3.2162 \(\int \frac {(a+b \sqrt {x})^{10}}{x^5} \, dx\)

Optimal. Leaf size=122 \[ -\frac {a^{10}}{4 x^4}-\frac {20 a^9 b}{7 x^{7/2}}-\frac {15 a^8 b^2}{x^3}-\frac {48 a^7 b^3}{x^{5/2}}-\frac {105 a^6 b^4}{x^2}-\frac {168 a^5 b^5}{x^{3/2}}-\frac {210 a^4 b^6}{x}-\frac {240 a^3 b^7}{\sqrt {x}}+45 a^2 b^8 \log (x)+20 a b^9 \sqrt {x}+b^{10} x \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 122, 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} \[ -\frac {15 a^8 b^2}{x^3}-\frac {48 a^7 b^3}{x^{5/2}}-\frac {105 a^6 b^4}{x^2}-\frac {168 a^5 b^5}{x^{3/2}}-\frac {210 a^4 b^6}{x}-\frac {240 a^3 b^7}{\sqrt {x}}+45 a^2 b^8 \log (x)-\frac {20 a^9 b}{7 x^{7/2}}-\frac {a^{10}}{4 x^4}+20 a b^9 \sqrt {x}+b^{10} x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10/x^5,x]

[Out]

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

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Mathematica [A]  time = 0.09, size = 122, normalized size = 1.00 \[ -\frac {a^{10}}{4 x^4}-\frac {20 a^9 b}{7 x^{7/2}}-\frac {15 a^8 b^2}{x^3}-\frac {48 a^7 b^3}{x^{5/2}}-\frac {105 a^6 b^4}{x^2}-\frac {168 a^5 b^5}{x^{3/2}}-\frac {210 a^4 b^6}{x}-\frac {240 a^3 b^7}{\sqrt {x}}+45 a^2 b^8 \log (x)+20 a b^9 \sqrt {x}+b^{10} x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10/x^5,x]

[Out]

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

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fricas [A]  time = 1.04, size = 117, normalized size = 0.96 \[ \frac {28 \, b^{10} x^{5} + 2520 \, a^{2} b^{8} x^{4} \log \left (\sqrt {x}\right ) - 5880 \, a^{4} b^{6} x^{3} - 2940 \, a^{6} b^{4} x^{2} - 420 \, a^{8} b^{2} x - 7 \, a^{10} + 16 \, {\left (35 \, a b^{9} x^{4} - 420 \, a^{3} b^{7} x^{3} - 294 \, a^{5} b^{5} x^{2} - 84 \, a^{7} b^{3} x - 5 \, a^{9} b\right )} \sqrt {x}}{28 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 110, normalized size = 0.90 \[ b^{10} x + 45 \, a^{2} b^{8} \log \left ({\left | x \right |}\right ) + 20 \, a b^{9} \sqrt {x} - \frac {6720 \, a^{3} b^{7} x^{\frac {7}{2}} + 5880 \, a^{4} b^{6} x^{3} + 4704 \, a^{5} b^{5} x^{\frac {5}{2}} + 2940 \, a^{6} b^{4} x^{2} + 1344 \, a^{7} b^{3} x^{\frac {3}{2}} + 420 \, a^{8} b^{2} x + 80 \, a^{9} b \sqrt {x} + 7 \, a^{10}}{28 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^5,x, algorithm="giac")

[Out]

b^10*x + 45*a^2*b^8*log(abs(x)) + 20*a*b^9*sqrt(x) - 1/28*(6720*a^3*b^7*x^(7/2) + 5880*a^4*b^6*x^3 + 4704*a^5*
b^5*x^(5/2) + 2940*a^6*b^4*x^2 + 1344*a^7*b^3*x^(3/2) + 420*a^8*b^2*x + 80*a^9*b*sqrt(x) + 7*a^10)/x^4

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maple [A]  time = 0.00, size = 109, normalized size = 0.89 \[ 45 a^{2} b^{8} \ln \relax (x )+b^{10} x +20 a \,b^{9} \sqrt {x}-\frac {240 a^{3} b^{7}}{\sqrt {x}}-\frac {210 a^{4} b^{6}}{x}-\frac {168 a^{5} b^{5}}{x^{\frac {3}{2}}}-\frac {105 a^{6} b^{4}}{x^{2}}-\frac {48 a^{7} b^{3}}{x^{\frac {5}{2}}}-\frac {15 a^{8} b^{2}}{x^{3}}-\frac {20 a^{9} b}{7 x^{\frac {7}{2}}}-\frac {a^{10}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^10/x^5,x)

[Out]

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

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maxima [A]  time = 0.89, size = 109, normalized size = 0.89 \[ b^{10} x + 45 \, a^{2} b^{8} \log \relax (x) + 20 \, a b^{9} \sqrt {x} - \frac {6720 \, a^{3} b^{7} x^{\frac {7}{2}} + 5880 \, a^{4} b^{6} x^{3} + 4704 \, a^{5} b^{5} x^{\frac {5}{2}} + 2940 \, a^{6} b^{4} x^{2} + 1344 \, a^{7} b^{3} x^{\frac {3}{2}} + 420 \, a^{8} b^{2} x + 80 \, a^{9} b \sqrt {x} + 7 \, a^{10}}{28 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^5,x, algorithm="maxima")

[Out]

b^10*x + 45*a^2*b^8*log(x) + 20*a*b^9*sqrt(x) - 1/28*(6720*a^3*b^7*x^(7/2) + 5880*a^4*b^6*x^3 + 4704*a^5*b^5*x
^(5/2) + 2940*a^6*b^4*x^2 + 1344*a^7*b^3*x^(3/2) + 420*a^8*b^2*x + 80*a^9*b*sqrt(x) + 7*a^10)/x^4

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mupad [B]  time = 0.05, size = 111, normalized size = 0.91 \[ b^{10}\,x-\frac {\frac {a^{10}}{4}+15\,a^8\,b^2\,x+\frac {20\,a^9\,b\,\sqrt {x}}{7}+105\,a^6\,b^4\,x^2+210\,a^4\,b^6\,x^3+48\,a^7\,b^3\,x^{3/2}+168\,a^5\,b^5\,x^{5/2}+240\,a^3\,b^7\,x^{7/2}}{x^4}+90\,a^2\,b^8\,\ln \left (\sqrt {x}\right )+20\,a\,b^9\,\sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^10/x^5,x)

[Out]

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

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sympy [A]  time = 2.01, size = 124, normalized size = 1.02 \[ - \frac {a^{10}}{4 x^{4}} - \frac {20 a^{9} b}{7 x^{\frac {7}{2}}} - \frac {15 a^{8} b^{2}}{x^{3}} - \frac {48 a^{7} b^{3}}{x^{\frac {5}{2}}} - \frac {105 a^{6} b^{4}}{x^{2}} - \frac {168 a^{5} b^{5}}{x^{\frac {3}{2}}} - \frac {210 a^{4} b^{6}}{x} - \frac {240 a^{3} b^{7}}{\sqrt {x}} + 45 a^{2} b^{8} \log {\relax (x )} + 20 a b^{9} \sqrt {x} + b^{10} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**10/x**5,x)

[Out]

-a**10/(4*x**4) - 20*a**9*b/(7*x**(7/2)) - 15*a**8*b**2/x**3 - 48*a**7*b**3/x**(5/2) - 105*a**6*b**4/x**2 - 16
8*a**5*b**5/x**(3/2) - 210*a**4*b**6/x - 240*a**3*b**7/sqrt(x) + 45*a**2*b**8*log(x) + 20*a*b**9*sqrt(x) + b**
10*x

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