3.20.3 \(\int \frac {(d+e x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [1903]

Optimal. Leaf size=139 \[ -\frac {1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}+\frac {e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac {e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 139, 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 = {640, 46} \begin {gather*} -\frac {e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}+\frac {e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac {1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}-\frac {e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-1/3*1/((c*d^2 - a*e^2)*(a*e + c*d*x)^3) + e/(2*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^2) - e^2/((c*d^2 - a*e^2)^3*(a
*e + c*d*x)) - (e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2)^4 + (e^3*Log[d + e*x])/(c*d^2 - a*e^2)^4

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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)^3}{\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)} \, dx\\ &=\int \left (\frac {c d}{\left (c d^2-a e^2\right ) (a e+c d x)^4}-\frac {c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)^3}+\frac {c d e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {c d e^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac {e^4}{\left (c d^2-a e^2\right )^4 (d+e x)}\right ) \, dx\\ &=-\frac {1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}+\frac {e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac {e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 117, normalized size = 0.84 \begin {gather*} -\frac {\frac {\left (c d^2-a e^2\right ) \left (11 a^2 e^4+a c d e^2 (-7 d+15 e x)+c^2 d^2 \left (2 d^2-3 d e x+6 e^2 x^2\right )\right )}{(a e+c d x)^3}+6 e^3 \log (a e+c d x)-6 e^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-1/6*(((c*d^2 - a*e^2)*(11*a^2*e^4 + a*c*d*e^2*(-7*d + 15*e*x) + c^2*d^2*(2*d^2 - 3*d*e*x + 6*e^2*x^2)))/(a*e
+ c*d*x)^3 + 6*e^3*Log[a*e + c*d*x] - 6*e^3*Log[d + e*x])/(c*d^2 - a*e^2)^4

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Maple [A]
time = 0.76, size = 135, normalized size = 0.97

