\(\int \frac {b+2 c x}{(b x+c x^2)^8} \, dx\) [168]

Optimal result

Integrand size = 18, antiderivative size = 15 \[ \int \frac {b+2 c x}{\left (b x+c x^2\right )^8} \, dx=-\frac {1}{7 \left (b x+c x^2\right )^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {643} \[ \int \frac {b+2 c x}{\left (b x+c x^2\right )^8} \, dx=-\frac {1}{7 \left (b x+c x^2\right )^7} \]

[In]

Int[(b + 2*c*x)/(b*x + c*x^2)^8,x]

[Out]

-1/7*1/(b*x + c*x^2)^7

Rule 643

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

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{7 \left (b x+c x^2\right )^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {b+2 c x}{\left (b x+c x^2\right )^8} \, dx=-\frac {1}{7 x^7 (b+c x)^7} \]

[In]

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

[Out]

-1/7*1/(x^7*(b + c*x)^7)

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87

method	result	size
gosper	\(-\frac {1}{7 x^{7} \left (c x +b \right )^{7}}\)	\(13\)
norman	\(-\frac {1}{7 x^{7} \left (c x +b \right )^{7}}\)	\(13\)
risch	\(-\frac {1}{7 x^{7} \left (c x +b \right )^{7}}\)	\(13\)
parallelrisch	\(-\frac {1}{7 x^{7} \left (c x +b \right )^{7}}\)	\(13\)
derivativedivides	\(-\frac {1}{7 \left (c \,x^{2}+b x \right )^{7}}\)	\(14\)
default	\(-\frac {1}{7 b^{7} x^{7}}-\frac {132 c^{6}}{b^{13} x}+\frac {66 c^{5}}{b^{12} x^{2}}-\frac {30 c^{4}}{b^{11} x^{3}}+\frac {12 c^{3}}{b^{10} x^{4}}-\frac {4 c^{2}}{b^{9} x^{5}}+\frac {c}{b^{8} x^{6}}+\frac {132 c^{7}}{b^{13} \left (c x +b \right )}+\frac {66 c^{7}}{b^{12} \left (c x +b \right )^{2}}+\frac {30 c^{7}}{b^{11} \left (c x +b \right )^{3}}+\frac {12 c^{7}}{b^{10} \left (c x +b \right )^{4}}+\frac {4 c^{7}}{b^{9} \left (c x +b \right )^{5}}+\frac {c^{7}}{b^{8} \left (c x +b \right )^{6}}+\frac {c^{7}}{7 b^{7} \left (c x +b \right )^{7}}\)	\(177\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (13) = 26\).

Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 5.40 \[ \int \frac {b+2 c x}{\left (b x+c x^2\right )^8} \, dx=-\frac {1}{7 \, {\left (c^{7} x^{14} + 7 \, b c^{6} x^{13} + 21 \, b^{2} c^{5} x^{12} + 35 \, b^{3} c^{4} x^{11} + 35 \, b^{4} c^{3} x^{10} + 21 \, b^{5} c^{2} x^{9} + 7 \, b^{6} c x^{8} + b^{7} x^{7}\right )}} \]

[In]

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

[Out]

-1/7/(c^7*x^14 + 7*b*c^6*x^13 + 21*b^2*c^5*x^12 + 35*b^3*c^4*x^11 + 35*b^4*c^3*x^10 + 21*b^5*c^2*x^9 + 7*b^6*c
*x^8 + b^7*x^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (14) = 28\).

Time = 0.46 (sec) , antiderivative size = 87, normalized size of antiderivative = 5.80 \[ \int \frac {b+2 c x}{\left (b x+c x^2\right )^8} \, dx=- \frac {1}{7 b^{7} x^{7} + 49 b^{6} c x^{8} + 147 b^{5} c^{2} x^{9} + 245 b^{4} c^{3} x^{10} + 245 b^{3} c^{4} x^{11} + 147 b^{2} c^{5} x^{12} + 49 b c^{6} x^{13} + 7 c^{7} x^{14}} \]

[In]

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

[Out]

-1/(7*b**7*x**7 + 49*b**6*c*x**8 + 147*b**5*c**2*x**9 + 245*b**4*c**3*x**10 + 245*b**3*c**4*x**11 + 147*b**2*c
**5*x**12 + 49*b*c**6*x**13 + 7*c**7*x**14)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {b+2 c x}{\left (b x+c x^2\right )^8} \, dx=-\frac {1}{7 \, {\left (c x^{2} + b x\right )}^{7}} \]

[In]

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

[Out]

-1/7/(c*x^2 + b*x)^7

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {b+2 c x}{\left (b x+c x^2\right )^8} \, dx=-\frac {1}{7 \, {\left (c x^{2} + b x\right )}^{7}} \]

[In]

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

[Out]

-1/7/(c*x^2 + b*x)^7

Mupad [B] (verification not implemented)

Time = 10.13 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {b+2 c x}{\left (b x+c x^2\right )^8} \, dx=-\frac {1}{7\,x^7\,{\left (b+c\,x\right )}^7} \]

[In]

int((b + 2*c*x)/(b*x + c*x^2)^8,x)

[Out]

-1/(7*x^7*(b + c*x)^7)