3.2.41 \(\int \frac {(a+b x)^{10}}{x^7} \, dx\) [141]

Optimal. Leaf size=119 \[ -\frac {a^{10}}{6 x^6}-\frac {2 a^9 b}{x^5}-\frac {45 a^8 b^2}{4 x^4}-\frac {40 a^7 b^3}{x^3}-\frac {105 a^6 b^4}{x^2}-\frac {252 a^5 b^5}{x}+120 a^3 b^7 x+\frac {45}{2} a^2 b^8 x^2+\frac {10}{3} a b^9 x^3+\frac {b^{10} x^4}{4}+210 a^4 b^6 \log (x) \]

[Out]

-1/6*a^10/x^6-2*a^9*b/x^5-45/4*a^8*b^2/x^4-40*a^7*b^3/x^3-105*a^6*b^4/x^2-252*a^5*b^5/x+120*a^3*b^7*x+45/2*a^2
*b^8*x^2+10/3*a*b^9*x^3+1/4*b^10*x^4+210*a^4*b^6*ln(x)

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Rubi [A]
time = 0.03, antiderivative size = 119, 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*} -\frac {a^{10}}{6 x^6}-\frac {2 a^9 b}{x^5}-\frac {45 a^8 b^2}{4 x^4}-\frac {40 a^7 b^3}{x^3}-\frac {105 a^6 b^4}{x^2}-\frac {252 a^5 b^5}{x}+210 a^4 b^6 \log (x)+120 a^3 b^7 x+\frac {45}{2} a^2 b^8 x^2+\frac {10}{3} a b^9 x^3+\frac {b^{10} x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^7,x]

[Out]

-1/6*a^10/x^6 - (2*a^9*b)/x^5 - (45*a^8*b^2)/(4*x^4) - (40*a^7*b^3)/x^3 - (105*a^6*b^4)/x^2 - (252*a^5*b^5)/x
+ 120*a^3*b^7*x + (45*a^2*b^8*x^2)/2 + (10*a*b^9*x^3)/3 + (b^10*x^4)/4 + 210*a^4*b^6*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)^{10}}{x^7} \, dx &=\int \left (120 a^3 b^7+\frac {a^{10}}{x^7}+\frac {10 a^9 b}{x^6}+\frac {45 a^8 b^2}{x^5}+\frac {120 a^7 b^3}{x^4}+\frac {210 a^6 b^4}{x^3}+\frac {252 a^5 b^5}{x^2}+\frac {210 a^4 b^6}{x}+45 a^2 b^8 x+10 a b^9 x^2+b^{10} x^3\right ) \, dx\\ &=-\frac {a^{10}}{6 x^6}-\frac {2 a^9 b}{x^5}-\frac {45 a^8 b^2}{4 x^4}-\frac {40 a^7 b^3}{x^3}-\frac {105 a^6 b^4}{x^2}-\frac {252 a^5 b^5}{x}+120 a^3 b^7 x+\frac {45}{2} a^2 b^8 x^2+\frac {10}{3} a b^9 x^3+\frac {b^{10} x^4}{4}+210 a^4 b^6 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 119, normalized size = 1.00 \begin {gather*} -\frac {a^{10}}{6 x^6}-\frac {2 a^9 b}{x^5}-\frac {45 a^8 b^2}{4 x^4}-\frac {40 a^7 b^3}{x^3}-\frac {105 a^6 b^4}{x^2}-\frac {252 a^5 b^5}{x}+120 a^3 b^7 x+\frac {45}{2} a^2 b^8 x^2+\frac {10}{3} a b^9 x^3+\frac {b^{10} x^4}{4}+210 a^4 b^6 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^7,x]

[Out]

-1/6*a^10/x^6 - (2*a^9*b)/x^5 - (45*a^8*b^2)/(4*x^4) - (40*a^7*b^3)/x^3 - (105*a^6*b^4)/x^2 - (252*a^5*b^5)/x
+ 120*a^3*b^7*x + (45*a^2*b^8*x^2)/2 + (10*a*b^9*x^3)/3 + (b^10*x^4)/4 + 210*a^4*b^6*Log[x]

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Mathics [A]
time = 2.58, size = 113, normalized size = 0.95 \begin {gather*} \frac {-a^5 \left (2 a^5+24 a^4 b x+135 a^3 b^2 x^2+480 a^2 b^3 x^3+1260 a b^4 x^4+3024 b^5 x^5\right )+b^6 x^6 \left (2520 a^4 \text {Log}\left [x\right ]+1440 a^3 b x+270 a^2 b^2 x^2+40 a b^3 x^3+3 b^4 x^4\right )}{12 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^10/x^7,x]')

