2} d^{3} e^{13}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {60 a^{4} c^{3} d^{5} e^{11}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {80 a^{3} c^{4} d^{7} e^{9}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {60 a^{2} c^{5} d^{9} e^{7}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {24 a c^{6} d^{11} e^{5}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 a c d e^{5} + \frac {4 c^{7} d^{13} e^{3}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 c^{2} d^{3} e^{3}}{8 c^{2} d^{2} e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {- 3 a^{3} e^{6} - 13 a^{2} c d^{2} e^{4} + 5 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 12 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 30 a c^{2} d^{2} e^{4} - 6 c^{3} d^{4} e^{2}\right ) + x \left (- 22 a^{2} c d e^{5} - 16 a c^{2} d^{3} e^{3} + 2 c^{3} d^{5} e\right )}{3 a^{7} d e^{11} - 12 a^{6} c d^{3} e^{9} + 18 a^{5} c^{2} d^{5} e^{7} - 12 a^{4} c^{3} d^{7} e^{5} + 3 a^{3} c^{4} d^{9} e^{3} + x^{4} \left (3 a^{4} c^{3} d^{3} e^{9} - 12 a^{3} c^{4} d^{5} e^{7} + 18 a^{2} c^{5} d^{7} e^{5} - 12 a c^{6} d^{9} e^{3} + 3 c^{7} d^{11} e\right ) + x^{3} \left (9 a^{5} c^{2} d^{2} e^{10} - 33 a^{4} c^{3} d^{4} e^{8} + 42 a^{3} c^{4} d^{6} e^{6} - 18 a^{2} c^{5} d^{8} e^{4} - 3 a c^{6} d^{10} e^{2} + 3 c^{7} d^{12}\right ) + x^{2} \left (9 a^{6} c d e^{11} - 27 a^{5} c^{2} d^{3} e^{9} + 18 a^{4} c^{3} d^{5} e^{7} + 18 a^{3} c^{4} d^{7} e^{5} - 27 a^{2} c^{5} d^{9} e^{3} + 9 a c^{6} d^{11} e\right ) + x \left (3 a^{7} e^{12} - 3 a^{6} c d^{2} e^{10} - 18 a^{5} c^{2} d^{4} e^{8} + 42 a^{4} c^{3} d^{6} e^{6} - 33 a^{3} c^{4} d^{8} e^{4} + 9 a^{2} c^{5} d^{10} e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*c*d*e**3*log(x + (-4*a**6*c*d*e**15/(a*e**2 - c*d**2)**5 + 24*a**5*c**2*d**3*e**13/(a*e**2 - c*d**2)**5 - 6
0*a**4*c**3*d**5*e**11/(a*e**2 - c*d**2)**5 + 80*a**3*c**4*d**7*e**9/(a*e**2 - c*d**2)**5 - 60*a**2*c**5*d**9*
e**7/(a*e**2 - c*d**2)**5 + 24*a*c**6*d**11*e**5/(a*e**2 - c*d**2)**5 + 4*a*c*d*e**5 - 4*c**7*d**13*e**3/(a*e*
*2 - c*d**2)**5 + 4*c**2*d**3*e**3)/(8*c**2*d**2*e**4))/(a*e**2 - c*d**2)**5 + 4*c*d*e**3*log(x + (4*a**6*c*d*
e**15/(a*e**2 - c*d**2)**5 - 24*a**5*c**2*d**3*e**13/(a*e**2 - c*d**2)**5 + 60*a**4*c**3*d**5*e**11/(a*e**2 -
c*d**2)**5 - 80*a**3*c**4*d**7*e**9/(a*e**2 - c*d**2)**5 + 60*a**2*c**5*d**9*e**7/(a*e**2 - c*d**2)**5 - 24*a*
c**6*d**11*e**5/(a*e**2 - c*d**2)**5 + 4*a*c*d*e**5 + 4*c**7*d**13*e**3/(a*e**2 - c*d**2)**5 + 4*c**2*d**3*e**
3)/(8*c**2*d**2*e**4))/(a*e**2 - c*d**2)**5 + (-3*a**3*e**6 - 13*a**2*c*d**2*e**4 + 5*a*c**2*d**4*e**2 - c**3*
d**6 - 12*c**3*d**3*e**3*x**3 + x**2*(-30*a*c**2*d**2*e**4 - 6*c**3*d**4*e**2) + x*(-22*a**2*c*d*e**5 - 16*a*c
**2*d**3*e**3 + 2*c**3*d**5*e))/(3*a**7*d*e**11 - 12*a**6*c*d**3*e**9 + 18*a**5*c**2*d**5*e**7 - 12*a**4*c**3*
d**7*e**5 + 3*a**3*c**4*d**9*e**3 + x**4*(3*a**4*c**3*d**3*e**9 - 12*a**3*c**4*d**5*e**7 + 18*a**2*c**5*d**7*e
**5 - 12*a*c**6*d**9*e**3 + 3*c**7*d**11*e) + x**3*(9*a**5*c**2*d**2*e**10 - 33*a**4*c**3*d**4*e**8 + 42*a**3*
c**4*d**6*e**6 - 18*a**2*c**5*d**8*e**4 - 3*a*c**6*d**10*e**2 + 3*c**7*d**12) + x**2*(9*a**6*c*d*e**11 - 27*a*
*5*c**2*d**3*e**9 + 18*a**4*c**3*d**5*e**7 + 18*a**3*c**4*d**7*e**5 - 27*a**2*c**5*d**9*e**3 + 9*a*c**6*d**11*
e) + x*(3*a**7*e**12 - 3*a**6*c*d**2*e**10 - 18*a**5*c**2*d**4*e**8 + 42*a**4*c**3*d**6*e**6 - 33*a**3*c**4*d*
*8*e**4 + 9*a**2*c**5*d**10*e**2))

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