3.2151 \(\int \frac {(a+b \sqrt {x})^5}{x^6} \, dx\)

Optimal. Leaf size=75 \[ -\frac {a^5}{5 x^5}-\frac {10 a^4 b}{9 x^{9/2}}-\frac {5 a^3 b^2}{2 x^4}-\frac {20 a^2 b^3}{7 x^{7/2}}-\frac {5 a b^4}{3 x^3}-\frac {2 b^5}{5 x^{5/2}} \]

[Out]

-1/5*a^5/x^5-10/9*a^4*b/x^(9/2)-5/2*a^3*b^2/x^4-20/7*a^2*b^3/x^(7/2)-5/3*a*b^4/x^3-2/5*b^5/x^(5/2)

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Rubi [A]  time = 0.03, antiderivative size = 75, 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} \[ -\frac {5 a^3 b^2}{2 x^4}-\frac {20 a^2 b^3}{7 x^{7/2}}-\frac {10 a^4 b}{9 x^{9/2}}-\frac {a^5}{5 x^5}-\frac {5 a b^4}{3 x^3}-\frac {2 b^5}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5/x^6,x]

[Out]

-a^5/(5*x^5) - (10*a^4*b)/(9*x^(9/2)) - (5*a^3*b^2)/(2*x^4) - (20*a^2*b^3)/(7*x^(7/2)) - (5*a*b^4)/(3*x^3) - (
2*b^5)/(5*x^(5/2))

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 )^5}{x^6} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^{11}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {a^5}{x^{11}}+\frac {5 a^4 b}{x^{10}}+\frac {10 a^3 b^2}{x^9}+\frac {10 a^2 b^3}{x^8}+\frac {5 a b^4}{x^7}+\frac {b^5}{x^6}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^5}{5 x^5}-\frac {10 a^4 b}{9 x^{9/2}}-\frac {5 a^3 b^2}{2 x^4}-\frac {20 a^2 b^3}{7 x^{7/2}}-\frac {5 a b^4}{3 x^3}-\frac {2 b^5}{5 x^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 65, normalized size = 0.87 \[ -\frac {126 a^5+700 a^4 b \sqrt {x}+1575 a^3 b^2 x+1800 a^2 b^3 x^{3/2}+1050 a b^4 x^2+252 b^5 x^{5/2}}{630 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5/x^6,x]

[Out]

-1/630*(126*a^5 + 700*a^4*b*Sqrt[x] + 1575*a^3*b^2*x + 1800*a^2*b^3*x^(3/2) + 1050*a*b^4*x^2 + 252*b^5*x^(5/2)
)/x^5

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fricas [A]  time = 1.32, size = 58, normalized size = 0.77 \[ -\frac {1050 \, a b^{4} x^{2} + 1575 \, a^{3} b^{2} x + 126 \, a^{5} + 4 \, {\left (63 \, b^{5} x^{2} + 450 \, a^{2} b^{3} x + 175 \, a^{4} b\right )} \sqrt {x}}{630 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x^6,x, algorithm="fricas")

[Out]

-1/630*(1050*a*b^4*x^2 + 1575*a^3*b^2*x + 126*a^5 + 4*(63*b^5*x^2 + 450*a^2*b^3*x + 175*a^4*b)*sqrt(x))/x^5

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giac [A]  time = 0.18, size = 57, normalized size = 0.76 \[ -\frac {252 \, b^{5} x^{\frac {5}{2}} + 1050 \, a b^{4} x^{2} + 1800 \, a^{2} b^{3} x^{\frac {3}{2}} + 1575 \, a^{3} b^{2} x + 700 \, a^{4} b \sqrt {x} + 126 \, a^{5}}{630 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x^6,x, algorithm="giac")

[Out]

-1/630*(252*b^5*x^(5/2) + 1050*a*b^4*x^2 + 1800*a^2*b^3*x^(3/2) + 1575*a^3*b^2*x + 700*a^4*b*sqrt(x) + 126*a^5
)/x^5

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maple [A]  time = 0.00, size = 58, normalized size = 0.77 \[ -\frac {2 b^{5}}{5 x^{\frac {5}{2}}}-\frac {5 a \,b^{4}}{3 x^{3}}-\frac {20 a^{2} b^{3}}{7 x^{\frac {7}{2}}}-\frac {5 a^{3} b^{2}}{2 x^{4}}-\frac {10 a^{4} b}{9 x^{\frac {9}{2}}}-\frac {a^{5}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^5/x^6,x)

[Out]

-1/5*a^5/x^5-10/9*a^4*b/x^(9/2)-5/2*a^3*b^2/x^4-20/7*a^2*b^3/x^(7/2)-5/3*a*b^4/x^3-2/5*b^5/x^(5/2)

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maxima [A]  time = 0.88, size = 57, normalized size = 0.76 \[ -\frac {252 \, b^{5} x^{\frac {5}{2}} + 1050 \, a b^{4} x^{2} + 1800 \, a^{2} b^{3} x^{\frac {3}{2}} + 1575 \, a^{3} b^{2} x + 700 \, a^{4} b \sqrt {x} + 126 \, a^{5}}{630 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x^6,x, algorithm="maxima")

[Out]

-1/630*(252*b^5*x^(5/2) + 1050*a*b^4*x^2 + 1800*a^2*b^3*x^(3/2) + 1575*a^3*b^2*x + 700*a^4*b*sqrt(x) + 126*a^5
)/x^5

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mupad [B]  time = 0.04, size = 57, normalized size = 0.76 \[ -\frac {126\,a^5+252\,b^5\,x^{5/2}+1575\,a^3\,b^2\,x+1050\,a\,b^4\,x^2+700\,a^4\,b\,\sqrt {x}+1800\,a^2\,b^3\,x^{3/2}}{630\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^5/x^6,x)

[Out]

-(126*a^5 + 252*b^5*x^(5/2) + 1575*a^3*b^2*x + 1050*a*b^4*x^2 + 700*a^4*b*x^(1/2) + 1800*a^2*b^3*x^(3/2))/(630
*x^5)

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sympy [A]  time = 2.21, size = 75, normalized size = 1.00 \[ - \frac {a^{5}}{5 x^{5}} - \frac {10 a^{4} b}{9 x^{\frac {9}{2}}} - \frac {5 a^{3} b^{2}}{2 x^{4}} - \frac {20 a^{2} b^{3}}{7 x^{\frac {7}{2}}} - \frac {5 a b^{4}}{3 x^{3}} - \frac {2 b^{5}}{5 x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**5/x**6,x)

[Out]

-a**5/(5*x**5) - 10*a**4*b/(9*x**(9/2)) - 5*a**3*b**2/(2*x**4) - 20*a**2*b**3/(7*x**(7/2)) - 5*a*b**4/(3*x**3)
- 2*b**5/(5*x**(5/2))

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