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

Optimal. Leaf size=226 \[ -\frac {6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {3 c^2 d^2 e}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac {c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (a e+c d x)^3}-\frac {10 c^2 d^2 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}+\frac {10 c^2 d^2 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^6}-\frac {4 c d e^3}{(d+e x) \left (c d^2-a e^2\right )^5}-\frac {e^3}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4} \]

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Rubi [A]  time = 0.21, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {626, 44} \begin {gather*} -\frac {6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {3 c^2 d^2 e}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac {c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (a e+c d x)^3}-\frac {10 c^2 d^2 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}+\frac {10 c^2 d^2 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^6}-\frac {4 c d e^3}{(d+e x) \left (c d^2-a e^2\right )^5}-\frac {e^3}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*d^2)/(3*(c*d^2 - a*e^2)^3*(a*e + c*d*x)^3) + (3*c^2*d^2*e)/(2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2) - (6*c^
2*d^2*e^2)/((c*d^2 - a*e^2)^5*(a*e + c*d*x)) - e^3/(2*(c*d^2 - a*e^2)^4*(d + e*x)^2) - (4*c*d*e^3)/((c*d^2 - a
*e^2)^5*(d + e*x)) - (10*c^2*d^2*e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2)^6 + (10*c^2*d^2*e^3*Log[d + e*x])/(c*d^
2 - a*e^2)^6

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}{\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)^3} \, dx\\ &=\int \left (\frac {c^3 d^3}{\left (c d^2-a e^2\right )^3 (a e+c d x)^4}-\frac {3 c^3 d^3 e}{\left (c d^2-a e^2\right )^4 (a e+c d x)^3}+\frac {6 c^3 d^3 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)^2}-\frac {10 c^3 d^3 e^3}{\left (c d^2-a e^2\right )^6 (a e+c d x)}+\frac {e^4}{\left (c d^2-a e^2\right )^4 (d+e x)^3}+\frac {4 c d e^4}{\left (c d^2-a e^2\right )^5 (d+e x)^2}+\frac {10 c^2 d^2 e^4}{\left (c d^2-a e^2\right )^6 (d+e x)}\right ) \, dx\\ &=-\frac {c^2 d^2}{3 \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 {6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac {e^3}{2 \left (c d^2-a e^2\right )^4 (d+e x)^2}-\frac {4 c d e^3}{\left (c d^2-a e^2\right )^5 (d+e x)}-\frac {10 c^2 d^2 