[Out]

(-a ^ 5 (2 a ^ 5 + 24 a ^ 4 b x + 135 a ^ 3 b ^ 2 x ^ 2 + 480 a ^ 2 b ^ 3 x ^ 3 + 1260 a b ^ 4 x ^ 4 + 3024 b
^ 5 x ^ 5) + b ^ 6 x ^ 6 (2520 a ^ 4 Log[x] + 1440 a ^ 3 b x + 270 a ^ 2 b ^ 2 x ^ 2 + 40 a b ^ 3 x ^ 3 + 3 b
^ 4 x ^ 4)) / (12 x ^ 6)

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Maple [A]
time = 0.08, size = 110, normalized size = 0.92

method	result	size
default	\(-\frac {a^{10}}{6 x^{6}}-\frac {2 a^{9} b}{x^{5}}-\frac {45 a^{8} b^{2}}{4 x^{4}}-\frac {40 a^{7} b^{3}}{x^{3}}-\frac {105 a^{6} b^{4}}{x^{2}}-\frac {252 a^{5} b^{5}}{x}+120 a^{3} b^{7} x +\frac {45 a^{2} b^{8} x^{2}}{2}+\frac {10 a \,b^{9} x^{3}}{3}+\frac {b^{10} x^{4}}{4}+210 a^{4} b^{6} \ln \left (x \right )\)	\(110\)
risch	\(\frac {b^{10} x^{4}}{4}+\frac {10 a \,b^{9} x^{3}}{3}+\frac {45 a^{2} b^{8} x^{2}}{2}+120 a^{3} b^{7} x +\frac {-252 a^{5} b^{5} x^{5}-105 a^{6} b^{4} x^{4}-40 a^{7} b^{3} x^{3}-\frac {45}{4} a^{8} b^{2} x^{2}-2 a^{9} b x -\frac {1}{6} a^{10}}{x^{6}}+210 a^{4} b^{6} \ln \left (x \right )\)	\(110\)
norman	\(\frac {-\frac {1}{6} a^{10}+\frac {1}{4} b^{10} x^{10}+\frac {10}{3} a \,b^{9} x^{9}+\frac {45}{2} a^{2} b^{8} x^{8}+120 a^{3} b^{7} x^{7}-252 a^{5} b^{5} x^{5}-105 a^{6} b^{4} x^{4}-40 a^{7} b^{3} x^{3}-\frac {45}{4} a^{8} b^{2} x^{2}-2 a^{9} b x}{x^{6}}+210 a^{4} b^{6} \ln \left (x \right )\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*a^10/x^6-2*a^9*b/x^5-45/4*a^8*b^2/x^4-40*a^7*b^3/x^3-105*a^6*b^4/x^2-252*a^5*b^5/x+120*a^3*b^7*x+45/2*a^2
*b^8*x^2+10/3*a*b^9*x^3+1/4*b^10*x^4+210*a^4*b^6*ln(x)

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Maxima [A]
time = 0.25, size = 110, normalized size = 0.92 \begin {gather*} \frac {1}{4} \, b^{10} x^{4} + \frac {10}{3} \, a b^{9} x^{3} + \frac {45}{2} \, a^{2} b^{8} x^{2} + 120 \, a^{3} b^{7} x + 210 \, a^{4} b^{6} \log \left (x\right ) - \frac {3024 \, a^{5} b^{5} x^{5} + 1260 \, a^{6} b^{4} x^{4} + 480 \, a^{7} b^{3} x^{3} + 135 \, a^{8} b^{2} x^{2} + 24 \, a^{9} b x + 2 \, a^{10}}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^7,x, algorithm="maxima")

[Out]

1/4*b^10*x^4 + 10/3*a*b^9*x^3 + 45/2*a^2*b^8*x^2 + 120*a^3*b^7*x + 210*a^4*b^6*log(x) - 1/12*(3024*a^5*b^5*x^5
+ 1260*a^6*b^4*x^4 + 480*a^7*b^3*x^3 + 135*a^8*b^2*x^2 + 24*a^9*b*x + 2*a^10)/x^6

