3.2.38 \(\int \frac {(a+b x)^{10}}{x^4} \, dx\)

Optimal. Leaf size=115 \[ -\frac {a^{10}}{3 x^3}-\frac {5 a^9 b}{x^2}-\frac {45 a^8 b^2}{x}+120 a^7 b^3 \log (x)+210 a^6 b^4 x+126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5+\frac {5}{3} a b^9 x^6+\frac {b^{10} x^7}{7} \]

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Rubi [A]  time = 0.05, antiderivative size = 115, 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} \begin {gather*} 126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5-\frac {45 a^8 b^2}{x}+210 a^6 b^4 x+120 a^7 b^3 \log (x)-\frac {5 a^9 b}{x^2}-\frac {a^{10}}{3 x^3}+\frac {5}{3} a b^9 x^6+\frac {b^{10} x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^4,x]

[Out]

-a^10/(3*x^3) - (5*a^9*b)/x^2 - (45*a^8*b^2)/x + 210*a^6*b^4*x + 126*a^5*b^5*x^2 + 70*a^4*b^6*x^3 + 30*a^3*b^7
*x^4 + 9*a^2*b^8*x^5 + (5*a*b^9*x^6)/3 + (b^10*x^7)/7 + 120*a^7*b^3*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)^{10}}{x^4} \, dx &=\int \left (210 a^6 b^4+\frac {a^{10}}{x^4}+\frac {10 a^9 b}{x^3}+\frac {45 a^8 b^2}{x^2}+\frac {120 a^7 b^3}{x}+252 a^5 b^5 x+210 a^4 b^6 x^2+120 a^3 b^7 x^3+45 a^2 b^8 x^4+10 a b^9 x^5+b^{10} x^6\right ) \, dx\\ &=-\frac {a^{10}}{3 x^3}-\frac {5 a^9 b}{x^2}-\frac {45 a^8 b^2}{x}+210 a^6 b^4 x+126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5+\frac {5}{3} a b^9 x^6+\frac {b^{10} x^7}{7}+120 a^7 b^3 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 115, normalized size = 1.00 \begin {gather*} -\frac {a^{10}}{3 x^3}-\frac {5 a^9 b}{x^2}-\frac {45 a^8 b^2}{x}+120 a^7 b^3 \log (x)+210 a^6 b^4 x+126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5+\frac {5}{3} a b^9 x^6+\frac {b^{10} x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^4,x]

[Out]

-1/3*a^10/x^3 - (5*a^9*b)/x^2 - (45*a^8*b^2)/x + 210*a^6*b^4*x + 126*a^5*b^5*x^2 + 70*a^4*b^6*x^3 + 30*a^3*b^7
*x^4 + 9*a^2*b^8*x^5 + (5*a*b^9*x^6)/3 + (b^10*x^7)/7 + 120*a^7*b^3*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^{10}}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^10/x^4,x]

[Out]

IntegrateAlgebraic[(a + b*x)^10/x^4, x]

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fricas [A]  time = 1.13, size = 114, normalized size = 0.99 \begin {gather*} \frac {3 \, b^{10} x^{10} + 35 \, a b^{9} x^{9} + 189 \, a^{2} b^{8} x^{8} + 630 \, a^{3} b^{7} x^{7} + 1470 \, a^{4} b^{6} x^{6} + 2646 \, a^{5} b^{5} x^{5} + 4410 \, a^{6} b^{4} x^{4} + 2520 \, a^{7} b^{3} x^{3} \log \relax (x) - 945 \, a^{8} b^{2} x^{2} - 105 \, a^{9} b x - 7 \, a^{10}}{21 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^4,x, algorithm="fricas")

[Out]

1/21*(3*b^10*x^10 + 35*a*b^9*x^9 + 189*a^2*b^8*x^8 + 630*a^3*b^7*x^7 + 1470*a^4*b^6*x^6 + 2646*a^5*b^5*x^5 + 4
410*a^6*b^4*x^4 + 2520*a^7*b^3*x^3*log(x) - 945*a^8*b^2*x^2 - 105*a^9*b*x - 7*a^10)/x^3

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giac [A]  time = 1.14, size = 109, normalized size = 0.95 \begin {gather*} \frac {1}{7} \, b^{10} x^{7} + \frac {5}{3} \, a b^{9} x^{6} + 9 \, a^{2} b^{8} x^{5} + 30 \, a^{3} b^{7} x^{4} + 70 \, a^{4} b^{6} x^{3} + 126 \, a^{5} b^{5} x^{2} + 210 \, a^{6} b^{4} x + 120 \, a^{7} b^{3} \log \left ({\left | x \right |}\right ) - \frac {135 \, a^{8} b^{2} x^{2} + 15 \, a^{9} b x + a^{10}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^4,x, algorithm="giac")

