3.221 \(\int \frac{1}{x^4 (a+b x)^7} \, dx\)

Optimal. Leaf size=157 \[ -\frac{56 b^3}{a^9 (a+b x)}-\frac{35 b^3}{2 a^8 (a+b x)^2}-\frac{20 b^3}{3 a^7 (a+b x)^3}-\frac{5 b^3}{2 a^6 (a+b x)^4}-\frac{4 b^3}{5 a^5 (a+b x)^5}-\frac{b^3}{6 a^4 (a+b x)^6}-\frac{28 b^2}{a^9 x}-\frac{84 b^3 \log (x)}{a^{10}}+\frac{84 b^3 \log (a+b x)}{a^{10}}+\frac{7 b}{2 a^8 x^2}-\frac{1}{3 a^7 x^3} \]

[Out]

-1/(3*a^7*x^3) + (7*b)/(2*a^8*x^2) - (28*b^2)/(a^9*x) - b^3/(6*a^4*(a + b*x)^6) - (4*b^3)/(5*a^5*(a + b*x)^5)
- (5*b^3)/(2*a^6*(a + b*x)^4) - (20*b^3)/(3*a^7*(a + b*x)^3) - (35*b^3)/(2*a^8*(a + b*x)^2) - (56*b^3)/(a^9*(a
+ b*x)) - (84*b^3*Log[x])/a^10 + (84*b^3*Log[a + b*x])/a^10

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Rubi [A]  time = 0.102711, antiderivative size = 157, 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} \[ -\frac{56 b^3}{a^9 (a+b x)}-\frac{35 b^3}{2 a^8 (a+b x)^2}-\frac{20 b^3}{3 a^7 (a+b x)^3}-\frac{5 b^3}{2 a^6 (a+b x)^4}-\frac{4 b^3}{5 a^5 (a+b x)^5}-\frac{b^3}{6 a^4 (a+b x)^6}-\frac{28 b^2}{a^9 x}-\frac{84 b^3 \log (x)}{a^{10}}+\frac{84 b^3 \log (a+b x)}{a^{10}}+\frac{7 b}{2 a^8 x^2}-\frac{1}{3 a^7 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x)^7),x]

[Out]

-1/(3*a^7*x^3) + (7*b)/(2*a^8*x^2) - (28*b^2)/(a^9*x) - b^3/(6*a^4*(a + b*x)^6) - (4*b^3)/(5*a^5*(a + b*x)^5)
- (5*b^3)/(2*a^6*(a + b*x)^4) - (20*b^3)/(3*a^7*(a + b*x)^3) - (35*b^3)/(2*a^8*(a + b*x)^2) - (56*b^3)/(a^9*(a
+ b*x)) - (84*b^3*Log[x])/a^10 + (84*b^3*Log[a + b*x])/a^10

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^4 (a+b x)^7} \, dx &=\int \left (\frac{1}{a^7 x^4}-\frac{7 b}{a^8 x^3}+\frac{28 b^2}{a^9 x^2}-\frac{84 b^3}{a^{10} x}+\frac{b^4}{a^4 (a+b x)^7}+\frac{4 b^4}{a^5 (a+b x)^6}+\frac{10 b^4}{a^6 (a+b x)^5}+\frac{20 b^4}{a^7 (a+b x)^4}+\frac{35 b^4}{a^8 (a+b x)^3}+\frac{56 b^4}{a^9 (a+b x)^2}+\frac{84 b^4}{a^{10} (a+b x)}\right ) \, dx\\ &=-\frac{1}{3 a^7 x^3}+\frac{7 b}{2 a^8 x^2}-\frac{28 b^2}{a^9 x}-\frac{b^3}{6 a^4 (a+b x)^6}-\frac{4 b^3}{5 a^5 (a+b x)^5}-\frac{5 b^3}{2 a^6 (a+b x)^4}-\frac{20 b^3}{3 a^7 (a+b x)^3}-\frac{35 b^3}{2 a^8 (a+b x)^2}-\frac{56 b^3}{a^9 (a+b x)}-\frac{84 b^3 \log (x)}{a^{10}}+\frac{84 b^3 \log (a+b x)}{a^{10}}\\ \end{align*}

Mathematica [A]  time = 0.236212, size = 123, normalized size = 0.78 \[ -\frac{\frac{a \left (360 a^6 b^2 x^2+6174 a^5 b^3 x^3+21924 a^4 b^4 x^4+35910 a^3 b^5 x^5+31080 a^2 b^6 x^6-45 a^7 b x+10 a^8+13860 a b^7 x^7+2520 b^8 x^8\right )}{x^3 (a+b x)^6}-2520 b^3 \log (a+b x)+2520 b^3 \log (x)}{30 a^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x)^7),x]

