3.2138 \(\int \frac {(a+b \sqrt {x})^3}{x^4} \, dx\)

Optimal. Leaf size=47 \[ -\frac {a^3}{3 x^3}-\frac {6 a^2 b}{5 x^{5/2}}-\frac {3 a b^2}{2 x^2}-\frac {2 b^3}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 47, 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} \[ -\frac {6 a^2 b}{5 x^{5/2}}-\frac {a^3}{3 x^3}-\frac {3 a b^2}{2 x^2}-\frac {2 b^3}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3/x^4,x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 41, normalized size = 0.87 \[ -\frac {10 a^3+36 a^2 b \sqrt {x}+45 a b^2 x+20 b^3 x^{3/2}}{30 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3/x^4,x]

[Out]

-1/30*(10*a^3 + 36*a^2*b*Sqrt[x] + 45*a*b^2*x + 20*b^3*x^(3/2))/x^3

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fricas [A]  time = 1.05, size = 36, normalized size = 0.77 \[ -\frac {45 \, a b^{2} x + 10 \, a^{3} + 4 \, {\left (5 \, b^{3} x + 9 \, a^{2} b\right )} \sqrt {x}}{30 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^4,x, algorithm="fricas")

[Out]

-1/30*(45*a*b^2*x + 10*a^3 + 4*(5*b^3*x + 9*a^2*b)*sqrt(x))/x^3

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giac [A]  time = 0.15, size = 35, normalized size = 0.74 \[ -\frac {20 \, b^{3} x^{\frac {3}{2}} + 45 \, a b^{2} x + 36 \, a^{2} b \sqrt {x} + 10 \, a^{3}}{30 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^4,x, algorithm="giac")

[Out]

-1/30*(20*b^3*x^(3/2) + 45*a*b^2*x + 36*a^2*b*sqrt(x) + 10*a^3)/x^3

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maple [A]  time = 0.00, size = 36, normalized size = 0.77 \[ -\frac {2 b^{3}}{3 x^{\frac {3}{2}}}-\frac {3 a \,b^{2}}{2 x^{2}}-\frac {6 a^{2} b}{5 x^{\frac {5}{2}}}-\frac {a^{3}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^3/x^4,x)

[Out]

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

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maxima [A]  time = 0.84, size = 35, normalized size = 0.74 \[ -\frac {20 \, b^{3} x^{\frac {3}{2}} + 45 \, a b^{2} x + 36 \, a^{2} b \sqrt {x} + 10 \, a^{3}}{30 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^4,x, algorithm="maxima")

[Out]

-1/30*(20*b^3*x^(3/2) + 45*a*b^2*x + 36*a^2*b*sqrt(x) + 10*a^3)/x^3

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mupad [B]  time = 1.07, size = 35, normalized size = 0.74 \[ -\frac {10\,a^3+20\,b^3\,x^{3/2}+36\,a^2\,b\,\sqrt {x}+45\,a\,b^2\,x}{30\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^3/x^4,x)

[Out]

-(10*a^3 + 20*b^3*x^(3/2) + 36*a^2*b*x^(1/2) + 45*a*b^2*x)/(30*x^3)

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sympy [A]  time = 1.10, size = 46, normalized size = 0.98 \[ - \frac {a^{3}}{3 x^{3}} - \frac {6 a^{2} b}{5 x^{\frac {5}{2}}} - \frac {3 a b^{2}}{2 x^{2}} - \frac {2 b^{3}}{3 x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**3/x**4,x)

[Out]

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

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