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}+\frac {10 c^2 d^2 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^6}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 206, normalized size = 0.91 \begin {gather*} \frac {-60 c^2 d^2 e^3 \log (a e+c d x)+\frac {36 c^2 d^2 e^2 \left (a e^2-c d^2\right )}{a e+c d x}+\frac {9 c^2 d^2 e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {2 c^2 d^2 \left (a e^2-c d^2\right )^3}{(a e+c d x)^3}+\frac {24 c d e^3 \left (a e^2-c d^2\right )}{d+e x}-\frac {3 e^3 \left (c d^2-a e^2\right )^2}{(d+e x)^2}+60 c^2 d^2 e^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c^2*d^2*(-(c*d^2) + a*e^2)^3)/(a*e + c*d*x)^3 + (9*c^2*d^2*e*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (36*c^2*
d^2*e^2*(-(c*d^2) + a*e^2))/(a*e + c*d*x) - (3*e^3*(c*d^2 - a*e^2)^2)/(d + e*x)^2 + (24*c*d*e^3*(-(c*d^2) + a*
e^2))/(d + e*x) - 60*c^2*d^2*e^3*Log[a*e + c*d*x] + 60*c^2*d^2*e^3*Log[d + e*x])/(6*(c*d^2 - a*e^2)^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.44, size = 1242, normalized size = 5.50

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*c^5*d^10 - 15*a*c^4*d^8*e^2 + 60*a^2*c^3*d^6*e^4 - 20*a^3*c^2*d^4*e^6 - 30*a^4*c*d^2*e^8 + 3*a^5*e^10
+ 60*(c^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 + 30*(3*c^5*d^7*e^3 + 2*a*c^4*d^5*e^5 - 5*a^2*c^3*d^3*e^7)*x^3 + 10*(2*
c^5*d^8*e^2 + 21*a*c^4*d^6*e^4 - 12*a^2*c^3*d^4*e^6 - 11*a^3*c^2*d^2*e^8)*x^2 - 5*(c^5*d^9*e - 12*a*c^4*d^7*e^
3 - 24*a^2*c^3*d^5*e^5 + 32*a^3*c^2*d^3*e^7 + 3*a^4*c*d*e^9)*x + 60*(c^5*d^5*e^5*x^5 + a^3*c^2*d^4*e^6 + (2*c^
5*d^6*e^4 + 3*a*c^4*d^4*e^6)*x^4 + (c^5*d^7*e^3 + 6*a*c^4*d^5*e^5 + 3*a^2*c^3*d^3*e^7)*x^3 + (3*a*c^4*d^6*e^4
+ 6*a^2*c^3*d^4*e^6 + a^3*c^2*d^2*e^8)*x^2 + (3*a^2*c^3*d^5*e^5 + 2*a^3*c^2*d^3*e^7)*x)*log(c*d*x + a*e) - 60*
(c^5*d^5*e^5*x^5 + a^3*c^2*d^4*e^6 + (2*c^5*d^6*e^4 + 3*a*c^4*d^4*e^6)*x^4 + (c^5*d^7*e^3 + 6*a*c^4*d^5*e^5 +
3*a^2*c^3*d^3*e^7)*x^3 + (3*a*c^4*d^6*e^4 + 6*a^2*c^3*d^4*e^6 + a^3*c^2*d^2*e^8)*x^2 + (3*a^2*c^3*d^5*e^5 + 2*
a^3*c^2*d^3*e^7)*x)*log(e*x + d))/(a^3*c^6*d^14*e^3 - 6*a^4*c^5*d^12*e^5 + 15*a^5*c^4*d^10*e^7 - 20*a^6*c^3*d^
8*e^9 + 15*a^7*c^2*d^6*e^11 - 6*a^8*c*d^4*e^13 + a^9*d^2*e^15 + (c^9*d^15*e^2 - 6*a*c^8*d^13*e^4 + 15*a^2*c^7*
d^11*e^6 - 20*a^3*c^6*d^9*e^8 + 15*a^4*c^5*d^7*e^10 - 6*a^5*c^4*d^5*e^12 + a^6*c^3*d^3*e^14)*x^5 + (2*c^9*d^16
*e - 9*a*c^8*d^14*e^3 + 12*a^2*c^7*d^12*e^5 + 5*a^3*c^6*d^10*e^7 - 30*a^4*c^5*d^8*e^9 + 33*a^5*c^4*d^6*e^11 -
