3.111 \(\int \frac{(a+b x)^7}{x^5} \, dx\)

Optimal. Leaf size=86 \[ -\frac{21 a^5 b^2}{2 x^2}-\frac{35 a^4 b^3}{x}+21 a^2 b^5 x+35 a^3 b^4 \log (x)-\frac{7 a^6 b}{3 x^3}-\frac{a^7}{4 x^4}+\frac{7}{2} a b^6 x^2+\frac{b^7 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0320514, antiderivative size = 86, 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} \[ -\frac{21 a^5 b^2}{2 x^2}-\frac{35 a^4 b^3}{x}+21 a^2 b^5 x+35 a^3 b^4 \log (x)-\frac{7 a^6 b}{3 x^3}-\frac{a^7}{4 x^4}+\frac{7}{2} a b^6 x^2+\frac{b^7 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^5,x]

[Out]

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

Mathematica [A]  time = 0.0154708, size = 86, normalized size = 1. \[ -\frac{21 a^5 b^2}{2 x^2}-\frac{35 a^4 b^3}{x}+21 a^2 b^5 x+35 a^3 b^4 \log (x)-\frac{7 a^6 b}{3 x^3}-\frac{a^7}{4 x^4}+\frac{7}{2} a b^6 x^2+\frac{b^7 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^5,x]

[Out]

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

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Maple [A]  time = 0.008, size = 77, normalized size = 0.9 \begin{align*} -{\frac{{a}^{7}}{4\,{x}^{4}}}-{\frac{7\,{a}^{6}b}{3\,{x}^{3}}}-{\frac{21\,{a}^{5}{b}^{2}}{2\,{x}^{2}}}-35\,{\frac{{a}^{4}{b}^{3}}{x}}+21\,{a}^{2}{b}^{5}x+{\frac{7\,a{b}^{6}{x}^{2}}{2}}+{\frac{{b}^{7}{x}^{3}}{3}}+35\,{a}^{3}{b}^{4}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^5,x)

[Out]

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

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Maxima [A]  time = 1.00811, size = 104, normalized size = 1.21 \begin{align*} \frac{1}{3} \, b^{7} x^{3} + \frac{7}{2} \, a b^{6} x^{2} + 21 \, a^{2} b^{5} x + 35 \, a^{3} b^{4} \log \left (x\right ) - \frac{420 \, a^{4} b^{3} x^{3} + 126 \, a^{5} b^{2} x^{2} + 28 \, a^{6} b x + 3 \, a^{7}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^5,x, algorithm="maxima")

[Out]

1/3*b^7*x^3 + 7/2*a*b^6*x^2 + 21*a^2*b^5*x + 35*a^3*b^4*log(x) - 1/12*(420*a^4*b^3*x^3 + 126*a^5*b^2*x^2 + 28*
a^6*b*x + 3*a^7)/x^4

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Fricas [A]  time = 1.76785, size = 185, normalized size = 2.15 \begin{align*} \frac{4 \, b^{7} x^{7} + 42 \, a b^{6} x^{6} + 252 \, a^{2} b^{5} x^{5} + 420 \, a^{3} b^{4} x^{4} \log \left (x\right ) - 420 \, a^{4} b^{3} x^{3} - 126 \, a^{5} b^{2} x^{2} - 28 \, a^{6} b x - 3 \, a^{7}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^5,x, algorithm="fricas")

[Out]

1/12*(4*b^7*x^7 + 42*a*b^6*x^6 + 252*a^2*b^5*x^5 + 420*a^3*b^4*x^4*log(x) - 420*a^4*b^3*x^3 - 126*a^5*b^2*x^2
- 28*a^6*b*x - 3*a^7)/x^4

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Sympy [A]  time = 0.592147, size = 83, normalized size = 0.97 \begin{align*} 35 a^{3} b^{4} \log{\left (x \right )} + 21 a^{2} b^{5} x + \frac{7 a b^{6} x^{2}}{2} + \frac{b^{7} x^{3}}{3} - \frac{3 a^{7} + 28 a^{6} b x + 126 a^{5} b^{2} x^{2} + 420 a^{4} b^{3} x^{3}}{12 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**5,x)

[Out]

35*a**3*b**4*log(x) + 21*a**2*b**5*x + 7*a*b**6*x**2/2 + b**7*x**3/3 - (3*a**7 + 28*a**6*b*x + 126*a**5*b**2*x
**2 + 420*a**4*b**3*x**3)/(12*x**4)

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Giac [A]  time = 1.17055, size = 105, normalized size = 1.22 \begin{align*} \frac{1}{3} \, b^{7} x^{3} + \frac{7}{2} \, a b^{6} x^{2} + 21 \, a^{2} b^{5} x + 35 \, a^{3} b^{4} \log \left ({\left | x \right |}\right ) - \frac{420 \, a^{4} b^{3} x^{3} + 126 \, a^{5} b^{2} x^{2} + 28 \, a^{6} b x + 3 \, a^{7}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^5,x, algorithm="giac")

[Out]

1/3*b^7*x^3 + 7/2*a*b^6*x^2 + 21*a^2*b^5*x + 35*a^3*b^4*log(abs(x)) - 1/12*(420*a^4*b^3*x^3 + 126*a^5*b^2*x^2
+ 28*a^6*b*x + 3*a^7)/x^4