\(\int \frac {1}{(a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5)^3} \, dx\) [68]

Optimal result

Integrand size = 51, antiderivative size = 14 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 b (a+b x)^{14}} \]

[Out]

-1/14/b/(b*x+a)^14

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2083, 32} \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 b (a+b x)^{14}} \]

[In]

Int[(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)^(-3),x]

[Out]

-1/14*1/(b*(a + b*x)^14)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2083

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^{15}} \, dx \\ & = -\frac {1}{14 b (a+b x)^{14}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 b (a+b x)^{14}} \]

[In]

Integrate[(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)^(-3),x]

[Out]

-1/14*1/(b*(a + b*x)^14)

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {1}{14 b \left (b x +a \right )^{14}}\)	\(13\)
norman	\(-\frac {1}{14 b \left (b x +a \right )^{14}}\)	\(13\)
risch	\(-\frac {1}{14 b \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )^{3} \left (b x +a \right )^{2}}\)	\(53\)
gosper	\(-\frac {1}{14 \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right ) \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 x \,a^{4} b +a^{5}\right )^{2} b}\)	\(97\)
parallelrisch	\(-\frac {1}{14 \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right ) \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 x \,a^{4} b +a^{5}\right )^{2} b}\)	\(97\)

[In]

int(1/(b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^3,x,method=_RETURNVERBOSE)

[Out]

-1/14/b/(b*x+a)^14

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 11.14 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 \, {\left (b^{15} x^{14} + 14 \, a b^{14} x^{13} + 91 \, a^{2} b^{13} x^{12} + 364 \, a^{3} b^{12} x^{11} + 1001 \, a^{4} b^{11} x^{10} + 2002 \, a^{5} b^{10} x^{9} + 3003 \, a^{6} b^{9} x^{8} + 3432 \, a^{7} b^{8} x^{7} + 3003 \, a^{8} b^{7} x^{6} + 2002 \, a^{9} b^{6} x^{5} + 1001 \, a^{10} b^{5} x^{4} + 364 \, a^{11} b^{4} x^{3} + 91 \, a^{12} b^{3} x^{2} + 14 \, a^{13} b^{2} x + a^{14} b\right )}} \]

[In]

integrate(1/(b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^3,x, algorithm="fricas")

[Out]

-1/14/(b^15*x^14 + 14*a*b^14*x^13 + 91*a^2*b^13*x^12 + 364*a^3*b^12*x^11 + 1001*a^4*b^11*x^10 + 2002*a^5*b^10*
x^9 + 3003*a^6*b^9*x^8 + 3432*a^7*b^8*x^7 + 3003*a^8*b^7*x^6 + 2002*a^9*b^6*x^5 + 1001*a^10*b^5*x^4 + 364*a^11
*b^4*x^3 + 91*a^12*b^3*x^2 + 14*a^13*b^2*x + a^14*b)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (12) = 24\).

Time = 0.43 (sec) , antiderivative size = 168, normalized size of antiderivative = 12.00 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=- \frac {1}{14 a^{14} b + 196 a^{13} b^{2} x + 1274 a^{12} b^{3} x^{2} + 5096 a^{11} b^{4} x^{3} + 14014 a^{10} b^{5} x^{4} + 28028 a^{9} b^{6} x^{5} + 42042 a^{8} b^{7} x^{6} + 48048 a^{7} b^{8} x^{7} + 42042 a^{6} b^{9} x^{8} + 28028 a^{5} b^{10} x^{9} + 14014 a^{4} b^{11} x^{10} + 5096 a^{3} b^{12} x^{11} + 1274 a^{2} b^{13} x^{12} + 196 a b^{14} x^{13} + 14 b^{15} x^{14}} \]

[In]

integrate(1/(b**5*x**5+5*a*b**4*x**4+10*a**2*b**3*x**3+10*a**3*b**2*x**2+5*a**4*b*x+a**5)**3,x)

[Out]

-1/(14*a**14*b + 196*a**13*b**2*x + 1274*a**12*b**3*x**2 + 5096*a**11*b**4*x**3 + 14014*a**10*b**5*x**4 + 2802
8*a**9*b**6*x**5 + 42042*a**8*b**7*x**6 + 48048*a**7*b**8*x**7 + 42042*a**6*b**9*x**8 + 28028*a**5*b**10*x**9
+ 14014*a**4*b**11*x**10 + 5096*a**3*b**12*x**11 + 1274*a**2*b**13*x**12 + 196*a*b**14*x**13 + 14*b**15*x**14)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (12) = 24\).

Time = 0.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 11.14 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 \, {\left (b^{15} x^{14} + 14 \, a b^{14} x^{13} + 91 \, a^{2} b^{13} x^{12} + 364 \, a^{3} b^{12} x^{11} + 1001 \, a^{4} b^{11} x^{10} + 2002 \, a^{5} b^{10} x^{9} + 3003 \, a^{6} b^{9} x^{8} + 3432 \, a^{7} b^{8} x^{7} + 3003 \, a^{8} b^{7} x^{6} + 2002 \, a^{9} b^{6} x^{5} + 1001 \, a^{10} b^{5} x^{4} + 364 \, a^{11} b^{4} x^{3} + 91 \, a^{12} b^{3} x^{2} + 14 \, a^{13} b^{2} x + a^{14} b\right )}} \]

[In]

integrate(1/(b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^3,x, algorithm="maxima")

[Out]

-1/14/(b^15*x^14 + 14*a*b^14*x^13 + 91*a^2*b^13*x^12 + 364*a^3*b^12*x^11 + 1001*a^4*b^11*x^10 + 2002*a^5*b^10*
x^9 + 3003*a^6*b^9*x^8 + 3432*a^7*b^8*x^7 + 3003*a^8*b^7*x^6 + 2002*a^9*b^6*x^5 + 1001*a^10*b^5*x^4 + 364*a^11
*b^4*x^3 + 91*a^12*b^3*x^2 + 14*a^13*b^2*x + a^14*b)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 \, {\left (b x + a\right )}^{14} b} \]

[In]

integrate(1/(b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^3,x, algorithm="giac")

[Out]

-1/14/((b*x + a)^14*b)

Mupad [B] (verification not implemented)

Time = 11.83 (sec) , antiderivative size = 158, normalized size of antiderivative = 11.29 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14\,a^{14}\,b+196\,a^{13}\,b^2\,x+1274\,a^{12}\,b^3\,x^2+5096\,a^{11}\,b^4\,x^3+14014\,a^{10}\,b^5\,x^4+28028\,a^9\,b^6\,x^5+42042\,a^8\,b^7\,x^6+48048\,a^7\,b^8\,x^7+42042\,a^6\,b^9\,x^8+28028\,a^5\,b^{10}\,x^9+14014\,a^4\,b^{11}\,x^{10}+5096\,a^3\,b^{12}\,x^{11}+1274\,a^2\,b^{13}\,x^{12}+196\,a\,b^{14}\,x^{13}+14\,b^{15}\,x^{14}} \]

[In]

int(1/(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)^3,x)

[Out]

-1/(14*a^14*b + 14*b^15*x^14 + 196*a^13*b^2*x + 196*a*b^14*x^13 + 1274*a^12*b^3*x^2 + 5096*a^11*b^4*x^3 + 1401
4*a^10*b^5*x^4 + 28028*a^9*b^6*x^5 + 42042*a^8*b^7*x^6 + 48048*a^7*b^8*x^7 + 42042*a^6*b^9*x^8 + 28028*a^5*b^1
0*x^9 + 14014*a^4*b^11*x^10 + 5096*a^3*b^12*x^11 + 1274*a^2*b^13*x^12)