3.219 \(\int \frac{1}{x^2 (a+b x)^7} \, dx\)

Optimal. Leaf size=117 \[ -\frac{6 b}{a^7 (a+b x)}-\frac{5 b}{2 a^6 (a+b x)^2}-\frac{4 b}{3 a^5 (a+b x)^3}-\frac{3 b}{4 a^4 (a+b x)^4}-\frac{2 b}{5 a^3 (a+b x)^5}-\frac{b}{6 a^2 (a+b x)^6}-\frac{7 b \log (x)}{a^8}+\frac{7 b \log (a+b x)}{a^8}-\frac{1}{a^7 x} \]

[Out]

-(1/(a^7*x)) - b/(6*a^2*(a + b*x)^6) - (2*b)/(5*a^3*(a + b*x)^5) - (3*b)/(4*a^4*(a + b*x)^4) - (4*b)/(3*a^5*(a
+ b*x)^3) - (5*b)/(2*a^6*(a + b*x)^2) - (6*b)/(a^7*(a + b*x)) - (7*b*Log[x])/a^8 + (7*b*Log[a + b*x])/a^8

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Rubi [A]  time = 0.0732826, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {44} \[ -\frac{6 b}{a^7 (a+b x)}-\frac{5 b}{2 a^6 (a+b x)^2}-\frac{4 b}{3 a^5 (a+b x)^3}-\frac{3 b}{4 a^4 (a+b x)^4}-\frac{2 b}{5 a^3 (a+b x)^5}-\frac{b}{6 a^2 (a+b x)^6}-\frac{7 b \log (x)}{a^8}+\frac{7 b \log (a+b x)}{a^8}-\frac{1}{a^7 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^7),x]

[Out]

-(1/(a^7*x)) - b/(6*a^2*(a + b*x)^6) - (2*b)/(5*a^3*(a + b*x)^5) - (3*b)/(4*a^4*(a + b*x)^4) - (4*b)/(3*a^5*(a
+ b*x)^3) - (5*b)/(2*a^6*(a + b*x)^2) - (6*b)/(a^7*(a + b*x)) - (7*b*Log[x])/a^8 + (7*b*Log[a + b*x])/a^8

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 (a+b x)^7} \, dx &=\int \left (\frac{1}{a^7 x^2}-\frac{7 b}{a^8 x}+\frac{b^2}{a^2 (a+b x)^7}+\frac{2 b^2}{a^3 (a+b x)^6}+\frac{3 b^2}{a^4 (a+b x)^5}+\frac{4 b^2}{a^5 (a+b x)^4}+\frac{5 b^2}{a^6 (a+b x)^3}+\frac{6 b^2}{a^7 (a+b x)^2}+\frac{7 b^2}{a^8 (a+b x)}\right ) \, dx\\ &=-\frac{1}{a^7 x}-\frac{b}{6 a^2 (a+b x)^6}-\frac{2 b}{5 a^3 (a+b x)^5}-\frac{3 b}{4 a^4 (a+b x)^4}-\frac{4 b}{3 a^5 (a+b x)^3}-\frac{5 b}{2 a^6 (a+b x)^2}-\frac{6 b}{a^7 (a+b x)}-\frac{7 b \log (x)}{a^8}+\frac{7 b \log (a+b x)}{a^8}\\ \end{align*}

Mathematica [A]  time = 0.164784, size = 97, normalized size = 0.83 \[ -\frac{\frac{a \left (3654 a^4 b^2 x^2+5985 a^3 b^3 x^3+5180 a^2 b^4 x^4+1029 a^5 b x+60 a^6+2310 a b^5 x^5+420 b^6 x^6\right )}{x (a+b x)^6}-420 b \log (a+b x)+420 b \log (x)}{60 a^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^7),x]

[Out]