16*a^6*c^3*d^4*e^13 + 3*a^7*c^2*d^2*e^15)*x^4 + (c^9*d^17 - 18*a^2*c^7*d^13*e^4 + 52*a^3*c^6*d^11*e^6 - 60*a^4
*c^5*d^9*e^8 + 24*a^5*c^4*d^7*e^10 + 10*a^6*c^3*d^5*e^12 - 12*a^7*c^2*d^3*e^14 + 3*a^8*c*d*e^16)*x^3 + (3*a*c^
8*d^16*e - 12*a^2*c^7*d^14*e^3 + 10*a^3*c^6*d^12*e^5 + 24*a^4*c^5*d^10*e^7 - 60*a^5*c^4*d^8*e^9 + 52*a^6*c^3*d
^6*e^11 - 18*a^7*c^2*d^4*e^13 + a^9*e^17)*x^2 + (3*a^2*c^7*d^15*e^2 - 16*a^3*c^6*d^13*e^4 + 33*a^4*c^5*d^11*e^
6 - 30*a^5*c^4*d^9*e^8 + 5*a^6*c^3*d^7*e^10 + 12*a^7*c^2*d^5*e^12 - 9*a^8*c*d^3*e^14 + 2*a^9*d*e^16)*x)

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giac [B]  time = 0.31, size = 586, normalized size = 2.59 \begin {gather*} -\frac {20 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\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 {60 \, c^{5} d^{6} x^{5} e^{5} + 150 \, c^{5} d^{7} x^{4} e^{4} + 110 \, c^{5} d^{8} x^{3} e^{3} + 15 \, c^{5} d^{9} x^{2} e^{2} - 3 \, c^{5} d^{10} x e + 2 \, c^{5} d^{11} - 60 \, a c^{4} d^{4} x^{5} e^{7} + 270 \, a c^{4} d^{6} x^{3} e^{5} + 270 \, a c^{4} d^{7} x^{2} e^{4} + 45 \, a c^{4} d^{8} x e^{3} - 15 \, a c^{4} d^{9} e^{2} - 150 \, a^{2} c^{3} d^{3} x^{4} e^{8} - 270 \, a^{2} c^{3} d^{4} x^{3} e^{7} + 180 \, a^{2} c^{3} d^{6} x e^{5} + 60 \, a^{2} c^{3} d^{7} e^{4} - 110 \, a^{3} c^{2} d^{2} x^{3} e^{9} - 270 \, a^{3} c^{2} d^{3} x^{2} e^{8} - 180 \, a^{3} c^{2} d^{4} x e^{7} - 20 \, a^{3} c^{2} d^{5} e^{6} - 15 \, a^{4} c d x^{2} e^{10} - 45 \, a^{4} c d^{2} x e^{9} - 30 \, a^{4} c d^{3} e^{8} + 3 \, a^{5} x e^{11} + 3 \, a^{5} d e^{10}}{6 \, {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20*(c^3*d^4*e^3 - a*c^2*d^2*e^5)*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)) - 1/6*(60*c^5*d^6*x^5*e^5 + 150*c^5*d^7*x^4*e^4 + 110
*c^5*d^8*x^3*e^3 + 15*c^5*d^9*x^2*e^2 - 3*c^5*d^10*x*e + 2*c^5*d^11 - 60*a*c^4*d^4*x^5*e^7 + 270*a*c^4*d^6*x^3
*e^5 + 270*a*c^4*d^7*x^2*e^4 + 45*a*c^4*d^8*x*e^3 - 15*a*c^4*d^9*e^2 - 150*a^2*c^3*d^3*x^4*e^8 - 270*a^2*c^3*d
^4*x^3*e^7 + 180*a^2*c^3*d^6*x*e^5 + 60*a^2*c^3*d^7*e^4 - 110*a^3*c^2*d^2*x^3*e^9 - 270*a^3*c^2*d^3*x^2*e^8 -
180*a^3*c^2*d^4*x*e^7 - 20*a^3*c^2*d^5*e^6 - 15*a^4*c*d*x^2*e^10 - 45*a^4*c*d^2*x*e^9 - 30*a^4*c*d^3*e^8 + 3*a
