3.136 \(\int \frac{(a+b x)^{10}}{x^2} \, dx\)

Optimal. Leaf size=115 \[ 60 a^7 b^3 x^2+70 a^6 b^4 x^3+63 a^5 b^5 x^4+42 a^4 b^6 x^5+20 a^3 b^7 x^6+\frac{45}{7} a^2 b^8 x^7+45 a^8 b^2 x+10 a^9 b \log (x)-\frac{a^{10}}{x}+\frac{5}{4} a b^9 x^8+\frac{b^{10} x^9}{9} \]

[Out]

-(a^10/x) + 45*a^8*b^2*x + 60*a^7*b^3*x^2 + 70*a^6*b^4*x^3 + 63*a^5*b^5*x^4 + 42*a^4*b^6*x^5 + 20*a^3*b^7*x^6
+ (45*a^2*b^8*x^7)/7 + (5*a*b^9*x^8)/4 + (b^10*x^9)/9 + 10*a^9*b*Log[x]

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Rubi [A]  time = 0.0468345, antiderivative size = 115, 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} \[ 60 a^7 b^3 x^2+70 a^6 b^4 x^3+63 a^5 b^5 x^4+42 a^4 b^6 x^5+20 a^3 b^7 x^6+\frac{45}{7} a^2 b^8 x^7+45 a^8 b^2 x+10 a^9 b \log (x)-\frac{a^{10}}{x}+\frac{5}{4} a b^9 x^8+\frac{b^{10} x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^2,x]

[Out]

-(a^10/x) + 45*a^8*b^2*x + 60*a^7*b^3*x^2 + 70*a^6*b^4*x^3 + 63*a^5*b^5*x^4 + 42*a^4*b^6*x^5 + 20*a^3*b^7*x^6
+ (45*a^2*b^8*x^7)/7 + (5*a*b^9*x^8)/4 + (b^10*x^9)/9 + 10*a^9*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)^{10}}{x^2} \, dx &=\int \left (45 a^8 b^2+\frac{a^{10}}{x^2}+\frac{10 a^9 b}{x}+120 a^7 b^3 x+210 a^6 b^4 x^2+252 a^5 b^5 x^3+210 a^4 b^6 x^4+120 a^3 b^7 x^5+45 a^2 b^8 x^6+10 a b^9 x^7+b^{10} x^8\right ) \, dx\\ &=-\frac{a^{10}}{x}+45 a^8 b^2 x+60 a^7 b^3 x^2+70 a^6 b^4 x^3+63 a^5 b^5 x^4+42 a^4 b^6 x^5+20 a^3 b^7 x^6+\frac{45}{7} a^2 b^8 x^7+\frac{5}{4} a b^9 x^8+\frac{b^{10} x^9}{9}+10 a^9 b \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0115629, size = 115, normalized size = 1. \[ 60 a^7 b^3 x^2+70 a^6 b^4 x^3+63 a^5 b^5 x^4+42 a^4 b^6 x^5+20 a^3 b^7 x^6+\frac{45}{7} a^2 b^8 x^7+45 a^8 b^2 x+10 a^9 b \log (x)-\frac{a^{10}}{x}+\frac{5}{4} a b^9 x^8+\frac{b^{10} x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^2,x]

[Out]

-(a^10/x) + 45*a^8*b^2*x + 60*a^7*b^3*x^2 + 70*a^6*b^4*x^3 + 63*a^5*b^5*x^4 + 42*a^4*b^6*x^5 + 20*a^3*b^7*x^6
+ (45*a^2*b^8*x^7)/7 + (5*a*b^9*x^8)/4 + (b^10*x^9)/9 + 10*a^9*b*Log[x]

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Maple [A]  time = 0.007, size = 110, normalized size = 1. \begin{align*} -{\frac{{a}^{10}}{x}}+45\,{a}^{8}{b}^{2}x+60\,{a}^{7}{b}^{3}{x}^{2}+70\,{a}^{6}{b}^{4}{x}^{3}+63\,{a}^{5}{b}^{5}{x}^{4}+42\,{a}^{4}{b}^{6}{x}^{5}+20\,{a}^{3}{b}^{7}{x}^{6}+{\frac{45\,{a}^{2}{b}^{8}{x}^{7}}{7}}+{\frac{5\,a{b}^{9}{x}^{8}}{4}}+{\frac{{b}^{10}{x}^{9}}{9}}+10\,{a}^{9}b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^2,x)

[Out]

-a^10/x+45*a^8*b^2*x+60*a^7*b^3*x^2+70*a^6*b^4*x^3+63*a^5*b^5*x^4+42*a^4*b^6*x^5+20*a^3*b^7*x^6+45/7*a^2*b^8*x
^7+5/4*a*b^9*x^8+1/9*b^10*x^9+10*a^9*b*ln(x)