-((a*(60*a^6 + 1029*a^5*b*x + 3654*a^4*b^2*x^2 + 5985*a^3*b^3*x^3 + 5180*a^2*b^4*x^4 + 2310*a*b^5*x^5 + 420*b^
6*x^6))/(x*(a + b*x)^6) + 420*b*Log[x] - 420*b*Log[a + b*x])/(60*a^8)

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Maple [A]  time = 0.011, size = 108, normalized size = 0.9 \begin{align*} -{\frac{1}{{a}^{7}x}}-{\frac{b}{6\,{a}^{2} \left ( bx+a \right ) ^{6}}}-{\frac{2\,b}{5\,{a}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{3\,b}{4\,{a}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{4\,b}{3\,{a}^{5} \left ( bx+a \right ) ^{3}}}-{\frac{5\,b}{2\,{a}^{6} \left ( bx+a \right ) ^{2}}}-6\,{\frac{b}{{a}^{7} \left ( bx+a \right ) }}-7\,{\frac{b\ln \left ( x \right ) }{{a}^{8}}}+7\,{\frac{b\ln \left ( bx+a \right ) }{{a}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x+a)^7,x)

[Out]

-1/a^7/x-1/6*b/a^2/(b*x+a)^6-2/5*b/a^3/(b*x+a)^5-3/4*b/a^4/(b*x+a)^4-4/3*b/a^5/(b*x+a)^3-5/2*b/a^6/(b*x+a)^2-6
*b/a^7/(b*x+a)-7*b*ln(x)/a^8+7*b*ln(b*x+a)/a^8

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Maxima [A]  time = 1.05639, size = 212, normalized size = 1.81 \begin{align*} -\frac{420 \, b^{6} x^{6} + 2310 \, a b^{5} x^{5} + 5180 \, a^{2} b^{4} x^{4} + 5985 \, a^{3} b^{3} x^{3} + 3654 \, a^{4} b^{2} x^{2} + 1029 \, a^{5} b x + 60 \, a^{6}}{60 \,{\left (a^{7} b^{6} x^{7} + 6 \, a^{8} b^{5} x^{6} + 15 \, a^{9} b^{4} x^{5} + 20 \, a^{10} b^{3} x^{4} + 15 \, a^{11} b^{2} x^{3} + 6 \, a^{12} b x^{2} + a^{13} x\right )}} + \frac{7 \, b \log \left (b x + a\right )}{a^{8}} - \frac{7 \, b \log \left (x\right )}{a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/60*(420*b^6*x^6 + 2310*a*b^5*x^5 + 5180*a^2*b^4*x^4 + 5985*a^3*b^3*x^3 + 3654*a^4*b^2*x^2 + 1029*a^5*b*x +
60*a^6)/(a^7*b^6*x^7 + 6*a^8*b^5*x^6 + 15*a^9*b^4*x^5 + 20*a^10*b^3*x^4 + 15*a^11*b^2*x^3 + 6*a^12*b*x^2 + a^1
3*x) + 7*b*log(b*x + a)/a^8 - 7*b*log(x)/a^8

