3.108 \(\int \frac {(a+b x)^7}{x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, 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} \[ \frac {35}{2} a^4 b^3 x^2+\frac {35}{3} a^3 b^4 x^3+\frac {21}{4} a^2 b^5 x^4+21 a^5 b^2 x+7 a^6 b \log (x)-\frac {a^7}{x}+\frac {7}{5} a b^6 x^5+\frac {b^7 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^2,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 86, normalized size = 1.00 \[ -\frac {a^7}{x}+7 a^6 b \log (x)+21 a^5 b^2 x+\frac {35}{2} a^4 b^3 x^2+\frac {35}{3} a^3 b^4 x^3+\frac {21}{4} a^2 b^5 x^4+\frac {7}{5} a b^6 x^5+\frac {b^7 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^2,x]

[Out]

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

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fricas [A]  time = 0.44, size = 81, normalized size = 0.94 \[ \frac {10 \, b^{7} x^{7} + 84 \, a b^{6} x^{6} + 315 \, a^{2} b^{5} x^{5} + 700 \, a^{3} b^{4} x^{4} + 1050 \, a^{4} b^{3} x^{3} + 1260 \, a^{5} b^{2} x^{2} + 420 \, a^{6} b x \log \relax (x) - 60 \, a^{7}}{60 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^2,x, algorithm="fricas")

[Out]

1/60*(10*b^7*x^7 + 84*a*b^6*x^6 + 315*a^2*b^5*x^5 + 700*a^3*b^4*x^4 + 1050*a^4*b^3*x^3 + 1260*a^5*b^2*x^2 + 42
0*a^6*b*x*log(x) - 60*a^7)/x

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giac [A]  time = 1.12, size = 77, normalized size = 0.90 \[ \frac {1}{6} \, b^{7} x^{6} + \frac {7}{5} \, a b^{6} x^{5} + \frac {21}{4} \, a^{2} b^{5} x^{4} + \frac {35}{3} \, a^{3} b^{4} x^{3} + \frac {35}{2} \, a^{4} b^{3} x^{2} + 21 \, a^{5} b^{2} x + 7 \, a^{6} b \log \left ({\left | x \right |}\right ) - \frac {a^{7}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 77, normalized size = 0.90 \[ \frac {b^{7} x^{6}}{6}+\frac {7 a \,b^{6} x^{5}}{5}+\frac {21 a^{2} b^{5} x^{4}}{4}+\frac {35 a^{3} b^{4} x^{3}}{3}+\frac {35 a^{4} b^{3} x^{2}}{2}+7 a^{6} b \ln \relax (x )+21 a^{5} b^{2} x -\frac {a^{7}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^2,x)

[Out]

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

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maxima [A]  time = 1.36, size = 76, normalized size = 0.88 \[ \frac {1}{6} \, b^{7} x^{6} + \frac {7}{5} \, a b^{6} x^{5} + \frac {21}{4} \, a^{2} b^{5} x^{4} + \frac {35}{3} \, a^{3} b^{4} x^{3} + \frac {35}{2} \, a^{4} b^{3} x^{2} + 21 \, a^{5} b^{2} x + 7 \, a^{6} b \log \relax (x) - \frac {a^{7}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 76, normalized size = 0.88 \[ \frac {b^7\,x^6}{6}-\frac {a^7}{x}+21\,a^5\,b^2\,x+\frac {7\,a\,b^6\,x^5}{5}+7\,a^6\,b\,\ln \relax (x)+\frac {35\,a^4\,b^3\,x^2}{2}+\frac {35\,a^3\,b^4\,x^3}{3}+\frac {21\,a^2\,b^5\,x^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/x^2,x)

[Out]

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

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sympy [A]  time = 0.20, size = 85, normalized size = 0.99 \[ - \frac {a^{7}}{x} + 7 a^{6} b \log {\relax (x )} + 21 a^{5} b^{2} x + \frac {35 a^{4} b^{3} x^{2}}{2} + \frac {35 a^{3} b^{4} x^{3}}{3} + \frac {21 a^{2} b^{5} x^{4}}{4} + \frac {7 a b^{6} x^{5}}{5} + \frac {b^{7} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**2,x)

[Out]

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

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