\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^5} \, dx\) [1858]

Optimal result

Integrand size = 35, antiderivative size = 94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=-\frac {c^2 d^2 \left (2 c d^2-3 a e^2\right ) x}{e^3}+\frac {c^3 d^3 x^2}{2 e^2}+\frac {\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=-\frac {c^2 d^2 x \left (2 c d^2-3 a e^2\right )}{e^3}+\frac {\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4}+\frac {c^3 d^3 x^2}{2 e^2} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^5,x]

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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*} \text {integral}& = \int \frac {(a e+c d x)^3}{(d+e x)^2} \, dx \\ & = \int \left (-\frac {c^2 d^2 \left (2 c d^2-3 a e^2\right )}{e^3}+\frac {c^3 d^3 x}{e^2}+\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^2}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)}\right ) \, dx \\ & = -\frac {c^2 d^2 \left (2 c d^2-3 a e^2\right ) x}{e^3}+\frac {c^3 d^3 x^2}{2 e^2}+\frac {\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {6 a^2 c d^2 e^4-2 a^3 e^6+6 a c^2 d^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+c^3 d^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+6 c d \left (c d^2-a e^2\right )^2 (d+e x) \log (d+e x)}{2 e^4 (d+e x)} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^5,x]

[Out]

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

Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.33

method	result	size
default	\(\frac {c^{2} d^{2} \left (\frac {1}{2} c d e \,x^{2}+3 a \,e^{2} x -2 c \,d^{2} x \right )}{e^{3}}-\frac {e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}{e^{4} \left (e x +d \right )}+\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{4}}\)	\(125\)
risch	\(\frac {c^{3} d^{3} x^{2}}{2 e^{2}}+\frac {3 d^{2} c^{2} a x}{e}-\frac {2 d^{4} c^{3} x}{e^{3}}-\frac {e^{2} a^{3}}{e x +d}+\frac {3 d^{2} a^{2} c}{e x +d}-\frac {3 d^{4} c^{2} a}{e^{2} \left (e x +d \right )}+\frac {c^{3} d^{6}}{e^{4} \left (e x +d \right )}+3 c d \ln \left (e x +d \right ) a^{2}-\frac {6 c^{2} d^{3} \ln \left (e x +d \right ) a}{e^{2}}+\frac {3 c^{3} d^{5} \ln \left (e x +d \right )}{e^{4}}\)	\(156\)
parallelrisch	\(\frac {x^{3} c^{3} d^{3} e^{3}+6 \ln \left (e x +d \right ) x \,a^{2} c d \,e^{5}-12 \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{3}+6 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +6 x^{2} a \,c^{2} d^{2} e^{4}-3 x^{2} c^{3} d^{4} e^{2}+6 \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{4}-12 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}+6 \ln \left (e x +d \right ) c^{3} d^{6}-2 e^{6} a^{3}+6 d^{2} e^{4} a^{2} c -12 d^{4} e^{2} c^{2} a +6 c^{3} d^{6}}{2 e^{4} \left (e x +d \right )}\)	\(198\)
norman	\(\frac {-\frac {d^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +15 d^{4} e^{2} c^{2} a -6 c^{3} d^{6}\right )}{e^{4}}-\frac {\left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +33 d^{4} e^{2} c^{2} a -11 c^{3} d^{6}\right ) x^{3}}{e}-\frac {3 d \left (2 e^{6} a^{3}-6 d^{2} e^{4} a^{2} c +46 d^{4} e^{2} c^{2} a -17 c^{3} d^{6}\right ) x^{2}}{2 e^{2}}-\frac {d^{2} \left (3 e^{6} a^{3}-9 d^{2} e^{4} a^{2} c +54 d^{4} e^{2} c^{2} a -21 c^{3} d^{6}\right ) x}{e^{3}}+\frac {e^{2} c^{3} d^{3} x^{6}}{2}+3 x^{5} e^{3} a \,c^{2} d^{2}}{\left (e x +d \right )^{4}}+\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{4}}\)	\(274\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (92) = 184\).

Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {c^{3} d^{3} e^{3} x^{3} + 2 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} - 3 \, {\left (c^{3} d^{4} e^{2} - 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x + 6 \, {\left (c^{3} d^{6} - 2 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x + d e^{4}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^5,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {c^{3} d^{3} x^{2}}{2 e^{2}} + \frac {3 c d \left (a e^{2} - c d^{2}\right )^{2} \log {\left (d + e x \right )}}{e^{4}} + x \left (\frac {3 a c^{2} d^{2}}{e} - \frac {2 c^{3} d^{4}}{e^{3}}\right ) + \frac {- a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}}{d e^{4} + e^{5} x} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**5,x)

[Out]

c**3*d**3*x**2/(2*e**2) + 3*c*d*(a*e**2 - c*d**2)**2*log(d + e*x)/e**4 + x*(3*a*c**2*d**2/e - 2*c**3*d**4/e**3
) + (-a**3*e**6 + 3*a**2*c*d**2*e**4 - 3*a*c**2*d**4*e**2 + c**3*d**6)/(d*e**4 + e**5*x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}}{e^{5} x + d e^{4}} + \frac {c^{3} d^{3} e x^{2} - 2 \, {\left (2 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} x}{2 \, e^{3}} + \frac {3 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \log \left (e x + d\right )}{e^{4}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^5,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {{\left (c^{3} d^{3} - \frac {6 \, {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )}}{{\left (e x + d\right )} e}\right )} {\left (e x + d\right )}^{2}}{2 \, e^{4}} - \frac {3 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{4}} + \frac {\frac {c^{3} d^{6} e^{2}}{e x + d} - \frac {3 \, a c^{2} d^{4} e^{4}}{e x + d} + \frac {3 \, a^{2} c d^{2} e^{6}}{e x + d} - \frac {a^{3} e^{8}}{e x + d}}{e^{6}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^5,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (3\,a^2\,c\,d\,e^4-6\,a\,c^2\,d^3\,e^2+3\,c^3\,d^5\right )}{e^4}-\frac {a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}{e\,\left (x\,e^4+d\,e^3\right )}-x\,\left (\frac {2\,c^3\,d^4}{e^3}-\frac {3\,a\,c^2\,d^2}{e}\right )+\frac {c^3\,d^3\,x^2}{2\,e^2} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3/(d + e*x)^5,x)

[Out]

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