^5*x*e^11 + 3*a^5*d*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 = 221, normalized size = 0.98 \begin {gather*} \frac {10 c^{2} d^{2} e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{6}}-\frac {10 c^{2} d^{2} e^{3} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{6}}+\frac {6 c^{2} d^{2} e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (c d x +a e \right )}+\frac {3 c^{2} d^{2} e}{2 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right )^{2}}+\frac {4 c d \,e^{3}}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (e x +d \right )}+\frac {c^{2} d^{2}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c d x +a e \right )^{3}}-\frac {e^{3}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (e x +d \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c^2*d^2/(a*e^2-c*d^2)^3/(c*d*x+a*e)^3-10*c^2*d^2/(a*e^2-c*d^2)^6*e^3*ln(c*d*x+a*e)+6*c^2*d^2/(a*e^2-c*d^2)
^5*e^2/(c*d*x+a*e)+3/2*c^2*d^2/(a*e^2-c*d^2)^4*e/(c*d*x+a*e)^2-1/2*e^3/(a*e^2-c*d^2)^4/(e*x+d)^2+10*c^2*d^2/(a
*e^2-c*d^2)^6*e^3*ln(e*x+d)+4*e^3/(a*e^2-c*d^2)^5*c*d/(e*x+d)

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maxima [B]  time = 1.68, size = 956, normalized size = 4.23 \begin {gather*} -\frac {10 \, c^{2} d^{2} e^{3} \log \left (c d x + a e\right )}{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}} + \frac {10 \, c^{2} d^{2} e^{3} \log \left (e x + d\right )}{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}} - \frac {60 \, c^{4} d^{4} e^{4} x^{4} + 2 \, c^{4} d^{8} - 13 \, a c^{3} d^{6} e^{2} + 47 \, a^{2} c^{2} d^{4} e^{4} + 27 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 30 \, {\left (3 \, c^{4} d^{5} e^{3} + 5 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, c^{4} d^{6} e^{2} + 23 \, a c^{3} d^{4} e^{4} + 11 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 5 \, {\left (c^{4} d^{7} e - 11 \, a c^{3} d^{5} e^{3} - 35 \, a^{2} c^{2} d^{3} e^{5} - 3 \, a^{3} c d e^{7}\right )} x}{6 \, {\left (a^{3} c^{5} d^{12} e^{3} - 5 \, a^{4} c^{4} d^{10} e^{5} + 10 \, a^{5} c^{3} d^{8} e^{7} - 10 \, a^{6} c^{2} d^{6} e^{9} + 5 \, a^{7} c d^{4} e^{11} - a^{8} d^{2} e^{13} + {\left (c^{8} d^{13} e^{2} - 5 \, a c^{7} d^{11} e^{4} + 10 \, a^{2} c^{6} d^{9} e^{6} - 10 \, a^{3} c^{5} d^{7} e^{8} + 5 \, a^{4} c^{4} d^{5} e^{10} - a^{5} c^{3} d^{3} e^{12}\right )} x^{5} + {\left (2 \, c^{8} d^{14} e - 7 \, a c^{7} d^{12} e^{3} + 5 \, a^{2} c^{6} d^{10} e^{5} + 10 \, a^{3} c^{5} d^{8} e^{7} - 20 \, a^{4} c^{4} d^{6} e^{9} + 13 \, a^{5} c^{3} d^{4} e^{11} - 3 \, a^{6} c^{2} d^{2} e^{13}\right )} x^{4} + {\left (c^{8} d^{15} + a c^{7} d^{13} e^{2} - 17 \, a^{2} c^{6} d^{11} e^{4} + 35 \, a^{3} c^{5} d^{9} e^{6} - 