3.3.20 \(\int \frac {1}{x^3 (a+b x)^7} \, dx\)

Optimal. Leaf size=144 \[ \frac {28 b^2 \log (x)}{a^9}-\frac {28 b^2 \log (a+b x)}{a^9}+\frac {21 b^2}{a^8 (a+b x)}+\frac {7 b}{a^8 x}+\frac {15 b^2}{2 a^7 (a+b x)^2}-\frac {1}{2 a^7 x^2}+\frac {10 b^2}{3 a^6 (a+b x)^3}+\frac {3 b^2}{2 a^5 (a+b x)^4}+\frac {3 b^2}{5 a^4 (a+b x)^5}+\frac {b^2}{6 a^3 (a+b x)^6} \]

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Rubi [A]  time = 0.09, antiderivative size = 144, 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 {21 b^2}{a^8 (a+b x)}+\frac {15 b^2}{2 a^7 (a+b x)^2}+\frac {10 b^2}{3 a^6 (a+b x)^3}+\frac {3 b^2}{2 a^5 (a+b x)^4}+\frac {3 b^2}{5 a^4 (a+b x)^5}+\frac {b^2}{6 a^3 (a+b x)^6}+\frac {28 b^2 \log (x)}{a^9}-\frac {28 b^2 \log (a+b x)}{a^9}+\frac {7 b}{a^8 x}-\frac {1}{2 a^7 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x)^7),x]

[Out]

-1/(2*a^7*x^2) + (7*b)/(a^8*x) + b^2/(6*a^3*(a + b*x)^6) + (3*b^2)/(5*a^4*(a + b*x)^5) + (3*b^2)/(2*a^5*(a + b
*x)^4) + (10*b^2)/(3*a^6*(a + b*x)^3) + (15*b^2)/(2*a^7*(a + b*x)^2) + (21*b^2)/(a^8*(a + b*x)) + (28*b^2*Log[
x])/a^9 - (28*b^2*Log[a + b*x])/a^9

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^3 (a+b x)^7} \, dx &=\int \left (\frac {1}{a^7 x^3}-\frac {7 b}{a^8 x^2}+\frac {28 b^2}{a^9 x}-\frac {b^3}{a^3 (a+b x)^7}-\frac {3 b^3}{a^4 (a+b x)^6}-\frac {6 b^3}{a^5 (a+b x)^5}-\frac {10 b^3}{a^6 (a+b x)^4}-\frac {15 b^3}{a^7 (a+b x)^3}-\frac {21 b^3}{a^8 (a+b x)^2}-\frac {28 b^3}{a^9 (a+b x)}\right ) \, dx\\ &=-\frac {1}{2 a^7 x^2}+\frac {7 b}{a^8 x}+\frac {b^2}{6 a^3 (a+b x)^6}+\frac {3 b^2}{5 a^4 (a+b x)^5}+\frac {3 b^2}{2 a^5 (a+b x)^4}+\frac {10 b^2}{3 a^6 (a+b x)^3}+\frac {15 b^2}{2 a^7 (a+b x)^2}+\frac {21 b^2}{a^8 (a+b x)}+\frac {28 b^2 \log (x)}{a^9}-\frac {28 b^2 \log (a+b x)}{a^9}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 112, normalized size = 0.78 \begin {gather*} \frac {\frac {a \left (-15 a^7+120 a^6 b x+2058 a^5 b^2 x^2+7308 a^4 b^3 x^3+11970 a^3 b^4 x^4+10360 a^2 b^5 x^5+4620 a b^6 x^6+840 b^7 x^7\right )}{x^2 (a+b x)^6}-840 b^2 \log (a+b x)+840 b^2 \log (x)}{30 a^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x)^7),x]

[Out]

((a*(-15*a^7 + 120*a^6*b*x + 2058*a^5*b^2*x^2 + 7308*a^4*b^3*x^3 + 11970*a^3*b^4*x^4 + 10360*a^2*b^5*x^5 + 462
0*a*b^6*x^6 + 840*b^7*x^7))/(x^2*(a + b*x)^6) + 840*b^2*Log[x] - 840*b^2*Log[a + b*x])/(30*a^9)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 (a+b x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^3*(a + b*x)^7),x]

[Out]

IntegrateAlgebraic[1/(x^3*(a + b*x)^7), x]

