3.2159 \(\int \frac {(a+b \sqrt {x})^{10}}{x^2} \, dx\)

Optimal. Leaf size=123 \[ -\frac {a^{10}}{x}-\frac {20 a^9 b}{\sqrt {x}}+45 a^8 b^2 \log (x)+240 a^7 b^3 \sqrt {x}+210 a^6 b^4 x+168 a^5 b^5 x^{3/2}+105 a^4 b^6 x^2+48 a^3 b^7 x^{5/2}+15 a^2 b^8 x^3+\frac {20}{7} a b^9 x^{7/2}+\frac {b^{10} x^4}{4} \]

[Out]

-a^10/x+210*a^6*b^4*x+168*a^5*b^5*x^(3/2)+105*a^4*b^6*x^2+48*a^3*b^7*x^(5/2)+15*a^2*b^8*x^3+20/7*a*b^9*x^(7/2)
+1/4*b^10*x^4+45*a^8*b^2*ln(x)-20*a^9*b/x^(1/2)+240*a^7*b^3*x^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ 168 a^5 b^5 x^{3/2}+105 a^4 b^6 x^2+48 a^3 b^7 x^{5/2}+15 a^2 b^8 x^3+240 a^7 b^3 \sqrt {x}+210 a^6 b^4 x+45 a^8 b^2 \log (x)-\frac {20 a^9 b}{\sqrt {x}}-\frac {a^{10}}{x}+\frac {20}{7} a b^9 x^{7/2}+\frac {b^{10} x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10/x^2,x]

[Out]

-(a^10/x) - (20*a^9*b)/Sqrt[x] + 240*a^7*b^3*Sqrt[x] + 210*a^6*b^4*x + 168*a^5*b^5*x^(3/2) + 105*a^4*b^6*x^2 +
48*a^3*b^7*x^(5/2) + 15*a^2*b^8*x^3 + (20*a*b^9*x^(7/2))/7 + (b^10*x^4)/4 + 45*a^8*b^2*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])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{10}}{x^3} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (120 a^7 b^3+\frac {a^{10}}{x^3}+\frac {10 a^9 b}{x^2}+\frac {45 a^8 b^2}{x}+210 a^6 b^4 x+252 a^5 b^5 x^2+210 a^4 b^6 x^3+120 a^3 b^7 x^4+45 a^2 b^8 x^5+10 a b^9 x^6+b^{10} x^7\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^{10}}{x}-\frac {20 a^9 b}{\sqrt {x}}+240 a^7 b^3 \sqrt {x}+210 a^6 b^4 x+168 a^5 b^5 x^{3/2}+105 a^4 b^6 x^2+48 a^3 b^7 x^{5/2}+15 a^2 b^8 x^3+\frac {20}{7} a b^9 x^{7/2}+\frac {b^{10} x^4}{4}+45 a^8 b^2 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 123, normalized size = 1.00 \[ -\frac {a^{10}}{x}-\frac {20 a^9 b}{\sqrt {x}}+45 a^8 b^2 \log (x)+240 a^7 b^3 \sqrt {x}+210 a^6 b^4 x+168 a^5 b^5 x^{3/2}+105 a^4 b^6 x^2+48 a^3 b^7 x^{5/2}+15 a^2 b^8 x^3+\frac {20}{7} a b^9 x^{7/2}+\frac {b^{10} x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10/x^2,x]

[Out]

-(a^10/x) - (20*a^9*b)/Sqrt[x] + 240*a^7*b^3*Sqrt[x] + 210*a^6*b^4*x + 168*a^5*b^5*x^(3/2) + 105*a^4*b^6*x^2 +
48*a^3*b^7*x^(5/2) + 15*a^2*b^8*x^3 + (20*a*b^9*x^(7/2))/7 + (b^10*x^4)/4 + 45*a^8*b^2*Log[x]

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fricas [A]  time = 0.63, size = 117, normalized size = 0.95 \[ \frac {7 \, b^{10} x^{5} + 420 \, a^{2} b^{8} x^{4} + 2940 \, a^{4} b^{6} x^{3} + 5880 \, a^{6} b^{4} x^{2} + 2520 \, a^{8} b^{2} x \log \left (\sqrt {x}\right ) - 28 \, a^{10} + 16 \, {\left (5 \, a b^{9} x^{4} + 84 \, a^{3} b^{7} x^{3} + 294 \, a^{5} b^{5} x^{2} + 420 \, a^{7} b^{3} x - 35 \, a^{9} b\right )} \sqrt {x}}{28 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^2,x, algorithm="fricas")

[Out]

1/28*(7*b^10*x^5 + 420*a^2*b^8*x^4 + 2940*a^4*b^6*x^3 + 5880*a^6*b^4*x^2 + 2520*a^8*b^2*x*log(sqrt(x)) - 28*a^
10 + 16*(5*a*b^9*x^4 + 84*a^3*b^7*x^3 + 294*a^5*b^5*x^2 + 420*a^7*b^3*x - 35*a^9*b)*sqrt(x))/x

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giac [A]  time = 0.17, size = 111, normalized size = 0.90 \[ \frac {1}{4} \, b^{10} x^{4} + \frac {20}{7} \, a b^{9} x^{\frac {7}{2}} + 15 \, a^{2} b^{8} x^{3} + 48 \, a^{3} b^{7} x^{\frac {5}{2}} + 105 \, a^{4} b^{6} x^{2} + 168 \, a^{5} b^{5} x^{\frac {3}{2}} + 210 \, a^{6} b^{4} x + 45 \, a^{8} b^{2} \log \left ({\left | x \right |}\right ) + 240 \, a^{7} b^{3} \sqrt {x} - \frac {20 \, a^{9} b \sqrt {x} + a^{10}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^2,x, algorithm="giac")