[Out]

-((a*(10*a^8 - 45*a^7*b*x + 360*a^6*b^2*x^2 + 6174*a^5*b^3*x^3 + 21924*a^4*b^4*x^4 + 35910*a^3*b^5*x^5 + 31080
*a^2*b^6*x^6 + 13860*a*b^7*x^7 + 2520*b^8*x^8))/(x^3*(a + b*x)^6) + 2520*b^3*Log[x] - 2520*b^3*Log[a + b*x])/(
30*a^10)

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Maple [A]  time = 0.014, size = 144, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{a}^{7}{x}^{3}}}+{\frac{7\,b}{2\,{a}^{8}{x}^{2}}}-28\,{\frac{{b}^{2}}{{a}^{9}x}}-{\frac{{b}^{3}}{6\,{a}^{4} \left ( bx+a \right ) ^{6}}}-{\frac{4\,{b}^{3}}{5\,{a}^{5} \left ( bx+a \right ) ^{5}}}-{\frac{5\,{b}^{3}}{2\,{a}^{6} \left ( bx+a \right ) ^{4}}}-{\frac{20\,{b}^{3}}{3\,{a}^{7} \left ( bx+a \right ) ^{3}}}-{\frac{35\,{b}^{3}}{2\,{a}^{8} \left ( bx+a \right ) ^{2}}}-56\,{\frac{{b}^{3}}{{a}^{9} \left ( bx+a \right ) }}-84\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{10}}}+84\,{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x+a)^7,x)

[Out]

-1/3/a^7/x^3+7/2*b/a^8/x^2-28*b^2/a^9/x-1/6*b^3/a^4/(b*x+a)^6-4/5*b^3/a^5/(b*x+a)^5-5/2*b^3/a^6/(b*x+a)^4-20/3
*b^3/a^7/(b*x+a)^3-35/2*b^3/a^8/(b*x+a)^2-56*b^3/a^9/(b*x+a)-84*b^3*ln(x)/a^10+84*b^3*ln(b*x+a)/a^10

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Maxima [A]  time = 1.1492, size = 250, normalized size = 1.59 \begin{align*} -\frac{2520 \, b^{8} x^{8} + 13860 \, a b^{7} x^{7} + 31080 \, a^{2} b^{6} x^{6} + 35910 \, a^{3} b^{5} x^{5} + 21924 \, a^{4} b^{4} x^{4} + 6174 \, a^{5} b^{3} x^{3} + 360 \, a^{6} b^{2} x^{2} - 45 \, a^{7} b x + 10 \, a^{8}}{30 \,{\left (a^{9} b^{6} x^{9} + 6 \, a^{10} b^{5} x^{8} + 15 \, a^{11} b^{4} x^{7} + 20 \, a^{12} b^{3} x^{6} + 15 \, a^{13} b^{2} x^{5} + 6 \, a^{14} b x^{4} + a^{15} x^{3}\right )}} + \frac{84 \, b^{3} \log \left (b x + a\right )}{a^{10}} - \frac{84 \, b^{3} \log \left (x\right )}{a^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/30*(2520*b^8*x^8 + 13860*a*b^7*x^7 + 31080*a^2*b^6*x^6 + 35910*a^3*b^5*x^5 + 21924*a^4*b^4*x^4 + 6174*a^5*b
^3*x^3 + 360*a^6*b^2*x^2 - 45*a^7*b*x + 10*a^8)/(a^9*b^6*x^9 + 6*a^10*b^5*x^8 + 15*a^11*b^4*x^7 + 20*a^12*b^3*
x^6 + 15*a^13*b^2*x^5 + 6*a^14*b*x^4 + a^15*x^3) + 84*b^3*log(b*x + a)/a^10 - 84*b^3*log(x)/a^10

