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

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

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Rubi [A]  time = 0.10, antiderivative size = 157, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.09, size = 123, normalized size = 0.78 \begin {gather*} -\frac {\frac {a \left (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\right )}{x^3 (a+b x)^6}-2520 b^3 \log (a+b x)+2520 b^3 \log (x)}{30 a^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x)^7),x]

[Out]

-1/30*((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])/a^10

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^4 (a+b x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^4*(a + b*x)^7),x]

[Out]

IntegrateAlgebraic[1/(x^4*(a + b*x)^7), x]

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fricas [B]  time = 1.29, size = 317, normalized size = 2.02 \begin {gather*} -\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 \relax (x)}{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 {gather*}

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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giac [A]  time = 1.25, size = 130, normalized size = 0.83 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.01, size = 144, normalized size = 0.92 \begin {gather*} -\frac {b^{3}}{6 \left (b x +a \right )^{6} a^{4}}-\frac {4 b^{3}}{5 \left (b x +a \right )^{5} a^{5}}-\frac {5 b^{3}}{2 \left (b x +a \right )^{4} a^{6}}-\frac {20 b^{3}}{3 \left (b x +a \right )^{3} a^{7}}-\frac {35 b^{3}}{2 \left (b x +a \right )^{2} a^{8}}-\frac {56 b^{3}}{\left (b x +a \right ) a^{9}}-\frac {84 b^{3} \ln \relax (x )}{a^{10}}+\frac {84 b^{3} \ln \left (b x +a \right )}{a^{10}}-\frac {28 b^{2}}{a^{9} x}+\frac {7 b}{2 a^{8} x^{2}}-\frac {1}{3 a^{7} x^{3}} \end {gather*}

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 = 1.59, size = 185, normalized size = 1.18 \begin {gather*} -\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 \relax (x)}{a^{10}} \end {gather*}

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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mupad [B]  time = 0.31, size = 179, normalized size = 1.14 \begin {gather*} \frac {168\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{10}}-\frac {\frac {1}{3\,a}+\frac {12\,b^2\,x^2}{a^3}+\frac {1029\,b^3\,x^3}{5\,a^4}+\frac {3654\,b^4\,x^4}{5\,a^5}+\frac {1197\,b^5\,x^5}{a^6}+\frac {1036\,b^6\,x^6}{a^7}+\frac {462\,b^7\,x^7}{a^8}+\frac {84\,b^8\,x^8}{a^9}-\frac {3\,b\,x}{2\,a^2}}{a^6\,x^3+6\,a^5\,b\,x^4+15\,a^4\,b^2\,x^5+20\,a^3\,b^3\,x^6+15\,a^2\,b^4\,x^7+6\,a\,b^5\,x^8+b^6\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x)^7),x)

[Out]

(168*b^3*atanh((2*b*x)/a + 1))/a^10 - (1/(3*a) + (12*b^2*x^2)/a^3 + (1029*b^3*x^3)/(5*a^4) + (3654*b^4*x^4)/(5
*a^5) + (1197*b^5*x^5)/a^6 + (1036*b^6*x^6)/a^7 + (462*b^7*x^7)/a^8 + (84*b^8*x^8)/a^9 - (3*b*x)/(2*a^2))/(a^6
*x^3 + b^6*x^9 + 6*a^5*b*x^4 + 6*a*b^5*x^8 + 15*a^4*b^2*x^5 + 20*a^3*b^3*x^6 + 15*a^2*b^4*x^7)

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sympy [A]  time = 0.99, size = 187, normalized size = 1.19 \begin {gather*} \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 {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{10}} \end {gather*}

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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