\(\int \frac {(d+e x)^4}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1529]

Optimal result

Integrand size = 26, antiderivative size = 28 \[ \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(d+e x)^5}{5 (b d-a e) (a+b x)^5} \]

[Out]

-1/5*(e*x+d)^5/(-a*e+b*d)/(b*x+a)^5

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 37} \[ \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(d+e x)^5}{5 (a+b x)^5 (b d-a e)} \]

[In]

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

[Out]

-1/5*(d + e*x)^5/((b*d - a*e)*(a + b*x)^5)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^4}{(a+b x)^6} \, dx \\ & = -\frac {(d+e x)^5}{5 (b d-a e) (a+b x)^5} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(28)=56\).

Time = 0.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.00 \[ \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {a^4 e^4+a^3 b e^3 (d+5 e x)+a^2 b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+a b^3 e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+b^4 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{5 b^5 (a+b x)^5} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(26)=52\).

Time = 2.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 6.39

method	result	size
risch	\(\frac {-\frac {e^{4} x^{4}}{b}-\frac {2 e^{3} \left (a e +b d \right ) x^{3}}{b^{2}}-\frac {2 e^{2} \left (a^{2} e^{2}+a b d e +b^{2} d^{2}\right ) x^{2}}{b^{3}}-\frac {e \left (a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{b^{4}}-\frac {e^{4} a^{4}+b \,e^{3} d \,a^{3}+b^{2} e^{2} d^{2} a^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}}{5 b^{5}}}{\left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}\)	\(179\)
norman	\(\frac {-\frac {e^{4} x^{4}}{b}+\frac {2 \left (-e^{4} a -b d \,e^{3}\right ) x^{3}}{b^{2}}+\frac {2 \left (-a^{2} e^{4}-a b d \,e^{3}-b^{2} d^{2} e^{2}\right ) x^{2}}{b^{3}}+\frac {\left (-e^{4} a^{3}-a^{2} d \,e^{3} b -a \,b^{2} d^{2} e^{2}-b^{3} d^{3} e \right ) x}{b^{4}}+\frac {-e^{4} a^{4}-b \,e^{3} d \,a^{3}-b^{2} e^{2} d^{2} a^{2}-a \,b^{3} d^{3} e -b^{4} d^{4}}{5 b^{5}}}{\left (b x +a \right )^{5}}\)	\(180\)
default	\(-\frac {2 e^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{b^{5} \left (b x +a \right )^{3}}-\frac {e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{5 b^{5} \left (b x +a \right )^{5}}+\frac {e \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{b^{5} \left (b x +a \right )^{4}}+\frac {2 e^{3} \left (a e -b d \right )}{b^{5} \left (b x +a \right )^{2}}-\frac {e^{4}}{b^{5} \left (b x +a \right )}\)	\(185\)
gosper	\(-\frac {5 b^{4} x^{4} e^{4}+10 x^{3} a \,b^{3} e^{4}+10 x^{3} b^{4} d \,e^{3}+10 x^{2} a^{2} b^{2} e^{4}+10 x^{2} a \,b^{3} d \,e^{3}+10 x^{2} b^{4} d^{2} e^{2}+5 x \,a^{3} b \,e^{4}+5 x \,a^{2} b^{2} d \,e^{3}+5 x a \,b^{3} d^{2} e^{2}+5 x \,b^{4} d^{3} e +e^{4} a^{4}+b \,e^{3} d \,a^{3}+b^{2} e^{2} d^{2} a^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}}{5 \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} b^{5}}\)	\(199\)
parallelrisch	\(\frac {-5 b^{4} x^{4} e^{4}-10 x^{3} a \,b^{3} e^{4}-10 x^{3} b^{4} d \,e^{3}-10 x^{2} a^{2} b^{2} e^{4}-10 x^{2} a \,b^{3} d \,e^{3}-10 x^{2} b^{4} d^{2} e^{2}-5 x \,a^{3} b \,e^{4}-5 x \,a^{2} b^{2} d \,e^{3}-5 x a \,b^{3} d^{2} e^{2}-5 x \,b^{4} d^{3} e -e^{4} a^{4}-b \,e^{3} d \,a^{3}-b^{2} e^{2} d^{2} a^{2}-a \,b^{3} d^{3} e -b^{4} d^{4}}{5 b^{5} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}\)	\(204\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (26) = 52\).

Time = 0.29 (sec) , antiderivative size = 215, normalized size of antiderivative = 7.68 \[ \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {5 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \, {\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \, {\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \, {\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )}} \]

[In]

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

[Out]

-1/5*(5*b^4*e^4*x^4 + b^4*d^4 + a*b^3*d^3*e + a^2*b^2*d^2*e^2 + a^3*b*d*e^3 + a^4*e^4 + 10*(b^4*d*e^3 + a*b^3*
e^4)*x^3 + 10*(b^4*d^2*e^2 + a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 5*(b^4*d^3*e + a*b^3*d^2*e^2 + a^2*b^2*d*e^3 + a
^3*b*e^4)*x)/(b^10*x^5 + 5*a*b^9*x^4 + 10*a^2*b^8*x^3 + 10*a^3*b^7*x^2 + 5*a^4*b^6*x + a^5*b^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (22) = 44\).