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Fricas [B]  time = 1.62415, size = 701, normalized size = 4.46 \begin{align*} -\frac{2520 \, a b^{8} x^{8} + 13860 \, a^{2} b^{7} x^{7} + 31080 \, a^{3} b^{6} x^{6} + 35910 \, a^{4} b^{5} x^{5} + 21924 \, a^{5} b^{4} x^{4} + 6174 \, a^{6} b^{3} x^{3} + 360 \, a^{7} b^{2} x^{2} - 45 \, a^{8} b x + 10 \, a^{9} - 2520 \,{\left (b^{9} x^{9} + 6 \, a b^{8} x^{8} + 15 \, a^{2} b^{7} x^{7} + 20 \, a^{3} b^{6} x^{6} + 15 \, a^{4} b^{5} x^{5} + 6 \, a^{5} b^{4} x^{4} + a^{6} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 2520 \,{\left (b^{9} x^{9} + 6 \, a b^{8} x^{8} + 15 \, a^{2} b^{7} x^{7} + 20 \, a^{3} b^{6} x^{6} + 15 \, a^{4} b^{5} x^{5} + 6 \, a^{5} b^{4} x^{4} + a^{6} b^{3} x^{3}\right )} \log \left (x\right )}{30 \,{\left (a^{10} b^{6} x^{9} + 6 \, a^{11} b^{5} x^{8} + 15 \, a^{12} b^{4} x^{7} + 20 \, a^{13} b^{3} x^{6} + 15 \, a^{14} b^{2} x^{5} + 6 \, a^{15} b x^{4} + a^{16} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/30*(2520*a*b^8*x^8 + 13860*a^2*b^7*x^7 + 31080*a^3*b^6*x^6 + 35910*a^4*b^5*x^5 + 21924*a^5*b^4*x^4 + 6174*a
^6*b^3*x^3 + 360*a^7*b^2*x^2 - 45*a^8*b*x + 10*a^9 - 2520*(b^9*x^9 + 6*a*b^8*x^8 + 15*a^2*b^7*x^7 + 20*a^3*b^6
*x^6 + 15*a^4*b^5*x^5 + 6*a^5*b^4*x^4 + a^6*b^3*x^3)*log(b*x + a) + 2520*(b^9*x^9 + 6*a*b^8*x^8 + 15*a^2*b^7*x
^7 + 20*a^3*b^6*x^6 + 15*a^4*b^5*x^5 + 6*a^5*b^4*x^4 + a^6*b^3*x^3)*log(x))/(a^10*b^6*x^9 + 6*a^11*b^5*x^8 + 1
5*a^12*b^4*x^7 + 20*a^13*b^3*x^6 + 15*a^14*b^2*x^5 + 6*a^15*b*x^4 + a^16*x^3)

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Sympy [A]  time = 1.70848, size = 187, normalized size = 1.19 \begin{align*} - \frac{10 a^{8} - 45 a^{7} b x + 360 a^{6} b^{2} x^{2} + 6174 a^{5} b^{3} x^{3} + 21924 a^{4} b^{4} x^{4} + 35910 a^{3} b^{5} x^{5} + 31080 a^{2} b^{6} x^{6} + 13860 a b^{7} x^{7} + 2520 b^{8} x^{8}}{30 a^{15} x^{3} + 180 a^{14} b x^{4} + 450 a^{13} b^{2} x^{5} + 600 a^{12} b^{3} x^{6} + 450 a^{11} b^{4} x^{7} + 180 a^{10} b^{5} x^{8} + 30 a^{9} b^{6} x^{9}} + \frac{84 b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x+a)**7,x)

[Out]

-(10*a**8 - 45*a**7*b*x + 360*a**6*b**2*x**2 + 6174*a**5*b**3*x**3 + 21924*a**4*b**4*x**4 + 35910*a**3*b**5*x*
*5 + 31080*a**2*b**6*x**6 + 13860*a*b**7*x**7 + 2520*b**8*x**8)/(30*a**15*x**3 + 180*a**14*b*x**4 + 450*a**13*
b**2*x**5 + 600*a**12*b**3*x**6 + 450*a**11*b**4*x**7 + 180*a**10*b**5*x**8 + 30*a**9*b**6*x**9) + 84*b**3*(-l
og(x) + log(a/b + x))/a**10

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Giac [A]  time = 1.19823, size = 176, normalized size = 1.12 \begin{align*} \frac{84 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{10}} - \frac{84 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{10}} - \frac{2520 \, a b^{8} x^{8} + 13860 \, a^{2} b^{7} x^{7} + 31080 \, a^{3} b^{6} x^{6} + 35910 \, a^{4} b^{5} x^{5} + 21924 \, a^{5} b^{4} x^{4} + 6174 \, a^{6} b^{3} x^{3} + 360 \, a^{7} b^{2} x^{2} - 45 \, a^{8} b x + 10 \, a^{9}}{30 \,{\left (b x + a\right )}^{6} a^{10} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^7,x, algorithm="giac")

[Out]

84*b^3*log(abs(b*x + a))/a^10 - 84*b^3*log(abs(x))/a^10 - 1/30*(2520*a*b^8*x^8 + 13860*a^2*b^7*x^7 + 31080*a^3
*b^6*x^6 + 35910*a^4*b^5*x^5 + 21924*a^5*b^4*x^4 + 6174*a^6*b^3*x^3 + 360*a^7*b^2*x^2 - 45*a^8*b*x + 10*a^9)/(
(b*x + a)^6*a^10*x^3)