[Out]

1/4*b^10*x^4 + 20/7*a*b^9*x^(7/2) + 15*a^2*b^8*x^3 + 48*a^3*b^7*x^(5/2) + 105*a^4*b^6*x^2 + 168*a^5*b^5*x^(3/2
) + 210*a^6*b^4*x + 45*a^8*b^2*log(abs(x)) + 240*a^7*b^3*sqrt(x) - (20*a^9*b*sqrt(x) + a^10)/x

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maple [A]  time = 0.00, size = 110, normalized size = 0.89 \[ \frac {b^{10} x^{4}}{4}+\frac {20 a \,b^{9} x^{\frac {7}{2}}}{7}+15 a^{2} b^{8} x^{3}+48 a^{3} b^{7} x^{\frac {5}{2}}+105 a^{4} b^{6} x^{2}+168 a^{5} b^{5} x^{\frac {3}{2}}+45 a^{8} b^{2} \ln \relax (x )+210 a^{6} b^{4} x +240 a^{7} b^{3} \sqrt {x}-\frac {20 a^{9} b}{\sqrt {x}}-\frac {a^{10}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^10/x^2,x)

[Out]

-a^10/x+210*a^6*b^4*x+168*a^5*b^5*x^(3/2)+105*a^4*b^6*x^2+48*a^3*b^7*x^(5/2)+15*a^2*b^8*x^3+20/7*a*b^9*x^(7/2)
+1/4*b^10*x^4+45*a^8*b^2*ln(x)-20*a^9*b/x^(1/2)+240*a^7*b^3*x^(1/2)

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maxima [A]  time = 0.81, size = 110, normalized size = 0.89 \[ \frac {1}{4} \, b^{10} x^{4} + \frac {20}{7} \, a b^{9} x^{\frac {7}{2}} + 15 \, a^{2} b^{8} x^{3} + 48 \, a^{3} b^{7} x^{\frac {5}{2}} + 105 \, a^{4} b^{6} x^{2} + 168 \, a^{5} b^{5} x^{\frac {3}{2}} + 210 \, a^{6} b^{4} x + 45 \, a^{8} b^{2} \log \relax (x) + 240 \, a^{7} b^{3} \sqrt {x} - \frac {20 \, a^{9} b \sqrt {x} + a^{10}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^2,x, algorithm="maxima")

[Out]

1/4*b^10*x^4 + 20/7*a*b^9*x^(7/2) + 15*a^2*b^8*x^3 + 48*a^3*b^7*x^(5/2) + 105*a^4*b^6*x^2 + 168*a^5*b^5*x^(3/2
) + 210*a^6*b^4*x + 45*a^8*b^2*log(x) + 240*a^7*b^3*sqrt(x) - (20*a^9*b*sqrt(x) + a^10)/x

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mupad [B]  time = 0.06, size = 112, normalized size = 0.91 \[ \frac {b^{10}\,x^4}{4}-\frac {a^{10}+20\,a^9\,b\,\sqrt {x}}{x}+90\,a^8\,b^2\,\ln \left (\sqrt {x}\right )+210\,a^6\,b^4\,x+\frac {20\,a\,b^9\,x^{7/2}}{7}+105\,a^4\,b^6\,x^2+15\,a^2\,b^8\,x^3+240\,a^7\,b^3\,\sqrt {x}+168\,a^5\,b^5\,x^{3/2}+48\,a^3\,b^7\,x^{5/2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^10/x^2,x)

[Out]

(b^10*x^4)/4 - (a^10 + 20*a^9*b*x^(1/2))/x + 90*a^8*b^2*log(x^(1/2)) + 210*a^6*b^4*x + (20*a*b^9*x^(7/2))/7 +
105*a^4*b^6*x^2 + 15*a^2*b^8*x^3 + 240*a^7*b^3*x^(1/2) + 168*a^5*b^5*x^(3/2) + 48*a^3*b^7*x^(5/2)

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sympy [A]  time = 2.18, size = 124, normalized size = 1.01 \[ - \frac {a^{10}}{x} - \frac {20 a^{9} b}{\sqrt {x}} + 45 a^{8} b^{2} \log {\relax (x )} + 240 a^{7} b^{3} \sqrt {x} + 210 a^{6} b^{4} x + 168 a^{5} b^{5} x^{\frac {3}{2}} + 105 a^{4} b^{6} x^{2} + 48 a^{3} b^{7} x^{\frac {5}{2}} + 15 a^{2} b^{8} x^{3} + \frac {20 a b^{9} x^{\frac {7}{2}}}{7} + \frac {b^{10} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**10/x**2,x)

[Out]

-a**10/x - 20*a**9*b/sqrt(x) + 45*a**8*b**2*log(x) + 240*a**7*b**3*sqrt(x) + 210*a**6*b**4*x + 168*a**5*b**5*x
**(3/2) + 105*a**4*b**6*x**2 + 48*a**3*b**7*x**(5/2) + 15*a**2*b**8*x**3 + 20*a*b**9*x**(7/2)/7 + b**10*x**4/4

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