3.2.7 \(\int \frac {(a+b x)^7}{x} \, dx\) [107]

Optimal. Leaf size=87 \[ 7 a^6 b x+\frac {21}{2} a^5 b^2 x^2+\frac {35}{3} a^4 b^3 x^3+\frac {35}{4} a^3 b^4 x^4+\frac {21}{5} a^2 b^5 x^5+\frac {7}{6} a b^6 x^6+\frac {b^7 x^7}{7}+a^7 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 87, 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 = {45} \begin {gather*} a^7 \log (x)+7 a^6 b x+\frac {21}{2} a^5 b^2 x^2+\frac {35}{3} a^4 b^3 x^3+\frac {35}{4} a^3 b^4 x^4+\frac {21}{5} a^2 b^5 x^5+\frac {7}{6} a b^6 x^6+\frac {b^7 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x,x]

[Out]

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

Rule 45

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

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Mathematica [A]
time = 0.00, size = 87, normalized size = 1.00 \begin {gather*} 7 a^6 b x+\frac {21}{2} a^5 b^2 x^2+\frac {35}{3} a^4 b^3 x^3+\frac {35}{4} a^3 b^4 x^4+\frac {21}{5} a^2 b^5 x^5+\frac {7}{6} a b^6 x^6+\frac {b^7 x^7}{7}+a^7 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x,x]

[Out]

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

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Mathics [A]
time = 1.94, size = 75, normalized size = 0.86 \begin {gather*} a^7 \text {Log}\left [x\right ]+7 a^6 b x+\frac {21 a^5 b^2 x^2}{2}+\frac {35 a^4 b^3 x^3}{3}+\frac {35 a^3 b^4 x^4}{4}+\frac {21 a^2 b^5 x^5}{5}+\frac {7 a b^6 x^6}{6}+\frac {b^7 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^7/x^1,x]')

[Out]

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

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Maple [A]
time = 0.08, size = 76, normalized size = 0.87

method	result	size
default	\(7 a^{6} b x +\frac {21 a^{5} b^{2} x^{2}}{2}+\frac {35 a^{4} b^{3} x^{3}}{3}+\frac {35 a^{3} b^{4} x^{4}}{4}+\frac {21 a^{2} b^{5} x^{5}}{5}+\frac {7 a \,b^{6} x^{6}}{6}+\frac {b^{7} x^{7}}{7}+a^{7} \ln \left (x \right )\)	\(76\)
norman	\(7 a^{6} b x +\frac {21 a^{5} b^{2} x^{2}}{2}+\frac {35 a^{4} b^{3} x^{3}}{3}+\frac {35 a^{3} b^{4} x^{4}}{4}+\frac {21 a^{2} b^{5} x^{5}}{5}+\frac {7 a \,b^{6} x^{6}}{6}+\frac {b^{7} x^{7}}{7}+a^{7} \ln \left (x \right )\)	\(76\)
risch	\(7 a^{6} b x +\frac {21 a^{5} b^{2} x^{2}}{2}+\frac {35 a^{4} b^{3} x^{3}}{3}+\frac {35 a^{3} b^{4} x^{4}}{4}+\frac {21 a^{2} b^{5} x^{5}}{5}+\frac {7 a \,b^{6} x^{6}}{6}+\frac {b^{7} x^{7}}{7}+a^{7} \ln \left (x \right )\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 75, normalized size = 0.86 \begin {gather*} \frac {1}{7} \, b^{7} x^{7} + \frac {7}{6} \, a b^{6} x^{6} + \frac {21}{5} \, a^{2} b^{5} x^{5} + \frac {35}{4} \, a^{3} b^{4} x^{4} + \frac {35}{3} \, a^{4} b^{3} x^{3} + \frac {21}{2} \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x + a^{7} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.31, size = 75, normalized size = 0.86 \begin {gather*} \frac {1}{7} \, b^{7} x^{7} + \frac {7}{6} \, a b^{6} x^{6} + \frac {21}{5} \, a^{2} b^{5} x^{5} + \frac {35}{4} \, a^{3} b^{4} x^{4} + \frac {35}{3} \, a^{4} b^{3} x^{3} + \frac {21}{2} \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x + a^{7} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.07, size = 88, normalized size = 1.01 \begin {gather*} a^{7} \log {\left (x \right )} + 7 a^{6} b x + \frac {21 a^{5} b^{2} x^{2}}{2} + \frac {35 a^{4} b^{3} x^{3}}{3} + \frac {35 a^{3} b^{4} x^{4}}{4} + \frac {21 a^{2} b^{5} x^{5}}{5} + \frac {7 a b^{6} x^{6}}{6} + \frac {b^{7} x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 88, normalized size = 1.01 \begin {gather*} \frac {1}{7} x^{7} b^{7}+\frac {7}{6} x^{6} b^{6} a+\frac {21}{5} x^{5} b^{5} a^{2}+\frac {35}{4} x^{4} b^{4} a^{3}+\frac {35}{3} x^{3} b^{3} a^{4}+\frac {21}{2} x^{2} b^{2} a^{5}+7 x b a^{6}+a^{7} \ln \left |x\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 75, normalized size = 0.86 \begin {gather*} a^7\,\ln \left (x\right )+\frac {b^7\,x^7}{7}+\frac {7\,a\,b^6\,x^6}{6}+\frac {21\,a^5\,b^2\,x^2}{2}+\frac {35\,a^4\,b^3\,x^3}{3}+\frac {35\,a^3\,b^4\,x^4}{4}+\frac {21\,a^2\,b^5\,x^5}{5}+7\,a^6\,b\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/x,x)

[Out]

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

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