method	result	size
default	\(\frac {e^{3} \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}+\frac {1}{3 \left (e^{2} a -c \,d^{2}\right ) \left (c d x +a e \right )^{3}}+\frac {e}{2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d x +a e \right )^{2}}+\frac {e^{2}}{\left (e^{2} a -c \,d^{2}\right )^{3} \left (c d x +a e \right )}-\frac {e^{3} \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}\)	\(135\)
risch	\(\frac {\frac {c^{2} d^{2} e^{2} x^{2}}{e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}+\frac {c d \left (5 e^{2} a -c \,d^{2}\right ) e x}{2 e^{6} a^{3}-6 e^{4} d^{2} a^{2} c +6 d^{4} e^{2} c^{2} a -2 d^{6} c^{3}}+\frac {11 a^{2} e^{4}-7 a c \,d^{2} e^{2}+2 c^{2} d^{4}}{6 e^{6} a^{3}-18 e^{4} d^{2} a^{2} c +18 d^{4} e^{2} c^{2} a -6 d^{6} c^{3}}}{\left (c d x +a e \right )^{3}}-\frac {e^{3} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {e^{3} \ln \left (-e x -d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\)	\(338\)
norman	\(\frac {\frac {c^{2} d^{2} e^{5} x^{5}}{e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}+\frac {11 a^{2} c^{3} d^{3} e^{4}-7 a \,c^{4} d^{5} e^{2}+2 c^{5} d^{7}}{6 c^{3} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}+\frac {\left (5 a \,c^{3} d^{3} e^{8}+5 e^{6} d^{5} c^{4}\right ) x^{4}}{2 e^{2} d^{2} c^{2} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}+\frac {\left (11 a^{2} c^{3} d^{3} e^{6}-2 a \,c^{4} d^{5} e^{4}+e^{2} d^{7} c^{5}\right ) x}{2 e d \,c^{3} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}+\frac {\left (11 a^{2} c^{3} d^{3} e^{8}+8 a \,c^{4} d^{5} e^{6}+e^{4} d^{7} c^{5}\right ) x^{2}}{2 d^{2} e^{2} c^{3} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}+\frac {\left (11 a^{2} c^{3} d^{3} e^{10}+38 a \,c^{4} d^{5} e^{8}+11 d^{7} e^{6} c^{5}\right ) x^{3}}{6 e^{3} d^{3} c^{3} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}+\frac {e^{3} \ln \left (e x +d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {e^{3} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\)	\(650\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (133) = 266\).
time = 0.33, size = 388, normalized size = 2.79 \begin {gather*} -\frac {e^{3} \log \left (c d x + a e\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 {e^{3} \log \left (x e + 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 {6 \, c^{2} d^{2} x^{2} e^{2} + 2 \, c^{2} d^{4} - 7 \, a c d^{2} e^{2} + 11 \, a^{2} e^{4} - 3 \, {\left (c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x}{6 \, {\left (a^{3} c^{3} d^{6} e^{3} - 3 \, a^{4} c^{2} d^{4} e^{5} + 3 \, a^{5} c d^{2} e^{7} - a^{6} e^{9} + {\left (c^{6} d^{9} - 3 \, a c^{5} d^{7} e^{2} + 3 \, a^{2} c^{4} d^{5} e^{4} - a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (a c^{5} d^{8} e - 3 \, a^{2} c^{4} d^{6} e^{3} + 3 \, a^{3} c^{3} d^{4} e^{5} - a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + 3 \, {\left (a^{2} c^{4} d^{7} e^{2} - 3 \, a^{3} c^{3} d^{5} e^{4} + 3 \, a^{4} c^{2} d^{3} e^{6} - a^{5} c d e^{8}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (133) = 266\).
time = 3.77, size = 487, normalized size = 3.50 \begin {gather*} \frac {3 \, c^{3} d^{5} x e - 2 \, c^{3} d^{6} - 18 \, a c^{2} d^{3} x e^{3} + 15 \, a^{2} c d x e^{5} + 11 \, a^{3} e^{6} + 6 \, {\left (a c^{2} d^{2} x^{2} - 3 \, a^{2} c d^{2}\right )} e^{4} - 3 \, {\left (2 \, c^{3} d^{4} x^{2} - 3 \, a c^{2} d^{4}\right )} e^{2} - 6 \, {\left (c^{3} d^{3} x^{3} e^{3} + 3 \, a c^{2} d^{2} x^{2} e^{4} + 3 \, a^{2} c d x e^{5} + a^{3} e^{6}\right )} \log \left (c d x + a e\right ) + 6 \, {\left (c^{3} d^{3} x^{3} e^{3} + 3 \, a c^{2} d^{2} x^{2} e^{4} + 3 \, a^{2} c d x e^{5} + a^{3} e^{6}\right )} \log \left (x e + d\right )}{6 \, {\left (c^{7} d^{11} x^{3} + 3 \, a c^{6} d^{10} x^{2} e + 3 \, a^{6} c d x e^{10} + a^{7} e^{11} + {\left (3 \, a^{5} c^{2} d^{2} x^{2} - 4 \, a^{6} c d^{2}\right )} e^{9} + {\left (a^{4} c^{3} d^{3} x^{3} - 12 \, a^{5} c^{2} d^{3} x\right )} e^{8} - 6 \, {\left (2 \, a^{4} c^{3} d^{4} x^{2} - a^{5} c^{2} d^{4}\right )} e^{7} - 2 \, {\left (2 \, a^{3} c^{4} d^{5} x^{3} - 9 \, a^{4} c^{3} d^{5} x\right )} e^{6} + 2 \, {\left (9 \, a^{3} c^{4} d^{6} x^{2} - 2 \, a^{4} c^{3} d^{6}\right )} e^{5} + 6 \, {\left (a^{2} c^{5} d^{7} x^{3} - 2 \, a^{3} c^{4} d^{7} x\right )} e^{4} - {\left (12 \, a^{2} c^{5} d^{8} x^{2} - a^{3} c^{4} d^{8}\right )} e^{3} - {\left (4 \, a c^{6} d^{9} x^{3} - 3 \, a^{2} c^{5} d^{9} x\right )} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

