3.22.39 \(\int \frac {(a+b \sqrt {x})^3}{x^5} \, dx\) [2139]

Optimal. Leaf size=45 \[ -\frac {a^3}{4 x^4}-\frac {6 a^2 b}{7 x^{7/2}}-\frac {a b^2}{x^3}-\frac {2 b^3}{5 x^{5/2}} \]

[Out]

-1/4*a^3/x^4-6/7*a^2*b/x^(7/2)-a*b^2/x^3-2/5*b^3/x^(5/2)

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Rubi [A]
time = 0.01, antiderivative size = 45, 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 = {272, 45} \begin {gather*} -\frac {a^3}{4 x^4}-\frac {6 a^2 b}{7 x^{7/2}}-\frac {a b^2}{x^3}-\frac {2 b^3}{5 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3/x^5,x]

[Out]

-1/4*a^3/x^4 - (6*a^2*b)/(7*x^(7/2)) - (a*b^2)/x^3 - (2*b^3)/(5*x^(5/2))

Rule 45

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 272

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

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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.91 \begin {gather*} \frac {-35 a^3-120 a^2 b \sqrt {x}-140 a b^2 x-56 b^3 x^{3/2}}{140 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3/x^5,x]

[Out]

(-35*a^3 - 120*a^2*b*Sqrt[x] - 140*a*b^2*x - 56*b^3*x^(3/2))/(140*x^4)

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Maple [A]
time = 0.19, size = 36, normalized size = 0.80

method	result	size
derivativedivides	\(-\frac {a^{3}}{4 x^{4}}-\frac {6 a^{2} b}{7 x^{\frac {7}{2}}}-\frac {a \,b^{2}}{x^{3}}-\frac {2 b^{3}}{5 x^{\frac {5}{2}}}\)	\(36\)
default	\(-\frac {a^{3}}{4 x^{4}}-\frac {6 a^{2} b}{7 x^{\frac {7}{2}}}-\frac {a \,b^{2}}{x^{3}}-\frac {2 b^{3}}{5 x^{\frac {5}{2}}}\)	\(36\)
trager	\(\frac {\left (x -1\right ) \left (a^{2} x^{3}+4 b^{2} x^{3}+a^{2} x^{2}+4 b^{2} x^{2}+a^{2} x +4 b^{2} x +a^{2}\right ) a}{4 x^{4}}-\frac {2 \left (7 b^{2} x +15 a^{2}\right ) b}{35 x^{\frac {7}{2}}}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^3/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*a^3/x^4-6/7*a^2*b/x^(7/2)-a*b^2/x^3-2/5*b^3/x^(5/2)

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Maxima [A]
time = 0.30, size = 35, normalized size = 0.78 \begin {gather*} -\frac {56 \, b^{3} x^{\frac {3}{2}} + 140 \, a b^{2} x + 120 \, a^{2} b \sqrt {x} + 35 \, a^{3}}{140 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^5,x, algorithm="maxima")

[Out]

-1/140*(56*b^3*x^(3/2) + 140*a*b^2*x + 120*a^2*b*sqrt(x) + 35*a^3)/x^4

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Fricas [A]
time = 0.36, size = 36, normalized size = 0.80 \begin {gather*} -\frac {140 \, a b^{2} x + 35 \, a^{3} + 8 \, {\left (7 \, b^{3} x + 15 \, a^{2} b\right )} \sqrt {x}}{140 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^5,x, algorithm="fricas")

[Out]

-1/140*(140*a*b^2*x + 35*a^3 + 8*(7*b^3*x + 15*a^2*b)*sqrt(x))/x^4

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Sympy [A]
time = 0.31, size = 42, normalized size = 0.93 \begin {gather*} - \frac {a^{3}}{4 x^{4}} - \frac {6 a^{2} b}{7 x^{\frac {7}{2}}} - \frac {a b^{2}}{x^{3}} - \frac {2 b^{3}}{5 x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**3/x**5,x)

[Out]

-a**3/(4*x**4) - 6*a**2*b/(7*x**(7/2)) - a*b**2/x**3 - 2*b**3/(5*x**(5/2))

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Giac [A]
time = 0.45, size = 35, normalized size = 0.78 \begin {gather*} -\frac {56 \, b^{3} x^{\frac {3}{2}} + 140 \, a b^{2} x + 120 \, a^{2} b \sqrt {x} + 35 \, a^{3}}{140 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^5,x, algorithm="giac")

[Out]

-1/140*(56*b^3*x^(3/2) + 140*a*b^2*x + 120*a^2*b*sqrt(x) + 35*a^3)/x^4

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Mupad [B]
time = 0.03, size = 35, normalized size = 0.78 \begin {gather*} -\frac {35\,a^3+56\,b^3\,x^{3/2}+120\,a^2\,b\,\sqrt {x}+140\,a\,b^2\,x}{140\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^3/x^5,x)

[Out]

-(35*a^3 + 56*b^3*x^(3/2) + 120*a^2*b*x^(1/2) + 140*a*b^2*x)/(140*x^4)

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