3.2240 \(\int \frac{x}{\sqrt{a+b \sqrt{x}}} \, dx\)
Optimal. Leaf size=84 \[ \frac{4 a^2 \left (a+b \sqrt{x}\right )^{3/2}}{b^4}-\frac{4 a^3 \sqrt{a+b \sqrt{x}}}{b^4}+\frac{4 \left (a+b \sqrt{x}\right )^{7/2}}{7 b^4}-\frac{12 a \left (a+b \sqrt{x}\right )^{5/2}}{5 b^4} \]
[Out]
(-4*a^3*Sqrt[a + b*Sqrt[x]])/b^4 + (4*a^2*(a + b*Sqrt[x])^(3/2))/b^4 - (12*a*(a + b*Sqrt[x])^(5/2))/(5*b^4) +
(4*(a + b*Sqrt[x])^(7/2))/(7*b^4)
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Rubi [A] time = 0.0382357, antiderivative size = 84, normalized size of antiderivative = 1.,
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} \[ \frac{4 a^2 \left (a+b \sqrt{x}\right )^{3/2}}{b^4}-\frac{4 a^3 \sqrt{a+b \sqrt{x}}}{b^4}+\frac{4 \left (a+b \sqrt{x}\right )^{7/2}}{7 b^4}-\frac{12 a \left (a+b \sqrt{x}\right )^{5/2}}{5 b^4} \]
Antiderivative was successfully verified.
[In]
Int[x/Sqrt[a + b*Sqrt[x]],x]
[Out]
(-4*a^3*Sqrt[a + b*Sqrt[x]])/b^4 + (4*a^2*(a + b*Sqrt[x])^(3/2))/b^4 - (12*a*(a + b*Sqrt[x])^(5/2))/(5*b^4) +
(4*(a + b*Sqrt[x])^(7/2))/(7*b^4)
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]]
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])
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a+b \sqrt{x}}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b x}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 \sqrt{a+b x}}+\frac{3 a^2 \sqrt{a+b x}}{b^3}-\frac{3 a (a+b x)^{3/2}}{b^3}+\frac{(a+b x)^{5/2}}{b^3}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 a^3 \sqrt{a+b \sqrt{x}}}{b^4}+\frac{4 a^2 \left (a+b \sqrt{x}\right )^{3/2}}{b^4}-\frac{12 a \left (a+b \sqrt{x}\right )^{5/2}}{5 b^4}+\frac{4 \left (a+b \sqrt{x}\right )^{7/2}}{7 b^4}\\ \end{align*}
Mathematica [A] time = 0.0282495, size = 54, normalized size = 0.64 \[ \frac{4 \sqrt{a+b \sqrt{x}} \left (8 a^2 b \sqrt{x}-16 a^3-6 a b^2 x+5 b^3 x^{3/2}\right )}{35 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x/Sqrt[a + b*Sqrt[x]],x]
[Out]
(4*Sqrt[a + b*Sqrt[x]]*(-16*a^3 + 8*a^2*b*Sqrt[x] - 6*a*b^2*x + 5*b^3*x^(3/2)))/(35*b^4)
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Maple [A] time = 0.002, size = 57, normalized size = 0.7 \begin{align*} 4\,{\frac{1/7\, \left ( a+b\sqrt{x} \right ) ^{7/2}-3/5\,a \left ( a+b\sqrt{x} \right ) ^{5/2}+{a}^{2} \left ( a+b\sqrt{x} \right ) ^{3/2}-{a}^{3}\sqrt{a+b\sqrt{x}}}{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(a+b*x^(1/2))^(1/2),x)
[Out]
4/b^4*(1/7*(a+b*x^(1/2))^(7/2)-3/5*a*(a+b*x^(1/2))^(5/2)+a^2*(a+b*x^(1/2))^(3/2)-a^3*(a+b*x^(1/2))^(1/2))
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Maxima [A] time = 0.969891, size = 86, normalized size = 1.02 \begin{align*} \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}}}{7 \, b^{4}} - \frac{12 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a}{5 \, b^{4}} + \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{2}}{b^{4}} - \frac{4 \, \sqrt{b \sqrt{x} + a} a^{3}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(a+b*x^(1/2))^(1/2),x, algorithm="maxima")
[Out]
4/7*(b*sqrt(x) + a)^(7/2)/b^4 - 12/5*(b*sqrt(x) + a)^(5/2)*a/b^4 + 4*(b*sqrt(x) + a)^(3/2)*a^2/b^4 - 4*sqrt(b*
sqrt(x) + a)*a^3/b^4
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Fricas [A] time = 1.30542, size = 111, normalized size = 1.32 \begin{align*} -\frac{4 \,{\left (6 \, a b^{2} x + 16 \, a^{3} -{\left (5 \, b^{3} x + 8 \, a^{2} b\right )} \sqrt{x}\right )} \sqrt{b \sqrt{x} + a}}{35 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(a+b*x^(1/2))^(1/2),x, algorithm="fricas")
[Out]
-4/35*(6*a*b^2*x + 16*a^3 - (5*b^3*x + 8*a^2*b)*sqrt(x))*sqrt(b*sqrt(x) + a)/b^4
