3.112 \(\int \frac{(a+b x)^7}{x^6} \, dx\)

Optimal. Leaf size=84 \[ -\frac{7 a^5 b^2}{x^3}-\frac{35 a^4 b^3}{2 x^2}-\frac{35 a^3 b^4}{x}+21 a^2 b^5 \log (x)-\frac{7 a^6 b}{4 x^4}-\frac{a^7}{5 x^5}+7 a b^6 x+\frac{b^7 x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.0321007, antiderivative size = 84, 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} \[ -\frac{7 a^5 b^2}{x^3}-\frac{35 a^4 b^3}{2 x^2}-\frac{35 a^3 b^4}{x}+21 a^2 b^5 \log (x)-\frac{7 a^6 b}{4 x^4}-\frac{a^7}{5 x^5}+7 a b^6 x+\frac{b^7 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^6,x]

[Out]

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

Mathematica [A]  time = 0.0084338, size = 84, normalized size = 1. \[ -\frac{7 a^5 b^2}{x^3}-\frac{35 a^4 b^3}{2 x^2}-\frac{35 a^3 b^4}{x}+21 a^2 b^5 \log (x)-\frac{7 a^6 b}{4 x^4}-\frac{a^7}{5 x^5}+7 a b^6 x+\frac{b^7 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^6,x]

[Out]

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

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Maple [A]  time = 0.007, size = 77, normalized size = 0.9 \begin{align*} -{\frac{{a}^{7}}{5\,{x}^{5}}}-{\frac{7\,{a}^{6}b}{4\,{x}^{4}}}-7\,{\frac{{a}^{5}{b}^{2}}{{x}^{3}}}-{\frac{35\,{a}^{4}{b}^{3}}{2\,{x}^{2}}}-35\,{\frac{{a}^{3}{b}^{4}}{x}}+7\,a{b}^{6}x+{\frac{{b}^{7}{x}^{2}}{2}}+21\,{a}^{2}{b}^{5}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^6,x)

[Out]

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

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Maxima [A]  time = 1.08316, size = 104, normalized size = 1.24 \begin{align*} \frac{1}{2} \, b^{7} x^{2} + 7 \, a b^{6} x + 21 \, a^{2} b^{5} \log \left (x\right ) - \frac{700 \, a^{3} b^{4} x^{4} + 350 \, a^{4} b^{3} x^{3} + 140 \, a^{5} b^{2} x^{2} + 35 \, a^{6} b x + 4 \, a^{7}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^6,x, algorithm="maxima")

[Out]

1/2*b^7*x^2 + 7*a*b^6*x + 21*a^2*b^5*log(x) - 1/20*(700*a^3*b^4*x^4 + 350*a^4*b^3*x^3 + 140*a^5*b^2*x^2 + 35*a
^6*b*x + 4*a^7)/x^5

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Fricas [A]  time = 1.81999, size = 188, normalized size = 2.24 \begin{align*} \frac{10 \, b^{7} x^{7} + 140 \, a b^{6} x^{6} + 420 \, a^{2} b^{5} x^{5} \log \left (x\right ) - 700 \, a^{3} b^{4} x^{4} - 350 \, a^{4} b^{3} x^{3} - 140 \, a^{5} b^{2} x^{2} - 35 \, a^{6} b x - 4 \, a^{7}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^6,x, algorithm="fricas")

[Out]

1/20*(10*b^7*x^7 + 140*a*b^6*x^6 + 420*a^2*b^5*x^5*log(x) - 700*a^3*b^4*x^4 - 350*a^4*b^3*x^3 - 140*a^5*b^2*x^
2 - 35*a^6*b*x - 4*a^7)/x^5

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Sympy [A]  time = 0.670363, size = 82, normalized size = 0.98 \begin{align*} 21 a^{2} b^{5} \log{\left (x \right )} + 7 a b^{6} x + \frac{b^{7} x^{2}}{2} - \frac{4 a^{7} + 35 a^{6} b x + 140 a^{5} b^{2} x^{2} + 350 a^{4} b^{3} x^{3} + 700 a^{3} b^{4} x^{4}}{20 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**6,x)

[Out]

21*a**2*b**5*log(x) + 7*a*b**6*x + b**7*x**2/2 - (4*a**7 + 35*a**6*b*x + 140*a**5*b**2*x**2 + 350*a**4*b**3*x*
*3 + 700*a**3*b**4*x**4)/(20*x**5)

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Giac [A]  time = 1.16802, size = 105, normalized size = 1.25 \begin{align*} \frac{1}{2} \, b^{7} x^{2} + 7 \, a b^{6} x + 21 \, a^{2} b^{5} \log \left ({\left | x \right |}\right ) - \frac{700 \, a^{3} b^{4} x^{4} + 350 \, a^{4} b^{3} x^{3} + 140 \, a^{5} b^{2} x^{2} + 35 \, a^{6} b x + 4 \, a^{7}}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^6,x, algorithm="giac")

[Out]

1/2*b^7*x^2 + 7*a*b^6*x + 21*a^2*b^5*log(abs(x)) - 1/20*(700*a^3*b^4*x^4 + 350*a^4*b^3*x^3 + 140*a^5*b^2*x^2 +
35*a^6*b*x + 4*a^7)/x^5