25 \, a^{4} c^{4} d^{7} e^{8} - a^{5} c^{3} d^{5} e^{10} + 9 \, a^{6} c^{2} d^{3} e^{12} - 3 \, a^{7} c d e^{14}\right )} x^{3} + {\left (3 \, a c^{7} d^{14} e - 9 \, a^{2} c^{6} d^{12} e^{3} + a^{3} c^{5} d^{10} e^{5} + 25 \, a^{4} c^{4} d^{8} e^{7} - 35 \, a^{5} c^{3} d^{6} e^{9} + 17 \, a^{6} c^{2} d^{4} e^{11} - a^{7} c d^{2} e^{13} - a^{8} e^{15}\right )} x^{2} + {\left (3 \, a^{2} c^{6} d^{13} e^{2} - 13 \, a^{3} c^{5} d^{11} e^{4} + 20 \, a^{4} c^{4} d^{9} e^{6} - 10 \, a^{5} c^{3} d^{7} e^{8} - 5 \, a^{6} c^{2} d^{5} e^{10} + 7 \, a^{7} c d^{3} e^{12} - 2 \, a^{8} d e^{14}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*c^2*d^2*e^3*log(c*d*x + a*e)/(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) + 10*c^2*d^2*e^3*log(e*x + d)/(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) - 1/6*(60*c^4*d^4*e^4
*x^4 + 2*c^4*d^8 - 13*a*c^3*d^6*e^2 + 47*a^2*c^2*d^4*e^4 + 27*a^3*c*d^2*e^6 - 3*a^4*e^8 + 30*(3*c^4*d^5*e^3 +
5*a*c^3*d^3*e^5)*x^3 + 10*(2*c^4*d^6*e^2 + 23*a*c^3*d^4*e^4 + 11*a^2*c^2*d^2*e^6)*x^2 - 5*(c^4*d^7*e - 11*a*c^
3*d^5*e^3 - 35*a^2*c^2*d^3*e^5 - 3*a^3*c*d*e^7)*x)/(a^3*c^5*d^12*e^3 - 5*a^4*c^4*d^10*e^5 + 10*a^5*c^3*d^8*e^7
- 10*a^6*c^2*d^6*e^9 + 5*a^7*c*d^4*e^11 - a^8*d^2*e^13 + (c^8*d^13*e^2 - 5*a*c^7*d^11*e^4 + 10*a^2*c^6*d^9*e^
6 - 10*a^3*c^5*d^7*e^8 + 5*a^4*c^4*d^5*e^10 - a^5*c^3*d^3*e^12)*x^5 + (2*c^8*d^14*e - 7*a*c^7*d^12*e^3 + 5*a^2
*c^6*d^10*e^5 + 10*a^3*c^5*d^8*e^7 - 20*a^4*c^4*d^6*e^9 + 13*a^5*c^3*d^4*e^11 - 3*a^6*c^2*d^2*e^13)*x^4 + (c^8
*d^15 + a*c^7*d^13*e^2 - 17*a^2*c^6*d^11*e^4 + 35*a^3*c^5*d^9*e^6 - 25*a^4*c^4*d^7*e^8 - a^5*c^3*d^5*e^10 + 9*
a^6*c^2*d^3*e^12 - 3*a^7*c*d*e^14)*x^3 + (3*a*c^7*d^14*e - 9*a^2*c^6*d^12*e^3 + a^3*c^5*d^10*e^5 + 25*a^4*c^4*
d^8*e^7 - 35*a^5*c^3*d^6*e^9 + 17*a^6*c^2*d^4*e^11 - a^7*c*d^2*e^13 - a^8*e^15)*x^2 + (3*a^2*c^6*d^13*e^2 - 13
*a^3*c^5*d^11*e^4 + 20*a^4*c^4*d^9*e^6 - 10*a^5*c^3*d^7*e^8 - 5*a^6*c^2*d^5*e^10 + 7*a^7*c*d^3*e^12 - 2*a^8*d*
e^14)*x)

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mupad [B]  time = 1.15, size = 891, normalized size = 3.94 \begin {gather*} \frac {\frac {-3\,a^4\,e^8+27\,a^3\,c\,d^2\,e^6+47\,a^2\,c^2\,d^4\,e^4-13\,a\,c^3\,d^6\,e^2+2\,c^4\,d^8}{6\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}+\frac {5\,e^2\,x^2\,\left (11\,a^2\,c^2\,d^2\,e^4+23\,a\,c^3\,d^4\,e^2+2\,c^4\,d^6\right )}{3\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}+\frac {5\,d\,e\,x\,\left (3\,a^3\,c\,e^6+35\,a^2\,c^2\,d^2\,e^4+11\,a\,c^3\,d^4\,e^2-c^4\,d^6\right )}{6\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}+\frac {5\,c\,e\,x^3\,\left (3\,c^3\,d^5\,e^2+5\,a\,c^2\,d^3\,e^4\right )}{a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}+\frac {10\,c^4\,d^4\,e^4\,x^4}{a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}}{x^2\,\left (a^3\,e^5+6\,a^2\,c\,d^2\,e^3+3\,a\,c^2\,d^4\,e\right )+x^3\,\left (3\,a^2\,c\,d\,e^4+6\,a\,c^2\,d^3\,e^2+c^3\,d^5\right )+x\,\left (2\,a^3\,d\,e^4+3\,c\,a^2\,d^3\,e^2\right )+x^4\,\left (2\,c^3\,d^4\,e+3\,a\,c^2\,d^2\,e^3\right )+a^3\,d^2\,e^3+c^3\,d^3\,e^2\,x^5}-\frac {20\,c^2\,d^2\,e^3\,\mathrm {atanh}\left (\frac {a^6\,e^{12}-4\,a^5\,c\,d^2\,e^{10}+5\,a^4\,c^2\,d^4\,e^8-5\,a^2\,c^4\,d^8\,e^4+4\,a\,c^5\,d^{10}\,e^2-c^6\,d^{12}}{{\left (a\,e^2-c\,d^2\right )}^6}+\frac {2\,c\,d\,e\,x\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}{{\left (a\,e^2-c\,d^2\right )}^6}\right )}{{\left (a\,e^2-c\,d^2\right )}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*c^4*d^8 - 3*a^4*e^8 - 13*a*c^3*d^6*e^2 + 27*a^3*c*d^2*e^6 + 47*a^2*c^2*d^4*e^4)/(6*(a^5*e^10 - c^5*d^10 +
5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6)) + (5*e^2*x^2*(2*c^4*d^6 + 23*a*c
^3*d^4*e^2 + 11*a^2*c^2*d^2*e^4))/(3*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6
*e^4 + 10*a^3*c^2*d^4*e^6)) + (5*d*e*x*(3*a^3*c*e^6 - c^4*d^6 + 11*a*c^3*d^4*e^2 + 35*a^2*c^2*d^2*e^4))/(6*(a^
5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6)) + (5*c*e*x^3
*(3*c^3*d^5*e^2 + 5*a*c^2*d^3*e^4))/(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*
e^4 + 10*a^3*c^2*d^4*e^6) + (10*c^4*d^4*e^4*x^4)/(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10
*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6))/(x^2*(a^3*e^5 + 6*a^2*c*d^2*e^3 + 3*a*c^2*d^4*e) + x^3*(c^3*d^5 + 6*a*
c^2*d^3*e^2 + 3*a^2*c*d*e^4) + x*(2*a^3*d*e^4 + 3*a^2*c*d^3*e^2) + x^4*(2*c^3*d^4*e + 3*a*c^2*d^2*e^3) + a^3*d
^2*e^3 + c^3*d^3*e^2*x^5) - (20*c^2*d^2*e^3*atanh((a^6*e^12 - c^6*d^12 + 4*a*c^5*d^10*e^2 - 4*a^5*c*d^2*e^10 -
5*a^2*c^4*d^8*e^4 + 5*a^4*c^2*d^4*e^8)/(a*e^2 - c*d^2)^6 + (2*c*d*e*x*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2
- 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6))/(a*e^2 - c*d^2)^6))/(a*e^2 - c*d^2)^6

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sympy [B]  time = 4.79, size = 1363, normalized size = 6.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*c**2*d**2*e**3*log(x + (-10*a**7*c**2*d**2*e**17/(a*e**2 - c*d**2)**6 + 70*a**6*c**3*d**4*e**15/(a*e**2 - c