1/6*(3*c^3*d^5*x*e - 2*c^3*d^6 - 18*a*c^2*d^3*x*e^3 + 15*a^2*c*d*x*e^5 + 11*a^3*e^6 + 6*(a*c^2*d^2*x^2 - 3*a^2
*c*d^2)*e^4 - 3*(2*c^3*d^4*x^2 - 3*a*c^2*d^4)*e^2 - 6*(c^3*d^3*x^3*e^3 + 3*a*c^2*d^2*x^2*e^4 + 3*a^2*c*d*x*e^5
+ a^3*e^6)*log(c*d*x + a*e) + 6*(c^3*d^3*x^3*e^3 + 3*a*c^2*d^2*x^2*e^4 + 3*a^2*c*d*x*e^5 + a^3*e^6)*log(x*e +
d))/(c^7*d^11*x^3 + 3*a*c^6*d^10*x^2*e + 3*a^6*c*d*x*e^10 + a^7*e^11 + (3*a^5*c^2*d^2*x^2 - 4*a^6*c*d^2)*e^9
+ (a^4*c^3*d^3*x^3 - 12*a^5*c^2*d^3*x)*e^8 - 6*(2*a^4*c^3*d^4*x^2 - a^5*c^2*d^4)*e^7 - 2*(2*a^3*c^4*d^5*x^3 -
9*a^4*c^3*d^5*x)*e^6 + 2*(9*a^3*c^4*d^6*x^2 - 2*a^4*c^3*d^6)*e^5 + 6*(a^2*c^5*d^7*x^3 - 2*a^3*c^4*d^7*x)*e^4 -
(12*a^2*c^5*d^8*x^2 - a^3*c^4*d^8)*e^3 - (4*a*c^6*d^9*x^3 - 3*a^2*c^5*d^9*x)*e^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (119) = 238\).
time = 1.16, size = 668, normalized size = 4.81 \begin {gather*} \frac {e^{3} \log {\left (x + \frac {- \frac {a^{5} e^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {5 a^{4} c d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {10 a^{3} c^{2} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {10 a^{2} c^{3} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {5 a c^{4} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + a e^{5} + \frac {c^{5} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + c d^{2} e^{3}}{2 c d e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {e^{3} \log {\left (x + \frac {\frac {a^{5} e^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {5 a^{4} c d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {10 a^{3} c^{2} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {10 a^{2} c^{3} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {5 a c^{4} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + a e^{5} - \frac {c^{5} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + c d^{2} e^{3}}{2 c d e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {11 a^{2} e^{4} - 7 a c d^{2} e^{2} + 2 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (15 a c d e^{3} - 3 c^{2} d^{3} e\right )}{6 a^{6} e^{9} - 18 a^{5} c d^{2} e^{7} + 18 a^{4} c^{2} d^{4} e^{5} - 6 a^{3} c^{3} d^{6} e^{3} + x^{3} \cdot \left (6 a^{3} c^{3} d^{3} e^{6} - 18 a^{2} c^{4} d^{5} e^{4} + 18 a c^{5} d^{7} e^{2} - 6 c^{6} d^{9}\right ) + x^{2} \cdot \left (18 a^{4} c^{2} d^{2} e^{7} - 54 a^{3} c^{3} d^{4} e^{5} + 54 a^{2} c^{4} d^{6} e^{3} - 18 a c^{5} d^{8} e\right ) + x \left (18 a^{5} c d e^{8} - 54 a^{4} c^{2} d^{3} e^{6} + 54 a^{3} c^{3} d^{5} e^{4} - 18 a^{2} c^{4} d^{7} e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

