\(\int \frac {x^2}{(a+b \sqrt {x})^2} \, dx\) [2200]

Optimal result

Integrand size = 15, antiderivative size = 83 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {2 a^5}{b^6 \left (a+b \sqrt {x}\right )}-\frac {8 a^3 \sqrt {x}}{b^5}+\frac {3 a^2 x}{b^4}-\frac {4 a x^{3/2}}{3 b^3}+\frac {x^2}{2 b^2}+\frac {10 a^4 \log \left (a+b \sqrt {x}\right )}{b^6} \]

[Out]

3*a^2*x/b^4-4/3*a*x^(3/2)/b^3+1/2*x^2/b^2+10*a^4*ln(a+b*x^(1/2))/b^6-8*a^3*x^(1/2)/b^5+2*a^5/b^6/(a+b*x^(1/2))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {2 a^5}{b^6 \left (a+b \sqrt {x}\right )}+\frac {10 a^4 \log \left (a+b \sqrt {x}\right )}{b^6}-\frac {8 a^3 \sqrt {x}}{b^5}+\frac {3 a^2 x}{b^4}-\frac {4 a x^{3/2}}{3 b^3}+\frac {x^2}{2 b^2} \]

[In]

Int[x^2/(a + b*Sqrt[x])^2,x]

[Out]

(2*a^5)/(b^6*(a + b*Sqrt[x])) - (8*a^3*Sqrt[x])/b^5 + (3*a^2*x)/b^4 - (4*a*x^(3/2))/(3*b^3) + x^2/(2*b^2) + (1
0*a^4*Log[a + b*Sqrt[x]])/b^6

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.14 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {12 a^5-48 a^4 b \sqrt {x}-30 a^3 b^2 x+10 a^2 b^3 x^{3/2}-5 a b^4 x^2+3 b^5 x^{5/2}}{6 b^6 \left (a+b \sqrt {x}\right )}+\frac {10 a^4 \log \left (a+b \sqrt {x}\right )}{b^6} \]

[In]

Integrate[x^2/(a + b*Sqrt[x])^2,x]

[Out]

(12*a^5 - 48*a^4*b*Sqrt[x] - 30*a^3*b^2*x + 10*a^2*b^3*x^(3/2) - 5*a*b^4*x^2 + 3*b^5*x^(5/2))/(6*b^6*(a + b*Sq
rt[x])) + (10*a^4*Log[a + b*Sqrt[x]])/b^6

Maple [A] (verified)

Time = 3.63 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {b^{3} x^{2}}{4}+\frac {2 a \,b^{2} x^{\frac {3}{2}}}{3}-\frac {3 a^{2} b x}{2}+4 a^{3} \sqrt {x}\right )}{b^{5}}+\frac {2 a^{5}}{b^{6} \left (a +b \sqrt {x}\right )}+\frac {10 a^{4} \ln \left (a +b \sqrt {x}\right )}{b^{6}}\)	\(73\)
default	\(-\frac {2 \left (-\frac {b^{3} x^{2}}{4}+\frac {2 a \,b^{2} x^{\frac {3}{2}}}{3}-\frac {3 a^{2} b x}{2}+4 a^{3} \sqrt {x}\right )}{b^{5}}+\frac {2 a^{5}}{b^{6} \left (a +b \sqrt {x}\right )}+\frac {10 a^{4} \ln \left (a +b \sqrt {x}\right )}{b^{6}}\)	\(73\)

[In]

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

[Out]

-2/b^5*(-1/4*b^3*x^2+2/3*a*b^2*x^(3/2)-3/2*a^2*b*x+4*a^3*x^(1/2))+2*a^5/b^6/(a+b*x^(1/2))+10*a^4*ln(a+b*x^(1/2
))/b^6

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.28 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {3 \, b^{6} x^{3} + 15 \, a^{2} b^{4} x^{2} - 18 \, a^{4} b^{2} x - 12 \, a^{6} + 60 \, {\left (a^{4} b^{2} x - a^{6}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (2 \, a b^{5} x^{2} + 10 \, a^{3} b^{3} x - 15 \, a^{5} b\right )} \sqrt {x}}{6 \, {\left (b^{8} x - a^{2} b^{6}\right )}} \]

[In]

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

[Out]

1/6*(3*b^6*x^3 + 15*a^2*b^4*x^2 - 18*a^4*b^2*x - 12*a^6 + 60*(a^4*b^2*x - a^6)*log(b*sqrt(x) + a) - 4*(2*a*b^5
*x^2 + 10*a^3*b^3*x - 15*a^5*b)*sqrt(x))/(b^8*x - a^2*b^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (80) = 160\).