[Out]

1/7*b^10*x^7 + 5/3*a*b^9*x^6 + 9*a^2*b^8*x^5 + 30*a^3*b^7*x^4 + 70*a^4*b^6*x^3 + 126*a^5*b^5*x^2 + 210*a^6*b^4
*x + 120*a^7*b^3*log(abs(x)) - 1/3*(135*a^8*b^2*x^2 + 15*a^9*b*x + a^10)/x^3

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maple [A]  time = 0.01, size = 110, normalized size = 0.96 \begin {gather*} \frac {b^{10} x^{7}}{7}+\frac {5 a \,b^{9} x^{6}}{3}+9 a^{2} b^{8} x^{5}+30 a^{3} b^{7} x^{4}+70 a^{4} b^{6} x^{3}+126 a^{5} b^{5} x^{2}+120 a^{7} b^{3} \ln \relax (x )+210 a^{6} b^{4} x -\frac {45 a^{8} b^{2}}{x}-\frac {5 a^{9} b}{x^{2}}-\frac {a^{10}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^4,x)

[Out]

-1/3*a^10/x^3-5*a^9*b/x^2-45*a^8*b^2/x+210*a^6*b^4*x+126*a^5*b^5*x^2+70*a^4*b^6*x^3+30*a^3*b^7*x^4+9*a^2*b^8*x
^5+5/3*a*b^9*x^6+1/7*b^10*x^7+120*a^7*b^3*ln(x)

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maxima [A]  time = 1.37, size = 108, normalized size = 0.94 \begin {gather*} \frac {1}{7} \, b^{10} x^{7} + \frac {5}{3} \, a b^{9} x^{6} + 9 \, a^{2} b^{8} x^{5} + 30 \, a^{3} b^{7} x^{4} + 70 \, a^{4} b^{6} x^{3} + 126 \, a^{5} b^{5} x^{2} + 210 \, a^{6} b^{4} x + 120 \, a^{7} b^{3} \log \relax (x) - \frac {135 \, a^{8} b^{2} x^{2} + 15 \, a^{9} b x + a^{10}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^4,x, algorithm="maxima")

[Out]

1/7*b^10*x^7 + 5/3*a*b^9*x^6 + 9*a^2*b^8*x^5 + 30*a^3*b^7*x^4 + 70*a^4*b^6*x^3 + 126*a^5*b^5*x^2 + 210*a^6*b^4
*x + 120*a^7*b^3*log(x) - 1/3*(135*a^8*b^2*x^2 + 15*a^9*b*x + a^10)/x^3

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mupad [B]  time = 0.06, size = 110, normalized size = 0.96 \begin {gather*} \frac {b^{10}\,x^7}{7}-\frac {\frac {a^{10}}{3}+5\,a^9\,b\,x+45\,a^8\,b^2\,x^2}{x^3}+210\,a^6\,b^4\,x+\frac {5\,a\,b^9\,x^6}{3}+126\,a^5\,b^5\,x^2+70\,a^4\,b^6\,x^3+30\,a^3\,b^7\,x^4+9\,a^2\,b^8\,x^5+120\,a^7\,b^3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^10/x^4,x)

[Out]

(b^10*x^7)/7 - (a^10/3 + 45*a^8*b^2*x^2 + 5*a^9*b*x)/x^3 + 210*a^6*b^4*x + (5*a*b^9*x^6)/3 + 126*a^5*b^5*x^2 +
70*a^4*b^6*x^3 + 30*a^3*b^7*x^4 + 9*a^2*b^8*x^5 + 120*a^7*b^3*log(x)

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sympy [A]  time = 0.34, size = 119, normalized size = 1.03 \begin {gather*} 120 a^{7} b^{3} \log {\relax (x )} + 210 a^{6} b^{4} x + 126 a^{5} b^{5} x^{2} + 70 a^{4} b^{6} x^{3} + 30 a^{3} b^{7} x^{4} + 9 a^{2} b^{8} x^{5} + \frac {5 a b^{9} x^{6}}{3} + \frac {b^{10} x^{7}}{7} + \frac {- a^{10} - 15 a^{9} b x - 135 a^{8} b^{2} x^{2}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**4,x)

[Out]

120*a**7*b**3*log(x) + 210*a**6*b**4*x + 126*a**5*b**5*x**2 + 70*a**4*b**6*x**3 + 30*a**3*b**7*x**4 + 9*a**2*b
**8*x**5 + 5*a*b**9*x**6/3 + b**10*x**7/7 + (-a**10 - 15*a**9*b*x - 135*a**8*b**2*x**2)/(3*x**3)

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