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

Optimal. Leaf size=85 \[ -\frac{21 a^5 b^2}{4 x^4}-\frac{35 a^4 b^3}{3 x^3}-\frac{35 a^3 b^4}{2 x^2}-\frac{21 a^2 b^5}{x}-\frac{7 a^6 b}{5 x^5}-\frac{a^7}{6 x^6}+7 a b^6 \log (x)+b^7 x \]

[Out]

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

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Rubi [A]  time = 0.0313059, antiderivative size = 85, 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} \[ -\frac{21 a^5 b^2}{4 x^4}-\frac{35 a^4 b^3}{3 x^3}-\frac{35 a^3 b^4}{2 x^2}-\frac{21 a^2 b^5}{x}-\frac{7 a^6 b}{5 x^5}-\frac{a^7}{6 x^6}+7 a b^6 \log (x)+b^7 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^7,x]

[Out]

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

Mathematica [A]  time = 0.0094263, size = 85, normalized size = 1. \[ -\frac{21 a^5 b^2}{4 x^4}-\frac{35 a^4 b^3}{3 x^3}-\frac{35 a^3 b^4}{2 x^2}-\frac{21 a^2 b^5}{x}-\frac{7 a^6 b}{5 x^5}-\frac{a^7}{6 x^6}+7 a b^6 \log (x)+b^7 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^7,x]

[Out]

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

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Maple [A]  time = 0.007, size = 76, normalized size = 0.9 \begin{align*} -{\frac{{a}^{7}}{6\,{x}^{6}}}-{\frac{7\,{a}^{6}b}{5\,{x}^{5}}}-{\frac{21\,{a}^{5}{b}^{2}}{4\,{x}^{4}}}-{\frac{35\,{a}^{4}{b}^{3}}{3\,{x}^{3}}}-{\frac{35\,{a}^{3}{b}^{4}}{2\,{x}^{2}}}-21\,{\frac{{a}^{2}{b}^{5}}{x}}+{b}^{7}x+7\,a{b}^{6}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01042, size = 103, normalized size = 1.21 \begin{align*} b^{7} x + 7 \, a b^{6} \log \left (x\right ) - \frac{1260 \, a^{2} b^{5} x^{5} + 1050 \, a^{3} b^{4} x^{4} + 700 \, a^{4} b^{3} x^{3} + 315 \, a^{5} b^{2} x^{2} + 84 \, a^{6} b x + 10 \, a^{7}}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^7*x + 7*a*b^6*log(x) - 1/60*(1260*a^2*b^5*x^5 + 1050*a^3*b^4*x^4 + 700*a^4*b^3*x^3 + 315*a^5*b^2*x^2 + 84*a^
6*b*x + 10*a^7)/x^6

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Fricas [A]  time = 1.70654, size = 192, normalized size = 2.26 \begin{align*} \frac{60 \, b^{7} x^{7} + 420 \, a b^{6} x^{6} \log \left (x\right ) - 1260 \, a^{2} b^{5} x^{5} - 1050 \, a^{3} b^{4} x^{4} - 700 \, a^{4} b^{3} x^{3} - 315 \, a^{5} b^{2} x^{2} - 84 \, a^{6} b x - 10 \, a^{7}}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*b^7*x^7 + 420*a*b^6*x^6*log(x) - 1260*a^2*b^5*x^5 - 1050*a^3*b^4*x^4 - 700*a^4*b^3*x^3 - 315*a^5*b^2*
x^2 - 84*a^6*b*x - 10*a^7)/x^6

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Sympy [A]  time = 0.734266, size = 80, normalized size = 0.94 \begin{align*} 7 a b^{6} \log{\left (x \right )} + b^{7} x - \frac{10 a^{7} + 84 a^{6} b x + 315 a^{5} b^{2} x^{2} + 700 a^{4} b^{3} x^{3} + 1050 a^{3} b^{4} x^{4} + 1260 a^{2} b^{5} x^{5}}{60 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7*a*b**6*log(x) + b**7*x - (10*a**7 + 84*a**6*b*x + 315*a**5*b**2*x**2 + 700*a**4*b**3*x**3 + 1050*a**3*b**4*x
**4 + 1260*a**2*b**5*x**5)/(60*x**6)

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Giac [A]  time = 1.16218, size = 104, normalized size = 1.22 \begin{align*} b^{7} x + 7 \, a b^{6} \log \left ({\left | x \right |}\right ) - \frac{1260 \, a^{2} b^{5} x^{5} + 1050 \, a^{3} b^{4} x^{4} + 700 \, a^{4} b^{3} x^{3} + 315 \, a^{5} b^{2} x^{2} + 84 \, a^{6} b x + 10 \, a^{7}}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^7*x + 7*a*b^6*log(abs(x)) - 1/60*(1260*a^2*b^5*x^5 + 1050*a^3*b^4*x^4 + 700*a^4*b^3*x^3 + 315*a^5*b^2*x^2 +
84*a^6*b*x + 10*a^7)/x^6