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fricas [B]  time = 1.33, size = 306, normalized size = 2.12 \begin {gather*} \frac {840 \, a b^{7} x^{7} + 4620 \, a^{2} b^{6} x^{6} + 10360 \, a^{3} b^{5} x^{5} + 11970 \, a^{4} b^{4} x^{4} + 7308 \, a^{5} b^{3} x^{3} + 2058 \, a^{6} b^{2} x^{2} + 120 \, a^{7} b x - 15 \, a^{8} - 840 \, {\left (b^{8} x^{8} + 6 \, a b^{7} x^{7} + 15 \, a^{2} b^{6} x^{6} + 20 \, a^{3} b^{5} x^{5} + 15 \, a^{4} b^{4} x^{4} + 6 \, a^{5} b^{3} x^{3} + a^{6} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 840 \, {\left (b^{8} x^{8} + 6 \, a b^{7} x^{7} + 15 \, a^{2} b^{6} x^{6} + 20 \, a^{3} b^{5} x^{5} + 15 \, a^{4} b^{4} x^{4} + 6 \, a^{5} b^{3} x^{3} + a^{6} b^{2} x^{2}\right )} \log \relax (x)}{30 \, {\left (a^{9} b^{6} x^{8} + 6 \, a^{10} b^{5} x^{7} + 15 \, a^{11} b^{4} x^{6} + 20 \, a^{12} b^{3} x^{5} + 15 \, a^{13} b^{2} x^{4} + 6 \, a^{14} b x^{3} + a^{15} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x+a)^7,x, algorithm="fricas")

[Out]

1/30*(840*a*b^7*x^7 + 4620*a^2*b^6*x^6 + 10360*a^3*b^5*x^5 + 11970*a^4*b^4*x^4 + 7308*a^5*b^3*x^3 + 2058*a^6*b
^2*x^2 + 120*a^7*b*x - 15*a^8 - 840*(b^8*x^8 + 6*a*b^7*x^7 + 15*a^2*b^6*x^6 + 20*a^3*b^5*x^5 + 15*a^4*b^4*x^4
+ 6*a^5*b^3*x^3 + a^6*b^2*x^2)*log(b*x + a) + 840*(b^8*x^8 + 6*a*b^7*x^7 + 15*a^2*b^6*x^6 + 20*a^3*b^5*x^5 + 1
5*a^4*b^4*x^4 + 6*a^5*b^3*x^3 + a^6*b^2*x^2)*log(x))/(a^9*b^6*x^8 + 6*a^10*b^5*x^7 + 15*a^11*b^4*x^6 + 20*a^12
*b^3*x^5 + 15*a^13*b^2*x^4 + 6*a^14*b*x^3 + a^15*x^2)

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giac [A]  time = 1.27, size = 119, normalized size = 0.83 \begin {gather*} -\frac {28 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{9}} + \frac {28 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{9}} + \frac {840 \, a b^{7} x^{7} + 4620 \, a^{2} b^{6} x^{6} + 10360 \, a^{3} b^{5} x^{5} + 11970 \, a^{4} b^{4} x^{4} + 7308 \, a^{5} b^{3} x^{3} + 2058 \, a^{6} b^{2} x^{2} + 120 \, a^{7} b x - 15 \, a^{8}}{30 \, {\left (b x + a\right )}^{6} a^{9} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x+a)^7,x, algorithm="giac")

[Out]

-28*b^2*log(abs(b*x + a))/a^9 + 28*b^2*log(abs(x))/a^9 + 1/30*(840*a*b^7*x^7 + 4620*a^2*b^6*x^6 + 10360*a^3*b^
5*x^5 + 11970*a^4*b^4*x^4 + 7308*a^5*b^3*x^3 + 2058*a^6*b^2*x^2 + 120*a^7*b*x - 15*a^8)/((b*x + a)^6*a^9*x^2)

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maple [A]  time = 0.01, size = 133, normalized size = 0.92 \begin {gather*} \frac {b^{2}}{6 \left (b x +a \right )^{6} a^{3}}+\frac {3 b^{2}}{5 \left (b x +a \right )^{5} a^{4}}+\frac {3 b^{2}}{2 \left (b x +a \right )^{4} a^{5}}+\frac {10 b^{2}}{3 \left (b x +a \right )^{3} a^{6}}+\frac {15 b^{2}}{2 \left (b x +a \right )^{2} a^{7}}+\frac {21 b^{2}}{\left (b x +a \right ) a^{8}}+\frac {28 b^{2} \ln \relax (x )}{a^{9}}-\frac {28 b^{2} \ln \left (b x +a \right )}{a^{9}}+\frac {7 b}{a^{8} x}-\frac {1}{2 a^{7} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x+a)^7,x)

[Out]

-1/2/a^7/x^2+7*b/a^8/x+1/6*b^2/a^3/(b*x+a)^6+3/5*b^2/a^4/(b*x+a)^5+3/2*b^2/a^5/(b*x+a)^4+10/3*b^2/a^6/(b*x+a)^
3+15/2*b^2/a^7/(b*x+a)^2+21*b^2/a^8/(b*x+a)+28*b^2*ln(x)/a^9-28*b^2*ln(b*x+a)/a^9