Time = 0.28 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.55 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^2} \, dx=\begin {cases} \frac {60 a^{5} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a b^{6} + 6 b^{7} \sqrt {x}} + \frac {60 a^{5}}{6 a b^{6} + 6 b^{7} \sqrt {x}} + \frac {60 a^{4} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a b^{6} + 6 b^{7} \sqrt {x}} - \frac {30 a^{3} b^{2} x}{6 a b^{6} + 6 b^{7} \sqrt {x}} + \frac {10 a^{2} b^{3} x^{\frac {3}{2}}}{6 a b^{6} + 6 b^{7} \sqrt {x}} - \frac {5 a b^{4} x^{2}}{6 a b^{6} + 6 b^{7} \sqrt {x}} + \frac {3 b^{5} x^{\frac {5}{2}}}{6 a b^{6} + 6 b^{7} \sqrt {x}} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3 a^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((60*a**5*log(a/b + sqrt(x))/(6*a*b**6 + 6*b**7*sqrt(x)) + 60*a**5/(6*a*b**6 + 6*b**7*sqrt(x)) + 60*a
**4*b*sqrt(x)*log(a/b + sqrt(x))/(6*a*b**6 + 6*b**7*sqrt(x)) - 30*a**3*b**2*x/(6*a*b**6 + 6*b**7*sqrt(x)) + 10
*a**2*b**3*x**(3/2)/(6*a*b**6 + 6*b**7*sqrt(x)) - 5*a*b**4*x**2/(6*a*b**6 + 6*b**7*sqrt(x)) + 3*b**5*x**(5/2)/
(6*a*b**6 + 6*b**7*sqrt(x)), Ne(b, 0)), (x**3/(3*a**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.14 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {10 \, a^{4} \log \left (b \sqrt {x} + a\right )}{b^{6}} + \frac {{\left (b \sqrt {x} + a\right )}^{4}}{2 \, b^{6}} - \frac {10 \, {\left (b \sqrt {x} + a\right )}^{3} a}{3 \, b^{6}} + \frac {10 \, {\left (b \sqrt {x} + a\right )}^{2} a^{2}}{b^{6}} - \frac {20 \, {\left (b \sqrt {x} + a\right )} a^{3}}{b^{6}} + \frac {2 \, a^{5}}{{\left (b \sqrt {x} + a\right )} b^{6}} \]

[In]

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

[Out]

10*a^4*log(b*sqrt(x) + a)/b^6 + 1/2*(b*sqrt(x) + a)^4/b^6 - 10/3*(b*sqrt(x) + a)^3*a/b^6 + 10*(b*sqrt(x) + a)^
2*a^2/b^6 - 20*(b*sqrt(x) + a)*a^3/b^6 + 2*a^5/((b*sqrt(x) + a)*b^6)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {10 \, a^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{6}} + \frac {2 \, a^{5}}{{\left (b \sqrt {x} + a\right )} b^{6}} + \frac {3 \, b^{6} x^{2} - 8 \, a b^{5} x^{\frac {3}{2}} + 18 \, a^{2} b^{4} x - 48 \, a^{3} b^{3} \sqrt {x}}{6 \, b^{8}} \]

[In]

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

[Out]

10*a^4*log(abs(b*sqrt(x) + a))/b^6 + 2*a^5/((b*sqrt(x) + a)*b^6) + 1/6*(3*b^6*x^2 - 8*a*b^5*x^(3/2) + 18*a^2*b
^4*x - 48*a^3*b^3*sqrt(x))/b^8

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^2} \, dx=\frac {x^2}{2\,b^2}+\frac {2\,a^5}{b\,\left (a\,b^5+b^6\,\sqrt {x}\right )}+\frac {3\,a^2\,x}{b^4}-\frac {4\,a\,x^{3/2}}{3\,b^3}+\frac {10\,a^4\,\ln \left (a+b\,\sqrt {x}\right )}{b^6}-\frac {8\,a^3\,\sqrt {x}}{b^5} \]

[In]

int(x^2/(a + b*x^(1/2))^2,x)

[Out]

x^2/(2*b^2) + (2*a^5)/(b*(a*b^5 + b^6*x^(1/2))) + (3*a^2*x)/b^4 - (4*a*x^(3/2))/(3*b^3) + (10*a^4*log(a + b*x^
(1/2)))/b^6 - (8*a^3*x^(1/2))/b^5