3.169 \(\int \frac {b+2 c x^2}{x^7 (b x+c x^3)^8} \, dx\)

Optimal. Leaf size=16 \[ -\frac {1}{14 x^{14} \left (b+c x^2\right )^7} \]

[Out]

-1/14/x^14/(c*x^2+b)^7

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1584, 446, 74} \[ -\frac {1}{14 x^{14} \left (b+c x^2\right )^7} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x^2)/(x^7*(b*x + c*x^3)^8),x]

[Out]

-1/(14*x^14*(b + c*x^2)^7)

Rule 74

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 446

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 1584

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*} \int \frac {b+2 c x^2}{x^7 \left (b x+c x^3\right )^8} \, dx &=\int \frac {b+2 c x^2}{x^{15} \left (b+c x^2\right )^8} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {b+2 c x}{x^8 (b+c x)^8} \, dx,x,x^2\right )\\ &=-\frac {1}{14 x^{14} \left (b+c x^2\right )^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 16, normalized size = 1.00 \[ -\frac {1}{14 x^{14} \left (b+c x^2\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x^2)/(x^7*(b*x + c*x^3)^8),x]

[Out]

-1/14*1/(x^14*(b + c*x^2)^7)

________________________________________________________________________________________

fricas [B]  time = 0.82, size = 81, normalized size = 5.06 \[ -\frac {1}{14 \, {\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 21 \, b^{2} c^{5} x^{24} + 35 \, b^{3} c^{4} x^{22} + 35 \, b^{4} c^{3} x^{20} + 21 \, b^{5} c^{2} x^{18} + 7 \, b^{6} c x^{16} + b^{7} x^{14}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x^2+b)/x^7/(c*x^3+b*x)^8,x, algorithm="fricas")

[Out]

-1/14/(c^7*x^28 + 7*b*c^6*x^26 + 21*b^2*c^5*x^24 + 35*b^3*c^4*x^22 + 35*b^4*c^3*x^20 + 21*b^5*c^2*x^18 + 7*b^6
*c*x^16 + b^7*x^14)

________________________________________________________________________________________

giac [A]  time = 0.41, size = 15, normalized size = 0.94 \[ -\frac {1}{14 \, {\left (c x^{4} + b x^{2}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x^2+b)/x^7/(c*x^3+b*x)^8,x, algorithm="giac")

[Out]

-1/14/(c*x^4 + b*x^2)^7

________________________________________________________________________________________

maple [B]  time = 0.02, size = 197, normalized size = 12.31 \[ -\frac {\left (-\frac {b^{6}}{7 \left (c \,x^{2}+b \right )^{7} c}-\frac {b^{5}}{\left (c \,x^{2}+b \right )^{6} c}-\frac {4 b^{4}}{\left (c \,x^{2}+b \right )^{5} c}-\frac {12 b^{3}}{\left (c \,x^{2}+b \right )^{4} c}-\frac {30 b^{2}}{\left (c \,x^{2}+b \right )^{3} c}-\frac {66 b}{\left (c \,x^{2}+b \right )^{2} c}-\frac {132}{\left (c \,x^{2}+b \right ) c}\right ) c^{8}}{2 b^{13}}-\frac {66 c^{6}}{b^{13} x^{2}}+\frac {33 c^{5}}{b^{12} x^{4}}-\frac {15 c^{4}}{b^{11} x^{6}}+\frac {6 c^{3}}{b^{10} x^{8}}-\frac {2 c^{2}}{b^{9} x^{10}}+\frac {c}{2 b^{8} x^{12}}-\frac {1}{14 b^{7} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x^2+b)/x^7/(c*x^3+b*x)^8,x)

[Out]

-1/2/b^13*c^8*(-1/7*b^6/c/(c*x^2+b)^7-b^5/c/(c*x^2+b)^6-132/c/(c*x^2+b)-66*b/c/(c*x^2+b)^2-4*b^4/c/(c*x^2+b)^5
-30*b^2/c/(c*x^2+b)^3-12*b^3/c/(c*x^2+b)^4)-1/14/b^7/x^14-66/b^13*c^6/x^2+33/b^12*c^5/x^4-15/b^11*c^4/x^6+6/b^
10*c^3/x^8-2/b^9*c^2/x^10+1/2/b^8*c/x^12

________________________________________________________________________________________

maxima [B]  time = 0.67, size = 81, normalized size = 5.06 \[ -\frac {1}{14 \, {\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 21 \, b^{2} c^{5} x^{24} + 35 \, b^{3} c^{4} x^{22} + 35 \, b^{4} c^{3} x^{20} + 21 \, b^{5} c^{2} x^{18} + 7 \, b^{6} c x^{16} + b^{7} x^{14}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x^2+b)/x^7/(c*x^3+b*x)^8,x, algorithm="maxima")

[Out]

-1/14/(c^7*x^28 + 7*b*c^6*x^26 + 21*b^2*c^5*x^24 + 35*b^3*c^4*x^22 + 35*b^4*c^3*x^20 + 21*b^5*c^2*x^18 + 7*b^6
*c*x^16 + b^7*x^14)

________________________________________________________________________________________

mupad [B]  time = 2.22, size = 14, normalized size = 0.88 \[ -\frac {1}{14\,x^{14}\,{\left (c\,x^2+b\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x^2)/(x^7*(b*x + c*x^3)^8),x)

[Out]

-1/(14*x^14*(b + c*x^2)^7)

________________________________________________________________________________________

sympy [B]  time = 1.43, size = 87, normalized size = 5.44 \[ - \frac {1}{14 b^{7} x^{14} + 98 b^{6} c x^{16} + 294 b^{5} c^{2} x^{18} + 490 b^{4} c^{3} x^{20} + 490 b^{3} c^{4} x^{22} + 294 b^{2} c^{5} x^{24} + 98 b c^{6} x^{26} + 14 c^{7} x^{28}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x**2+b)/x**7/(c*x**3+b*x)**8,x)

[Out]

-1/(14*b**7*x**14 + 98*b**6*c*x**16 + 294*b**5*c**2*x**18 + 490*b**4*c**3*x**20 + 490*b**3*c**4*x**22 + 294*b*
*2*c**5*x**24 + 98*b*c**6*x**26 + 14*c**7*x**28)

________________________________________________________________________________________