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Sympy [B] time = 2.33505, size = 1872, normalized size = 22.29 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(a+b*x**(1/2))**(1/2),x)
[Out]
-64*a**(47/2)*x**8*sqrt(1 + b*sqrt(x)/a)/(35*a**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9
+ 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) + 64*a**(
47/2)*x**8/(35*a**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 5
25*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) - 352*a**(45/2)*b*x**(17/2)*sqrt(1 + b*
sqrt(x)/a)/(35*a**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 5
25*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) + 384*a**(45/2)*b*x**(17/2)/(35*a**20*b
**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x**10 +
210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) - 792*a**(43/2)*b**2*x**9*sqrt(1 + b*sqrt(x)/a)/(35*a**20*b**
4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x**10 + 21
0*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) + 960*a**(43/2)*b**2*x**9/(35*a**20*b**4*x**8 + 210*a**19*b**5*
x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2) +
35*a**14*b**10*x**11) - 924*a**(41/2)*b**3*x**(19/2)*sqrt(1 + b*sqrt(x)/a)/(35*a**20*b**4*x**8 + 210*a**19*b*
*5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2
) + 35*a**14*b**10*x**11) + 1280*a**(41/2)*b**3*x**(19/2)/(35*a**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525
*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10
*x**11) - 560*a**(39/2)*b**4*x**10*sqrt(1 + b*sqrt(x)/a)/(35*a**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*
a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*
x**11) + 960*a**(39/2)*b**4*x**10/(35*a**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a
**17*b**7*x**(19/2) + 525*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) - 84*a**(37/2)*b
**5*x**(21/2)*sqrt(1 + b*sqrt(x)/a)/(35*a**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700
*a**17*b**7*x**(19/2) + 525*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) + 384*a**(37/2
)*b**5*x**(21/2)/(35*a**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/
2) + 525*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) + 168*a**(35/2)*b**6*x**11*sqrt(1
+ b*sqrt(x)/a)/(35*a**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2
) + 525*a**16*b**8*x**10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) + 64*a**(35/2)*b**6*x**11/(35*a**2
0*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x**10
+ 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) + 188*a**(33/2)*b**7*x**(23/2)*sqrt(1 + b*sqrt(x)/a)/(35*a
**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x*
*10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) + 96*a**(31/2)*b**8*x**12*sqrt(1 + b*sqrt(x)/a)/(35*a**
20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x**1
0 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) + 20*a**(29/2)*b**9*x**(25/2)*sqrt(1 + b*sqrt(x)/a)/(35*a
**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**18*b**6*x**9 + 700*a**17*b**7*x**(19/2) + 525*a**16*b**8*x*
*10 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11)
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Giac [A] time = 1.09634, size = 77, normalized size = 0.92 \begin{align*} \frac{4 \,{\left (5 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} - 21 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b \sqrt{x} + a} a^{3}\right )}}{35 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(a+b*x^(1/2))^(1/2),x, algorithm="giac")
[Out]
4/35*(5*(b*sqrt(x) + a)^(7/2) - 21*(b*sqrt(x) + a)^(5/2)*a + 35*(b*sqrt(x) + a)^(3/2)*a^2 - 35*sqrt(b*sqrt(x)
+ a)*a^3)/b^4