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

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

[Out]

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

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Rubi [A]  time = 0.0327259, 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} \[ \frac{35}{2} a^3 b^4 x^2+7 a^2 b^5 x^3+35 a^4 b^3 x+21 a^5 b^2 \log (x)-\frac{7 a^6 b}{x}-\frac{a^7}{2 x^2}+\frac{7}{4} a b^6 x^4+\frac{b^7 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0060226, size = 84, normalized size = 1. \[ \frac{35}{2} a^3 b^4 x^2+7 a^2 b^5 x^3+35 a^4 b^3 x+21 a^5 b^2 \log (x)-\frac{7 a^6 b}{x}-\frac{a^7}{2 x^2}+\frac{7}{4} a b^6 x^4+\frac{b^7 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^3,x]

[Out]

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

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Maple [A]  time = 0.004, size = 77, normalized size = 0.9 \begin{align*} -{\frac{{a}^{7}}{2\,{x}^{2}}}-7\,{\frac{{a}^{6}b}{x}}+35\,{a}^{4}{b}^{3}x+{\frac{35\,{a}^{3}{b}^{4}{x}^{2}}{2}}+7\,{a}^{2}{b}^{5}{x}^{3}+{\frac{7\,a{b}^{6}{x}^{4}}{4}}+{\frac{{b}^{7}{x}^{5}}{5}}+21\,{a}^{5}{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02658, size = 101, normalized size = 1.2 \begin{align*} \frac{1}{5} \, b^{7} x^{5} + \frac{7}{4} \, a b^{6} x^{4} + 7 \, a^{2} b^{5} x^{3} + \frac{35}{2} \, a^{3} b^{4} x^{2} + 35 \, a^{4} b^{3} x + 21 \, a^{5} b^{2} \log \left (x\right ) - \frac{14 \, a^{6} b x + a^{7}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^7*x^5 + 7/4*a*b^6*x^4 + 7*a^2*b^5*x^3 + 35/2*a^3*b^4*x^2 + 35*a^4*b^3*x + 21*a^5*b^2*log(x) - 1/2*(14*a^
6*b*x + a^7)/x^2

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Fricas [A]  time = 1.80903, size = 188, normalized size = 2.24 \begin{align*} \frac{4 \, b^{7} x^{7} + 35 \, a b^{6} x^{6} + 140 \, a^{2} b^{5} x^{5} + 350 \, a^{3} b^{4} x^{4} + 700 \, a^{4} b^{3} x^{3} + 420 \, a^{5} b^{2} x^{2} \log \left (x\right ) - 140 \, a^{6} b x - 10 \, a^{7}}{20 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.448175, size = 83, normalized size = 0.99 \begin{align*} 21 a^{5} b^{2} \log{\left (x \right )} + 35 a^{4} b^{3} x + \frac{35 a^{3} b^{4} x^{2}}{2} + 7 a^{2} b^{5} x^{3} + \frac{7 a b^{6} x^{4}}{4} + \frac{b^{7} x^{5}}{5} - \frac{a^{7} + 14 a^{6} b x}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21*a**5*b**2*log(x) + 35*a**4*b**3*x + 35*a**3*b**4*x**2/2 + 7*a**2*b**5*x**3 + 7*a*b**6*x**4/4 + b**7*x**5/5
- (a**7 + 14*a**6*b*x)/(2*x**2)

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Giac [A]  time = 1.16313, size = 103, normalized size = 1.23 \begin{align*} \frac{1}{5} \, b^{7} x^{5} + \frac{7}{4} \, a b^{6} x^{4} + 7 \, a^{2} b^{5} x^{3} + \frac{35}{2} \, a^{3} b^{4} x^{2} + 35 \, a^{4} b^{3} x + 21 \, a^{5} b^{2} \log \left ({\left | x \right |}\right ) - \frac{14 \, a^{6} b x + a^{7}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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