e**3*log(x + (-a**5*e**13/(a*e**2 - c*d**2)**4 + 5*a**4*c*d**2*e**11/(a*e**2 - c*d**2)**4 - 10*a**3*c**2*d**4*
e**9/(a*e**2 - c*d**2)**4 + 10*a**2*c**3*d**6*e**7/(a*e**2 - c*d**2)**4 - 5*a*c**4*d**8*e**5/(a*e**2 - c*d**2)
**4 + a*e**5 + c**5*d**10*e**3/(a*e**2 - c*d**2)**4 + c*d**2*e**3)/(2*c*d*e**4))/(a*e**2 - c*d**2)**4 - e**3*l
og(x + (a**5*e**13/(a*e**2 - c*d**2)**4 - 5*a**4*c*d**2*e**11/(a*e**2 - c*d**2)**4 + 10*a**3*c**2*d**4*e**9/(a
*e**2 - c*d**2)**4 - 10*a**2*c**3*d**6*e**7/(a*e**2 - c*d**2)**4 + 5*a*c**4*d**8*e**5/(a*e**2 - c*d**2)**4 + a
*e**5 - c**5*d**10*e**3/(a*e**2 - c*d**2)**4 + c*d**2*e**3)/(2*c*d*e**4))/(a*e**2 - c*d**2)**4 + (11*a**2*e**4
- 7*a*c*d**2*e**2 + 2*c**2*d**4 + 6*c**2*d**2*e**2*x**2 + x*(15*a*c*d*e**3 - 3*c**2*d**3*e))/(6*a**6*e**9 - 1
8*a**5*c*d**2*e**7 + 18*a**4*c**2*d**4*e**5 - 6*a**3*c**3*d**6*e**3 + x**3*(6*a**3*c**3*d**3*e**6 - 18*a**2*c*
*4*d**5*e**4 + 18*a*c**5*d**7*e**2 - 6*c**6*d**9) + x**2*(18*a**4*c**2*d**2*e**7 - 54*a**3*c**3*d**4*e**5 + 54
*a**2*c**4*d**6*e**3 - 18*a*c**5*d**8*e) + x*(18*a**5*c*d*e**8 - 54*a**4*c**2*d**3*e**6 + 54*a**3*c**3*d**5*e*
*4 - 18*a**2*c**4*d**7*e**2))

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Giac [A]
time = 0.61, size = 260, normalized size = 1.87 \begin {gather*} -\frac {c d e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}} + \frac {e^{4} \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} - \frac {2 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{6 \, {\left (c d^{2} - a e^{2}\right )}^{4} {\left (c d x + a e\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

-c*d*e^3*log(abs(c*d*x + a*e))/(c^5*d^9 - 4*a*c^4*d^7*e^2 + 6*a^2*c^3*d^5*e^4 - 4*a^3*c^2*d^3*e^6 + a^4*c*d*e^
8) + e^4*log(abs(x*e + d))/(c^4*d^8*e - 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 - 4*a^3*c*d^2*e^7 + a^4*e^9) - 1/6
*(2*c^3*d^6 - 9*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 - 11*a^3*e^6 + 6*(c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 - 3*(c^3*d
^5*e - 6*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*x)/((c*d^2 - a*e^2)^4*(c*d*x + a*e)^3)

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Mupad [B]
time = 0.76, size = 372, normalized size = 2.68 \begin {gather*} \frac {\frac {11\,a^2\,e^4-7\,a\,c\,d^2\,e^2+2\,c^2\,d^4}{6\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}-\frac {e\,x\,\left (c^2\,d^3-5\,a\,c\,d\,e^2\right )}{2\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}+\frac {c^2\,d^2\,e^2\,x^2}{a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3}-\frac {2\,e^3\,\mathrm {atanh}\left (\frac {a^4\,e^8-2\,a^3\,c\,d^2\,e^6+2\,a\,c^3\,d^6\,e^2-c^4\,d^8}{{\left (a\,e^2-c\,d^2\right )}^4}+\frac {2\,c\,d\,e\,x\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^4}\right )}{{\left (a\,e^2-c\,d^2\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)

[Out]

((11*a^2*e^4 + 2*c^2*d^4 - 7*a*c*d^2*e^2)/(6*(a^3*e^6 - c^3*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4)) - (e*x*(
c^2*d^3 - 5*a*c*d*e^2))/(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)) + (c^2*d^2*e^2*x^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))/(a^3*e^3 + c^3*d^3*x^3 + 3*a^2*c*d*e^2*x + 3*a*c^2*d^2*e*x^
2) - (2*e^3*atanh((a^4*e^8 - c^4*d^8 + 2*a*c^3*d^6*e^2 - 2*a^3*c*d^2*e^6)/(a*e^2 - c*d^2)^4 + (2*c*d*e*x*(a^3*
e^6 - c^3*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4))/(a*e^2 - c*d^2)^4))/(a*e^2 - c*d^2)^4

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