### 3.125 $$\int \frac{b+2 c x}{(b x+c x^2)^8} \, dx$$

Optimal. Leaf size=15 $-\frac{1}{7 \left (b x+c x^2\right )^7}$

[Out]

-1/(7*(b*x + c*x^2)^7)

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Rubi [A]  time = 0.0040906, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.056, Rules used = {629} $-\frac{1}{7 \left (b x+c x^2\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b + 2*c*x)/(b*x + c*x^2)^8,x]

[Out]

-1/(7*(b*x + c*x^2)^7)

Rule 629

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*} \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}\\ \end{align*}

Mathematica [A]  time = 0.0195546, size = 14, normalized size = 0.93 $-\frac{1}{7 x^7 (b+c x)^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(7*x^7*(b + c*x)^7)

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Maple [B]  time = 0.017, size = 177, normalized size = 11.8 \begin{align*} -{\frac{1}{7\,{b}^{7}{x}^{7}}}-132\,{\frac{{c}^{6}}{{b}^{13}x}}+66\,{\frac{{c}^{5}}{{b}^{12}{x}^{2}}}-30\,{\frac{{c}^{4}}{{b}^{11}{x}^{3}}}+12\,{\frac{{c}^{3}}{{b}^{10}{x}^{4}}}-4\,{\frac{{c}^{2}}{{b}^{9}{x}^{5}}}+{\frac{c}{{b}^{8}{x}^{6}}}+132\,{\frac{{c}^{7}}{{b}^{13} \left ( cx+b \right ) }}+66\,{\frac{{c}^{7}}{{b}^{12} \left ( cx+b \right ) ^{2}}}+30\,{\frac{{c}^{7}}{{b}^{11} \left ( cx+b \right ) ^{3}}}+12\,{\frac{{c}^{7}}{{b}^{10} \left ( cx+b \right ) ^{4}}}+4\,{\frac{{c}^{7}}{{b}^{9} \left ( cx+b \right ) ^{5}}}+{\frac{{c}^{7}}{{b}^{8} \left ( cx+b \right ) ^{6}}}+{\frac{{c}^{7}}{7\,{b}^{7} \left ( cx+b \right ) ^{7}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.981226, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{7 \,{\left (c x^{2} + b x\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/(c*x^2 + b*x)^7

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Fricas [B]  time = 1.18938, size = 171, normalized size = 11.4 \begin{align*} -\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 )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 4.08241, size = 87, normalized size = 5.8 \begin{align*} - \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.11225, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{7 \,{\left (c x^{2} + b x\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/(c*x^2 + b*x)^7