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Fricas [B]  time = 1.63639, size = 628, normalized size = 5.37 \begin{align*} -\frac{420 \, a b^{6} x^{6} + 2310 \, a^{2} b^{5} x^{5} + 5180 \, a^{3} b^{4} x^{4} + 5985 \, a^{4} b^{3} x^{3} + 3654 \, a^{5} b^{2} x^{2} + 1029 \, a^{6} b x + 60 \, a^{7} - 420 \,{\left (b^{7} x^{7} + 6 \, a b^{6} x^{6} + 15 \, a^{2} b^{5} x^{5} + 20 \, a^{3} b^{4} x^{4} + 15 \, a^{4} b^{3} x^{3} + 6 \, a^{5} b^{2} x^{2} + a^{6} b x\right )} \log \left (b x + a\right ) + 420 \,{\left (b^{7} x^{7} + 6 \, a b^{6} x^{6} + 15 \, a^{2} b^{5} x^{5} + 20 \, a^{3} b^{4} x^{4} + 15 \, a^{4} b^{3} x^{3} + 6 \, a^{5} b^{2} x^{2} + a^{6} b x\right )} \log \left (x\right )}{60 \,{\left (a^{8} b^{6} x^{7} + 6 \, a^{9} b^{5} x^{6} + 15 \, a^{10} b^{4} x^{5} + 20 \, a^{11} b^{3} x^{4} + 15 \, a^{12} b^{2} x^{3} + 6 \, a^{13} b x^{2} + a^{14} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/60*(420*a*b^6*x^6 + 2310*a^2*b^5*x^5 + 5180*a^3*b^4*x^4 + 5985*a^4*b^3*x^3 + 3654*a^5*b^2*x^2 + 1029*a^6*b*
x + 60*a^7 - 420*(b^7*x^7 + 6*a*b^6*x^6 + 15*a^2*b^5*x^5 + 20*a^3*b^4*x^4 + 15*a^4*b^3*x^3 + 6*a^5*b^2*x^2 + a
^6*b*x)*log(b*x + a) + 420*(b^7*x^7 + 6*a*b^6*x^6 + 15*a^2*b^5*x^5 + 20*a^3*b^4*x^4 + 15*a^4*b^3*x^3 + 6*a^5*b
^2*x^2 + a^6*b*x)*log(x))/(a^8*b^6*x^7 + 6*a^9*b^5*x^6 + 15*a^10*b^4*x^5 + 20*a^11*b^3*x^4 + 15*a^12*b^2*x^3 +
6*a^13*b*x^2 + a^14*x)

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Sympy [A]  time = 1.34076, size = 160, normalized size = 1.37 \begin{align*} - \frac{60 a^{6} + 1029 a^{5} b x + 3654 a^{4} b^{2} x^{2} + 5985 a^{3} b^{3} x^{3} + 5180 a^{2} b^{4} x^{4} + 2310 a b^{5} x^{5} + 420 b^{6} x^{6}}{60 a^{13} x + 360 a^{12} b x^{2} + 900 a^{11} b^{2} x^{3} + 1200 a^{10} b^{3} x^{4} + 900 a^{9} b^{4} x^{5} + 360 a^{8} b^{5} x^{6} + 60 a^{7} b^{6} x^{7}} + \frac{7 b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x+a)**7,x)

[Out]

-(60*a**6 + 1029*a**5*b*x + 3654*a**4*b**2*x**2 + 5985*a**3*b**3*x**3 + 5180*a**2*b**4*x**4 + 2310*a*b**5*x**5
+ 420*b**6*x**6)/(60*a**13*x + 360*a**12*b*x**2 + 900*a**11*b**2*x**3 + 1200*a**10*b**3*x**4 + 900*a**9*b**4*
x**5 + 360*a**8*b**5*x**6 + 60*a**7*b**6*x**7) + 7*b*(-log(x) + log(a/b + x))/a**8

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Giac [A]  time = 1.1868, size = 140, normalized size = 1.2 \begin{align*} \frac{7 \, b \log \left ({\left | b x + a \right |}\right )}{a^{8}} - \frac{7 \, b \log \left ({\left | x \right |}\right )}{a^{8}} - \frac{420 \, a b^{6} x^{6} + 2310 \, a^{2} b^{5} x^{5} + 5180 \, a^{3} b^{4} x^{4} + 5985 \, a^{4} b^{3} x^{3} + 3654 \, a^{5} b^{2} x^{2} + 1029 \, a^{6} b x + 60 \, a^{7}}{60 \,{\left (b x + a\right )}^{6} a^{8} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^7,x, algorithm="giac")

[Out]

7*b*log(abs(b*x + a))/a^8 - 7*b*log(abs(x))/a^8 - 1/60*(420*a*b^6*x^6 + 2310*a^2*b^5*x^5 + 5180*a^3*b^4*x^4 +
5985*a^4*b^3*x^3 + 3654*a^5*b^2*x^2 + 1029*a^6*b*x + 60*a^7)/((b*x + a)^6*a^8*x)