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maxima [A]  time = 1.55, size = 174, normalized size = 1.21 \begin {gather*} \frac {840 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 10360 \, a^{2} b^{5} x^{5} + 11970 \, a^{3} b^{4} x^{4} + 7308 \, a^{4} b^{3} x^{3} + 2058 \, a^{5} b^{2} x^{2} + 120 \, a^{6} b x - 15 \, a^{7}}{30 \, {\left (a^{8} b^{6} x^{8} + 6 \, a^{9} b^{5} x^{7} + 15 \, a^{10} b^{4} x^{6} + 20 \, a^{11} b^{3} x^{5} + 15 \, a^{12} b^{2} x^{4} + 6 \, a^{13} b x^{3} + a^{14} x^{2}\right )}} - \frac {28 \, b^{2} \log \left (b x + a\right )}{a^{9}} + \frac {28 \, b^{2} \log \relax (x)}{a^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x+a)^7,x, algorithm="maxima")

[Out]

1/30*(840*b^7*x^7 + 4620*a*b^6*x^6 + 10360*a^2*b^5*x^5 + 11970*a^3*b^4*x^4 + 7308*a^4*b^3*x^3 + 2058*a^5*b^2*x
^2 + 120*a^6*b*x - 15*a^7)/(a^8*b^6*x^8 + 6*a^9*b^5*x^7 + 15*a^10*b^4*x^6 + 20*a^11*b^3*x^5 + 15*a^12*b^2*x^4
+ 6*a^13*b*x^3 + a^14*x^2) - 28*b^2*log(b*x + a)/a^9 + 28*b^2*log(x)/a^9

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mupad [B]  time = 0.21, size = 167, normalized size = 1.16 \begin {gather*} \frac {\frac {343\,b^2\,x^2}{5\,a^3}-\frac {1}{2\,a}+\frac {1218\,b^3\,x^3}{5\,a^4}+\frac {399\,b^4\,x^4}{a^5}+\frac {1036\,b^5\,x^5}{3\,a^6}+\frac {154\,b^6\,x^6}{a^7}+\frac {28\,b^7\,x^7}{a^8}+\frac {4\,b\,x}{a^2}}{a^6\,x^2+6\,a^5\,b\,x^3+15\,a^4\,b^2\,x^4+20\,a^3\,b^3\,x^5+15\,a^2\,b^4\,x^6+6\,a\,b^5\,x^7+b^6\,x^8}-\frac {56\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x)^7),x)

[Out]

((343*b^2*x^2)/(5*a^3) - 1/(2*a) + (1218*b^3*x^3)/(5*a^4) + (399*b^4*x^4)/a^5 + (1036*b^5*x^5)/(3*a^6) + (154*
b^6*x^6)/a^7 + (28*b^7*x^7)/a^8 + (4*b*x)/a^2)/(a^6*x^2 + b^6*x^8 + 6*a^5*b*x^3 + 6*a*b^5*x^7 + 15*a^4*b^2*x^4
+ 20*a^3*b^3*x^5 + 15*a^2*b^4*x^6) - (56*b^2*atanh((2*b*x)/a + 1))/a^9

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sympy [A]  time = 0.84, size = 175, normalized size = 1.22 \begin {gather*} \frac {- 15 a^{7} + 120 a^{6} b x + 2058 a^{5} b^{2} x^{2} + 7308 a^{4} b^{3} x^{3} + 11970 a^{3} b^{4} x^{4} + 10360 a^{2} b^{5} x^{5} + 4620 a b^{6} x^{6} + 840 b^{7} x^{7}}{30 a^{14} x^{2} + 180 a^{13} b x^{3} + 450 a^{12} b^{2} x^{4} + 600 a^{11} b^{3} x^{5} + 450 a^{10} b^{4} x^{6} + 180 a^{9} b^{5} x^{7} + 30 a^{8} b^{6} x^{8}} + \frac {28 b^{2} \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b*x+a)**7,x)

[Out]

(-15*a**7 + 120*a**6*b*x + 2058*a**5*b**2*x**2 + 7308*a**4*b**3*x**3 + 11970*a**3*b**4*x**4 + 10360*a**2*b**5*
x**5 + 4620*a*b**6*x**6 + 840*b**7*x**7)/(30*a**14*x**2 + 180*a**13*b*x**3 + 450*a**12*b**2*x**4 + 600*a**11*b
**3*x**5 + 450*a**10*b**4*x**6 + 180*a**9*b**5*x**7 + 30*a**8*b**6*x**8) + 28*b**2*(log(x) - log(a/b + x))/a**
9

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