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Maxima [A]  time = 1.01666, size = 147, normalized size = 1.28 \begin{align*} \frac{1}{9} \, b^{10} x^{9} + \frac{5}{4} \, a b^{9} x^{8} + \frac{45}{7} \, a^{2} b^{8} x^{7} + 20 \, a^{3} b^{7} x^{6} + 42 \, a^{4} b^{6} x^{5} + 63 \, a^{5} b^{5} x^{4} + 70 \, a^{6} b^{4} x^{3} + 60 \, a^{7} b^{3} x^{2} + 45 \, a^{8} b^{2} x + 10 \, a^{9} b \log \left (x\right ) - \frac{a^{10}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^2,x, algorithm="maxima")

[Out]

1/9*b^10*x^9 + 5/4*a*b^9*x^8 + 45/7*a^2*b^8*x^7 + 20*a^3*b^7*x^6 + 42*a^4*b^6*x^5 + 63*a^5*b^5*x^4 + 70*a^6*b^
4*x^3 + 60*a^7*b^3*x^2 + 45*a^8*b^2*x + 10*a^9*b*log(x) - a^10/x

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Fricas [A]  time = 1.7833, size = 285, normalized size = 2.48 \begin{align*} \frac{28 \, b^{10} x^{10} + 315 \, a b^{9} x^{9} + 1620 \, a^{2} b^{8} x^{8} + 5040 \, a^{3} b^{7} x^{7} + 10584 \, a^{4} b^{6} x^{6} + 15876 \, a^{5} b^{5} x^{5} + 17640 \, a^{6} b^{4} x^{4} + 15120 \, a^{7} b^{3} x^{3} + 11340 \, a^{8} b^{2} x^{2} + 2520 \, a^{9} b x \log \left (x\right ) - 252 \, a^{10}}{252 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^2,x, algorithm="fricas")

[Out]

1/252*(28*b^10*x^10 + 315*a*b^9*x^9 + 1620*a^2*b^8*x^8 + 5040*a^3*b^7*x^7 + 10584*a^4*b^6*x^6 + 15876*a^5*b^5*
x^5 + 17640*a^6*b^4*x^4 + 15120*a^7*b^3*x^3 + 11340*a^8*b^2*x^2 + 2520*a^9*b*x*log(x) - 252*a^10)/x

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Sympy [A]  time = 0.484801, size = 117, normalized size = 1.02 \begin{align*} - \frac{a^{10}}{x} + 10 a^{9} b \log{\left (x \right )} + 45 a^{8} b^{2} x + 60 a^{7} b^{3} x^{2} + 70 a^{6} b^{4} x^{3} + 63 a^{5} b^{5} x^{4} + 42 a^{4} b^{6} x^{5} + 20 a^{3} b^{7} x^{6} + \frac{45 a^{2} b^{8} x^{7}}{7} + \frac{5 a b^{9} x^{8}}{4} + \frac{b^{10} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**2,x)

[Out]

-a**10/x + 10*a**9*b*log(x) + 45*a**8*b**2*x + 60*a**7*b**3*x**2 + 70*a**6*b**4*x**3 + 63*a**5*b**5*x**4 + 42*
a**4*b**6*x**5 + 20*a**3*b**7*x**6 + 45*a**2*b**8*x**7/7 + 5*a*b**9*x**8/4 + b**10*x**9/9

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Giac [A]  time = 1.15766, size = 149, normalized size = 1.3 \begin{align*} \frac{1}{9} \, b^{10} x^{9} + \frac{5}{4} \, a b^{9} x^{8} + \frac{45}{7} \, a^{2} b^{8} x^{7} + 20 \, a^{3} b^{7} x^{6} + 42 \, a^{4} b^{6} x^{5} + 63 \, a^{5} b^{5} x^{4} + 70 \, a^{6} b^{4} x^{3} + 60 \, a^{7} b^{3} x^{2} + 45 \, a^{8} b^{2} x + 10 \, a^{9} b \log \left ({\left | x \right |}\right ) - \frac{a^{10}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^2,x, algorithm="giac")

[Out]

1/9*b^10*x^9 + 5/4*a*b^9*x^8 + 45/7*a^2*b^8*x^7 + 20*a^3*b^7*x^6 + 42*a^4*b^6*x^5 + 63*a^5*b^5*x^4 + 70*a^6*b^
4*x^3 + 60*a^7*b^3*x^2 + 45*a^8*b^2*x + 10*a^9*b*log(abs(x)) - a^10/x