*d**2)**6 - 210*a**5*c**4*d**6*e**13/(a*e**2 - c*d**2)**6 + 350*a**4*c**5*d**8*e**11/(a*e**2 - c*d**2)**6 - 35
0*a**3*c**6*d**10*e**9/(a*e**2 - c*d**2)**6 + 210*a**2*c**7*d**12*e**7/(a*e**2 - c*d**2)**6 - 70*a*c**8*d**14*
e**5/(a*e**2 - c*d**2)**6 + 10*a*c**2*d**2*e**5 + 10*c**9*d**16*e**3/(a*e**2 - c*d**2)**6 + 10*c**3*d**4*e**3)
/(20*c**3*d**3*e**4))/(a*e**2 - c*d**2)**6 - 10*c**2*d**2*e**3*log(x + (10*a**7*c**2*d**2*e**17/(a*e**2 - c*d*
*2)**6 - 70*a**6*c**3*d**4*e**15/(a*e**2 - c*d**2)**6 + 210*a**5*c**4*d**6*e**13/(a*e**2 - c*d**2)**6 - 350*a*
*4*c**5*d**8*e**11/(a*e**2 - c*d**2)**6 + 350*a**3*c**6*d**10*e**9/(a*e**2 - c*d**2)**6 - 210*a**2*c**7*d**12*
e**7/(a*e**2 - c*d**2)**6 + 70*a*c**8*d**14*e**5/(a*e**2 - c*d**2)**6 + 10*a*c**2*d**2*e**5 - 10*c**9*d**16*e*
*3/(a*e**2 - c*d**2)**6 + 10*c**3*d**4*e**3)/(20*c**3*d**3*e**4))/(a*e**2 - c*d**2)**6 + (-3*a**4*e**8 + 27*a*
*3*c*d**2*e**6 + 47*a**2*c**2*d**4*e**4 - 13*a*c**3*d**6*e**2 + 2*c**4*d**8 + 60*c**4*d**4*e**4*x**4 + x**3*(1
50*a*c**3*d**3*e**5 + 90*c**4*d**5*e**3) + x**2*(110*a**2*c**2*d**2*e**6 + 230*a*c**3*d**4*e**4 + 20*c**4*d**6
*e**2) + x*(15*a**3*c*d*e**7 + 175*a**2*c**2*d**3*e**5 + 55*a*c**3*d**5*e**3 - 5*c**4*d**7*e))/(6*a**8*d**2*e*
*13 - 30*a**7*c*d**4*e**11 + 60*a**6*c**2*d**6*e**9 - 60*a**5*c**3*d**8*e**7 + 30*a**4*c**4*d**10*e**5 - 6*a**
3*c**5*d**12*e**3 + x**5*(6*a**5*c**3*d**3*e**12 - 30*a**4*c**4*d**5*e**10 + 60*a**3*c**5*d**7*e**8 - 60*a**2*
c**6*d**9*e**6 + 30*a*c**7*d**11*e**4 - 6*c**8*d**13*e**2) + x**4*(18*a**6*c**2*d**2*e**13 - 78*a**5*c**3*d**4
*e**11 + 120*a**4*c**4*d**6*e**9 - 60*a**3*c**5*d**8*e**7 - 30*a**2*c**6*d**10*e**5 + 42*a*c**7*d**12*e**3 - 1
2*c**8*d**14*e) + x**3*(18*a**7*c*d*e**14 - 54*a**6*c**2*d**3*e**12 + 6*a**5*c**3*d**5*e**10 + 150*a**4*c**4*d
**7*e**8 - 210*a**3*c**5*d**9*e**6 + 102*a**2*c**6*d**11*e**4 - 6*a*c**7*d**13*e**2 - 6*c**8*d**15) + x**2*(6*
a**8*e**15 + 6*a**7*c*d**2*e**13 - 102*a**6*c**2*d**4*e**11 + 210*a**5*c**3*d**6*e**9 - 150*a**4*c**4*d**8*e**
7 - 6*a**3*c**5*d**10*e**5 + 54*a**2*c**6*d**12*e**3 - 18*a*c**7*d**14*e) + x*(12*a**8*d*e**14 - 42*a**7*c*d**
3*e**12 + 30*a**6*c**2*d**5*e**10 + 60*a**5*c**3*d**7*e**8 - 120*a**4*c**4*d**9*e**6 + 78*a**3*c**5*d**11*e**4
- 18*a**2*c**6*d**13*e**2))

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