3.114 \(\int \frac{(a+b x)^7}{x^8} \, dx\)

Optimal. Leaf size=89 \[ -\frac{21 a^5 b^2}{5 x^5}-\frac{35 a^4 b^3}{4 x^4}-\frac{35 a^3 b^4}{3 x^3}-\frac{21 a^2 b^5}{2 x^2}-\frac{7 a^6 b}{6 x^6}-\frac{a^7}{7 x^7}-\frac{7 a b^6}{x}+b^7 \log (x) \]

[Out]

-a^7/(7*x^7) - (7*a^6*b)/(6*x^6) - (21*a^5*b^2)/(5*x^5) - (35*a^4*b^3)/(4*x^4) - (35*a^3*b^4)/(3*x^3) - (21*a^
2*b^5)/(2*x^2) - (7*a*b^6)/x + b^7*Log[x]

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Rubi [A]  time = 0.0332283, antiderivative size = 89, 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 = {43} \[ -\frac{21 a^5 b^2}{5 x^5}-\frac{35 a^4 b^3}{4 x^4}-\frac{35 a^3 b^4}{3 x^3}-\frac{21 a^2 b^5}{2 x^2}-\frac{7 a^6 b}{6 x^6}-\frac{a^7}{7 x^7}-\frac{7 a b^6}{x}+b^7 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^8,x]

[Out]

-a^7/(7*x^7) - (7*a^6*b)/(6*x^6) - (21*a^5*b^2)/(5*x^5) - (35*a^4*b^3)/(4*x^4) - (35*a^3*b^4)/(3*x^3) - (21*a^
2*b^5)/(2*x^2) - (7*a*b^6)/x + b^7*Log[x]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^7}{x^8} \, dx &=\int \left (\frac{a^7}{x^8}+\frac{7 a^6 b}{x^7}+\frac{21 a^5 b^2}{x^6}+\frac{35 a^4 b^3}{x^5}+\frac{35 a^3 b^4}{x^4}+\frac{21 a^2 b^5}{x^3}+\frac{7 a b^6}{x^2}+\frac{b^7}{x}\right ) \, dx\\ &=-\frac{a^7}{7 x^7}-\frac{7 a^6 b}{6 x^6}-\frac{21 a^5 b^2}{5 x^5}-\frac{35 a^4 b^3}{4 x^4}-\frac{35 a^3 b^4}{3 x^3}-\frac{21 a^2 b^5}{2 x^2}-\frac{7 a b^6}{x}+b^7 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.009578, size = 89, normalized size = 1. \[ -\frac{21 a^5 b^2}{5 x^5}-\frac{35 a^4 b^3}{4 x^4}-\frac{35 a^3 b^4}{3 x^3}-\frac{21 a^2 b^5}{2 x^2}-\frac{7 a^6 b}{6 x^6}-\frac{a^7}{7 x^7}-\frac{7 a b^6}{x}+b^7 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^8,x]

[Out]

-a^7/(7*x^7) - (7*a^6*b)/(6*x^6) - (21*a^5*b^2)/(5*x^5) - (35*a^4*b^3)/(4*x^4) - (35*a^3*b^4)/(3*x^3) - (21*a^
2*b^5)/(2*x^2) - (7*a*b^6)/x + b^7*Log[x]

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Maple [A]  time = 0.008, size = 78, normalized size = 0.9 \begin{align*} -{\frac{{a}^{7}}{7\,{x}^{7}}}-{\frac{7\,{a}^{6}b}{6\,{x}^{6}}}-{\frac{21\,{a}^{5}{b}^{2}}{5\,{x}^{5}}}-{\frac{35\,{a}^{4}{b}^{3}}{4\,{x}^{4}}}-{\frac{35\,{a}^{3}{b}^{4}}{3\,{x}^{3}}}-{\frac{21\,{a}^{2}{b}^{5}}{2\,{x}^{2}}}-7\,{\frac{a{b}^{6}}{x}}+{b}^{7}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^8,x)

[Out]

-1/7*a^7/x^7-7/6*a^6*b/x^6-21/5*a^5*b^2/x^5-35/4*a^4*b^3/x^4-35/3*a^3*b^4/x^3-21/2*a^2*b^5/x^2-7*a*b^6/x+b^7*l
n(x)

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Maxima [A]  time = 1.03378, size = 105, normalized size = 1.18 \begin{align*} b^{7} \log \left (x\right ) - \frac{2940 \, a b^{6} x^{6} + 4410 \, a^{2} b^{5} x^{5} + 4900 \, a^{3} b^{4} x^{4} + 3675 \, a^{4} b^{3} x^{3} + 1764 \, a^{5} b^{2} x^{2} + 490 \, a^{6} b x + 60 \, a^{7}}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^8,x, algorithm="maxima")

[Out]

b^7*log(x) - 1/420*(2940*a*b^6*x^6 + 4410*a^2*b^5*x^5 + 4900*a^3*b^4*x^4 + 3675*a^4*b^3*x^3 + 1764*a^5*b^2*x^2
+ 490*a^6*b*x + 60*a^7)/x^7

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Fricas [A]  time = 1.73739, size = 200, normalized size = 2.25 \begin{align*} \frac{420 \, b^{7} x^{7} \log \left (x\right ) - 2940 \, a b^{6} x^{6} - 4410 \, a^{2} b^{5} x^{5} - 4900 \, a^{3} b^{4} x^{4} - 3675 \, a^{4} b^{3} x^{3} - 1764 \, a^{5} b^{2} x^{2} - 490 \, a^{6} b x - 60 \, a^{7}}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^8,x, algorithm="fricas")

[Out]

1/420*(420*b^7*x^7*log(x) - 2940*a*b^6*x^6 - 4410*a^2*b^5*x^5 - 4900*a^3*b^4*x^4 - 3675*a^4*b^3*x^3 - 1764*a^5
*b^2*x^2 - 490*a^6*b*x - 60*a^7)/x^7

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Sympy [A]  time = 0.78602, size = 82, normalized size = 0.92 \begin{align*} b^{7} \log{\left (x \right )} - \frac{60 a^{7} + 490 a^{6} b x + 1764 a^{5} b^{2} x^{2} + 3675 a^{4} b^{3} x^{3} + 4900 a^{3} b^{4} x^{4} + 4410 a^{2} b^{5} x^{5} + 2940 a b^{6} x^{6}}{420 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**8,x)

[Out]

b**7*log(x) - (60*a**7 + 490*a**6*b*x + 1764*a**5*b**2*x**2 + 3675*a**4*b**3*x**3 + 4900*a**3*b**4*x**4 + 4410
*a**2*b**5*x**5 + 2940*a*b**6*x**6)/(420*x**7)

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Giac [A]  time = 1.18989, size = 107, normalized size = 1.2 \begin{align*} b^{7} \log \left ({\left | x \right |}\right ) - \frac{2940 \, a b^{6} x^{6} + 4410 \, a^{2} b^{5} x^{5} + 4900 \, a^{3} b^{4} x^{4} + 3675 \, a^{4} b^{3} x^{3} + 1764 \, a^{5} b^{2} x^{2} + 490 \, a^{6} b x + 60 \, a^{7}}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^8,x, algorithm="giac")

[Out]

b^7*log(abs(x)) - 1/420*(2940*a*b^6*x^6 + 4410*a^2*b^5*x^5 + 4900*a^3*b^4*x^4 + 3675*a^4*b^3*x^3 + 1764*a^5*b^
2*x^2 + 490*a^6*b*x + 60*a^7)/x^7