\(\int \frac {b+2 c x^3}{x^{14} (b x+c x^4)^8} \, dx\) [170]

Optimal result

Integrand size = 23, antiderivative size = 16 \[ \int \frac {b+2 c x^3}{x^{14} \left (b x+c x^4\right )^8} \, dx=-\frac {1}{21 x^{21} \left (b+c x^3\right )^7} \]

[Out]

-1/21/x^21/(c*x^3+b)^7

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1598, 457, 75} \[ \int \frac {b+2 c x^3}{x^{14} \left (b x+c x^4\right )^8} \, dx=-\frac {1}{21 x^{21} \left (b+c x^3\right )^7} \]

[In]

Int[(b + 2*c*x^3)/(x^14*(b*x + c*x^4)^8),x]

[Out]

-1/21*1/(x^21*(b + c*x^3)^7)

Rule 75

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

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {b+2 c x^3}{x^{22} \left (b+c x^3\right )^8} \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {b+2 c x}{x^8 (b+c x)^8} \, dx,x,x^3\right ) \\ & = -\frac {1}{21 x^{21} \left (b+c x^3\right )^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {b+2 c x^3}{x^{14} \left (b x+c x^4\right )^8} \, dx=-\frac {1}{21 x^{21} \left (b+c x^3\right )^7} \]

[In]

Integrate[(b + 2*c*x^3)/(x^14*(b*x + c*x^4)^8),x]

[Out]

-1/21*1/(x^21*(b + c*x^3)^7)

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {1}{21 x^{21} \left (c \,x^{3}+b \right )^{7}}\)	\(15\)
risch	\(-\frac {1}{21 x^{21} \left (c \,x^{3}+b \right )^{7}}\)	\(15\)
parallelrisch	\(-\frac {1}{21 x^{21} \left (c \,x^{3}+b \right )^{7}}\)	\(15\)
default	\(-\frac {1}{21 b^{7} x^{21}}-\frac {44 c^{6}}{b^{13} x^{3}}+\frac {22 c^{5}}{b^{12} x^{6}}-\frac {10 c^{4}}{b^{11} x^{9}}+\frac {4 c^{3}}{b^{10} x^{12}}-\frac {4 c^{2}}{3 b^{9} x^{15}}+\frac {c}{3 b^{8} x^{18}}-\frac {c^{8} \left (-\frac {66 b}{c \left (c \,x^{3}+b \right )^{2}}-\frac {4 b^{4}}{c \left (c \,x^{3}+b \right )^{5}}-\frac {132}{c \left (c \,x^{3}+b \right )}-\frac {12 b^{3}}{c \left (c \,x^{3}+b \right )^{4}}-\frac {30 b^{2}}{c \left (c \,x^{3}+b \right )^{3}}-\frac {b^{6}}{7 c \left (c \,x^{3}+b \right )^{7}}-\frac {b^{5}}{c \left (c \,x^{3}+b \right )^{6}}\right )}{3 b^{13}}\)	\(197\)

[In]

int((2*c*x^3+b)/x^14/(c*x^4+b*x)^8,x,method=_RETURNVERBOSE)

[Out]

-1/21/x^21/(c*x^3+b)^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (14) = 28\).

Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 5.06 \[ \int \frac {b+2 c x^3}{x^{14} \left (b x+c x^4\right )^8} \, dx=-\frac {1}{21 \, {\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 21 \, b^{2} c^{5} x^{36} + 35 \, b^{3} c^{4} x^{33} + 35 \, b^{4} c^{3} x^{30} + 21 \, b^{5} c^{2} x^{27} + 7 \, b^{6} c x^{24} + b^{7} x^{21}\right )}} \]

[In]

integrate((2*c*x^3+b)/x^14/(c*x^4+b*x)^8,x, algorithm="fricas")

[Out]

-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 21*b^2*c^5*x^36 + 35*b^3*c^4*x^33 + 35*b^4*c^3*x^30 + 21*b^5*c^2*x^27 + 7*b^6
*c*x^24 + b^7*x^21)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (15) = 30\).

Time = 0.93 (sec) , antiderivative size = 87, normalized size of antiderivative = 5.44 \[ \int \frac {b+2 c x^3}{x^{14} \left (b x+c x^4\right )^8} \, dx=- \frac {1}{21 b^{7} x^{21} + 147 b^{6} c x^{24} + 441 b^{5} c^{2} x^{27} + 735 b^{4} c^{3} x^{30} + 735 b^{3} c^{4} x^{33} + 441 b^{2} c^{5} x^{36} + 147 b c^{6} x^{39} + 21 c^{7} x^{42}} \]

[In]

integrate((2*c*x**3+b)/x**14/(c*x**4+b*x)**8,x)

[Out]

-1/(21*b**7*x**21 + 147*b**6*c*x**24 + 441*b**5*c**2*x**27 + 735*b**4*c**3*x**30 + 735*b**3*c**4*x**33 + 441*b
**2*c**5*x**36 + 147*b*c**6*x**39 + 21*c**7*x**42)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (14) = 28\).

Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 5.06 \[ \int \frac {b+2 c x^3}{x^{14} \left (b x+c x^4\right )^8} \, dx=-\frac {1}{21 \, {\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 21 \, b^{2} c^{5} x^{36} + 35 \, b^{3} c^{4} x^{33} + 35 \, b^{4} c^{3} x^{30} + 21 \, b^{5} c^{2} x^{27} + 7 \, b^{6} c x^{24} + b^{7} x^{21}\right )}} \]

[In]

integrate((2*c*x^3+b)/x^14/(c*x^4+b*x)^8,x, algorithm="maxima")

[Out]

-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 21*b^2*c^5*x^36 + 35*b^3*c^4*x^33 + 35*b^4*c^3*x^30 + 21*b^5*c^2*x^27 + 7*b^6
*c*x^24 + b^7*x^21)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {b+2 c x^3}{x^{14} \left (b x+c x^4\right )^8} \, dx=-\frac {1}{21 \, {\left (c x^{6} + b x^{3}\right )}^{7}} \]

[In]

integrate((2*c*x^3+b)/x^14/(c*x^4+b*x)^8,x, algorithm="giac")

[Out]

-1/21/(c*x^6 + b*x^3)^7

Mupad [B] (verification not implemented)

Time = 12.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {b+2 c x^3}{x^{14} \left (b x+c x^4\right )^8} \, dx=-\frac {1}{21\,x^{21}\,{\left (c\,x^3+b\right )}^7} \]

[In]

int((b + 2*c*x^3)/(x^14*(b*x + c*x^4)^8),x)

[Out]

-1/(21*x^21*(b + c*x^3)^7)