3.22.29 \(\int \frac {(a+b \sqrt {x})^2}{x^5} \, dx\) [2129]

Optimal. Leaf size=32 \[ -\frac {a^2}{4 x^4}-\frac {4 a b}{7 x^{7/2}}-\frac {b^2}{3 x^3} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, 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^2}{4 x^4}-\frac {4 a b}{7 x^{7/2}}-\frac {b^2}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^2/x^5,x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.88 \begin {gather*} \frac {-21 a^2-48 a b \sqrt {x}-28 b^2 x}{84 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^2/x^5,x]

[Out]

(-21*a^2 - 48*a*b*Sqrt[x] - 28*b^2*x)/(84*x^4)

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Maple [A]
time = 0.21, size = 25, normalized size = 0.78

method	result	size
derivativedivides	\(-\frac {a^{2}}{4 x^{4}}-\frac {4 a b}{7 x^{\frac {7}{2}}}-\frac {b^{2}}{3 x^{3}}\)	\(25\)
default	\(-\frac {a^{2}}{4 x^{4}}-\frac {4 a b}{7 x^{\frac {7}{2}}}-\frac {b^{2}}{3 x^{3}}\)	\(25\)
trager	\(\frac {\left (x -1\right ) \left (3 a^{2} x^{3}+4 b^{2} x^{3}+3 a^{2} x^{2}+4 b^{2} x^{2}+3 a^{2} x +4 b^{2} x +3 a^{2}\right )}{12 x^{4}}-\frac {4 a b}{7 x^{\frac {7}{2}}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 24, normalized size = 0.75 \begin {gather*} -\frac {28 \, b^{2} x + 48 \, a b \sqrt {x} + 21 \, a^{2}}{84 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/84*(28*b^2*x + 48*a*b*sqrt(x) + 21*a^2)/x^4

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Fricas [A]
time = 0.38, size = 24, normalized size = 0.75 \begin {gather*} -\frac {28 \, b^{2} x + 48 \, a b \sqrt {x} + 21 \, a^{2}}{84 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/84*(28*b^2*x + 48*a*b*sqrt(x) + 21*a^2)/x^4

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Sympy [A]
time = 0.31, size = 29, normalized size = 0.91 \begin {gather*} - \frac {a^{2}}{4 x^{4}} - \frac {4 a b}{7 x^{\frac {7}{2}}} - \frac {b^{2}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.91, size = 24, normalized size = 0.75 \begin {gather*} -\frac {28 \, b^{2} x + 48 \, a b \sqrt {x} + 21 \, a^{2}}{84 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/84*(28*b^2*x + 48*a*b*sqrt(x) + 21*a^2)/x^4

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Mupad [B]
time = 0.03, size = 24, normalized size = 0.75 \begin {gather*} -\frac {28\,b^2\,x+21\,a^2+48\,a\,b\,\sqrt {x}}{84\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^2/x^5,x)

[Out]

-(28*b^2*x + 21*a^2 + 48*a*b*x^(1/2))/(84*x^4)

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