Time = 4.54 (sec) , antiderivative size = 236, normalized size of antiderivative = 8.43 \[ \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {- a^{4} e^{4} - a^{3} b d e^{3} - a^{2} b^{2} d^{2} e^{2} - a b^{3} d^{3} e - b^{4} d^{4} - 5 b^{4} e^{4} x^{4} + x^{3} \left (- 10 a b^{3} e^{4} - 10 b^{4} d e^{3}\right ) + x^{2} \left (- 10 a^{2} b^{2} e^{4} - 10 a b^{3} d e^{3} - 10 b^{4} d^{2} e^{2}\right ) + x \left (- 5 a^{3} b e^{4} - 5 a^{2} b^{2} d e^{3} - 5 a b^{3} d^{2} e^{2} - 5 b^{4} d^{3} e\right )}{5 a^{5} b^{5} + 25 a^{4} b^{6} x + 50 a^{3} b^{7} x^{2} + 50 a^{2} b^{8} x^{3} + 25 a b^{9} x^{4} + 5 b^{10} x^{5}} \]

[In]

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

[Out]

(-a**4*e**4 - a**3*b*d*e**3 - a**2*b**2*d**2*e**2 - a*b**3*d**3*e - b**4*d**4 - 5*b**4*e**4*x**4 + x**3*(-10*a
*b**3*e**4 - 10*b**4*d*e**3) + x**2*(-10*a**2*b**2*e**4 - 10*a*b**3*d*e**3 - 10*b**4*d**2*e**2) + x*(-5*a**3*b
*e**4 - 5*a**2*b**2*d*e**3 - 5*a*b**3*d**2*e**2 - 5*b**4*d**3*e))/(5*a**5*b**5 + 25*a**4*b**6*x + 50*a**3*b**7
*x**2 + 50*a**2*b**8*x**3 + 25*a*b**9*x**4 + 5*b**10*x**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (26) = 52\).

Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 7.68 \[ \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {5 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \, {\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \, {\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \, {\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )}} \]

[In]

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

[Out]

-1/5*(5*b^4*e^4*x^4 + b^4*d^4 + a*b^3*d^3*e + a^2*b^2*d^2*e^2 + a^3*b*d*e^3 + a^4*e^4 + 10*(b^4*d*e^3 + a*b^3*
e^4)*x^3 + 10*(b^4*d^2*e^2 + a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 5*(b^4*d^3*e + a*b^3*d^2*e^2 + a^2*b^2*d*e^3 + a
^3*b*e^4)*x)/(b^10*x^5 + 5*a*b^9*x^4 + 10*a^2*b^8*x^3 + 10*a^3*b^7*x^2 + 5*a^4*b^6*x + a^5*b^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 6.43 \[ \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {5 \, b^{4} e^{4} x^{4} + 10 \, b^{4} d e^{3} x^{3} + 10 \, a b^{3} e^{4} x^{3} + 10 \, b^{4} d^{2} e^{2} x^{2} + 10 \, a b^{3} d e^{3} x^{2} + 10 \, a^{2} b^{2} e^{4} x^{2} + 5 \, b^{4} d^{3} e x + 5 \, a b^{3} d^{2} e^{2} x + 5 \, a^{2} b^{2} d e^{3} x + 5 \, a^{3} b e^{4} x + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4}}{5 \, {\left (b x + a\right )}^{5} b^{5}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 203, normalized size of antiderivative = 7.25 \[ \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\frac {a^4\,e^4+a^3\,b\,d\,e^3+a^2\,b^2\,d^2\,e^2+a\,b^3\,d^3\,e+b^4\,d^4}{5\,b^5}+\frac {e^4\,x^4}{b}+\frac {2\,e^3\,x^3\,\left (a\,e+b\,d\right )}{b^2}+\frac {e\,x\,\left (a^3\,e^3+a^2\,b\,d\,e^2+a\,b^2\,d^2\,e+b^3\,d^3\right )}{b^4}+\frac {2\,e^2\,x^2\,\left (a^2\,e^2+a\,b\,d\,e+b^2\,d^2\right )}{b^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \]

[In]

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

[Out]

-((a^4*e^4 + b^4*d^4 + a^2*b^2*d^2*e^2 + a*b^3*d^3*e + a^3*b*d*e^3)/(5*b^5) + (e^4*x^4)/b + (2*e^3*x^3*(a*e +
b*d))/b^2 + (e*x*(a^3*e^3 + b^3*d^3 + a*b^2*d^2*e + a^2*b*d*e^2))/b^4 + (2*e^2*x^2*(a^2*e^2 + b^2*d^2 + a*b*d*
e))/b^3)/(a^5 + b^5*x^5 + 5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x)