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Fricas [A]
time = 0.30, size = 114, normalized size = 0.96 \begin {gather*} \frac {3 \, b^{10} x^{10} + 40 \, a b^{9} x^{9} + 270 \, a^{2} b^{8} x^{8} + 1440 \, a^{3} b^{7} x^{7} + 2520 \, a^{4} b^{6} x^{6} \log \left (x\right ) - 3024 \, a^{5} b^{5} x^{5} - 1260 \, a^{6} b^{4} x^{4} - 480 \, a^{7} b^{3} x^{3} - 135 \, a^{8} b^{2} x^{2} - 24 \, a^{9} b x - 2 \, a^{10}}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^7,x, algorithm="fricas")

[Out]

1/12*(3*b^10*x^10 + 40*a*b^9*x^9 + 270*a^2*b^8*x^8 + 1440*a^3*b^7*x^7 + 2520*a^4*b^6*x^6*log(x) - 3024*a^5*b^5
*x^5 - 1260*a^6*b^4*x^4 - 480*a^7*b^3*x^3 - 135*a^8*b^2*x^2 - 24*a^9*b*x - 2*a^10)/x^6

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Sympy [A]
time = 0.24, size = 122, normalized size = 1.03 \begin {gather*} 210 a^{4} b^{6} \log {\left (x \right )} + 120 a^{3} b^{7} x + \frac {45 a^{2} b^{8} x^{2}}{2} + \frac {10 a b^{9} x^{3}}{3} + \frac {b^{10} x^{4}}{4} + \frac {- 2 a^{10} - 24 a^{9} b x - 135 a^{8} b^{2} x^{2} - 480 a^{7} b^{3} x^{3} - 1260 a^{6} b^{4} x^{4} - 3024 a^{5} b^{5} x^{5}}{12 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**7,x)

[Out]

210*a**4*b**6*log(x) + 120*a**3*b**7*x + 45*a**2*b**8*x**2/2 + 10*a*b**9*x**3/3 + b**10*x**4/4 + (-2*a**10 - 2
4*a**9*b*x - 135*a**8*b**2*x**2 - 480*a**7*b**3*x**3 - 1260*a**6*b**4*x**4 - 3024*a**5*b**5*x**5)/(12*x**6)

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Giac [A]
time = 0.00, size = 126, normalized size = 1.06 \begin {gather*} \frac {1}{4} x^{4} b^{10}+\frac {10}{3} x^{3} b^{9} a+\frac {45}{2} x^{2} b^{8} a^{2}+120 x b^{7} a^{3}+\frac {\frac {1}{12} \left (-3024 b^{5} a^{5} x^{5}-1260 b^{4} a^{6} x^{4}-480 b^{3} a^{7} x^{3}-135 b^{2} a^{8} x^{2}-24 b a^{9} x-2 a^{10}\right )}{x^{6}}+210 b^{6} a^{4} \ln \left |x\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^7,x)

[Out]

1/4*b^10*x^4 + 10/3*a*b^9*x^3 + 45/2*a^2*b^8*x^2 + 120*a^3*b^7*x + 210*a^4*b^6*log(abs(x)) - 1/12*(3024*a^5*b^
5*x^5 + 1260*a^6*b^4*x^4 + 480*a^7*b^3*x^3 + 135*a^8*b^2*x^2 + 24*a^9*b*x + 2*a^10)/x^6

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Mupad [B]
time = 0.05, size = 110, normalized size = 0.92 \begin {gather*} \frac {b^{10}\,x^4}{4}-\frac {\frac {a^{10}}{6}+2\,a^9\,b\,x+\frac {45\,a^8\,b^2\,x^2}{4}+40\,a^7\,b^3\,x^3+105\,a^6\,b^4\,x^4+252\,a^5\,b^5\,x^5}{x^6}+120\,a^3\,b^7\,x+\frac {10\,a\,b^9\,x^3}{3}+\frac {45\,a^2\,b^8\,x^2}{2}+210\,a^4\,b^6\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^10/x^7,x)

[Out]

(b^10*x^4)/4 - (a^10/6 + (45*a^8*b^2*x^2)/4 + 40*a^7*b^3*x^3 + 105*a^6*b^4*x^4 + 252*a^5*b^5*x^5 + 2*a^9*b*x)/
x^6 + 120*a^3*b^7*x + (10*a*b^9*x^3)/3 + (45*a^2*b^8*x^2)/2 + 210*a^4*b^6*log(x)

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