3.2234 \(\int \sqrt{a+b \sqrt{x}} x \, dx\)
Optimal. Leaf size=88 \[ \frac{12 a^2 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^4}-\frac{4 a^3 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^4}+\frac{4 \left (a+b \sqrt{x}\right )^{9/2}}{9 b^4}-\frac{12 a \left (a+b \sqrt{x}\right )^{7/2}}{7 b^4} \]
[Out]
(-4*a^3*(a + b*Sqrt[x])^(3/2))/(3*b^4) + (12*a^2*(a + b*Sqrt[x])^(5/2))/(5*b^4) - (12*a*(a + b*Sqrt[x])^(7/2))
/(7*b^4) + (4*(a + b*Sqrt[x])^(9/2))/(9*b^4)
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Rubi [A] time = 0.037432, antiderivative size = 88, 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} \[ \frac{12 a^2 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^4}-\frac{4 a^3 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^4}+\frac{4 \left (a+b \sqrt{x}\right )^{9/2}}{9 b^4}-\frac{12 a \left (a+b \sqrt{x}\right )^{7/2}}{7 b^4} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + b*Sqrt[x]]*x,x]
[Out]
(-4*a^3*(a + b*Sqrt[x])^(3/2))/(3*b^4) + (12*a^2*(a + b*Sqrt[x])^(5/2))/(5*b^4) - (12*a*(a + b*Sqrt[x])^(7/2))
/(7*b^4) + (4*(a + b*Sqrt[x])^(9/2))/(9*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 \sqrt{a+b \sqrt{x}} x \, dx &=2 \operatorname{Subst}\left (\int x^3 \sqrt{a+b x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^3 \sqrt{a+b x}}{b^3}+\frac{3 a^2 (a+b x)^{3/2}}{b^3}-\frac{3 a (a+b x)^{5/2}}{b^3}+\frac{(a+b x)^{7/2}}{b^3}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 a^3 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^4}+\frac{12 a^2 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^4}-\frac{12 a \left (a+b \sqrt{x}\right )^{7/2}}{7 b^4}+\frac{4 \left (a+b \sqrt{x}\right )^{9/2}}{9 b^4}\\ \end{align*}
Mathematica [A] time = 0.0254862, size = 54, normalized size = 0.61 \[ \frac{4 \left (a+b \sqrt{x}\right )^{3/2} \left (24 a^2 b \sqrt{x}-16 a^3-30 a b^2 x+35 b^3 x^{3/2}\right )}{315 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + b*Sqrt[x]]*x,x]
[Out]
(4*(a + b*Sqrt[x])^(3/2)*(-16*a^3 + 24*a^2*b*Sqrt[x] - 30*a*b^2*x + 35*b^3*x^(3/2)))/(315*b^4)
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Maple [A] time = 0.002, size = 58, normalized size = 0.7 \begin{align*} 4\,{\frac{1/9\, \left ( a+b\sqrt{x} \right ) ^{9/2}-3/7\,a \left ( a+b\sqrt{x} \right ) ^{7/2}+3/5\,{a}^{2} \left ( a+b\sqrt{x} \right ) ^{5/2}-1/3\,{a}^{3} \left ( a+b\sqrt{x} \right ) ^{3/2}}{{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*(1/9*(a+b*x^(1/2))^(9/2)-3/7*a*(a+b*x^(1/2))^(7/2)+3/5*a^2*(a+b*x^(1/2))^(5/2)-1/3*a^3*(a+b*x^(1/2))^(3/
2))
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Maxima [A] time = 0.966574, size = 86, normalized size = 0.98 \begin{align*} \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{9}{2}}}{9 \, b^{4}} - \frac{12 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} a}{7 \, b^{4}} + \frac{12 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{4}} - \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{3}}{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]
4/9*(b*sqrt(x) + a)^(9/2)/b^4 - 12/7*(b*sqrt(x) + a)^(7/2)*a/b^4 + 12/5*(b*sqrt(x) + a)^(5/2)*a^2/b^4 - 4/3*(b
*sqrt(x) + a)^(3/2)*a^3/b^4
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Fricas [A] time = 1.29542, size = 134, normalized size = 1.52 \begin{align*} \frac{4 \,{\left (35 \, b^{4} x^{2} - 6 \, a^{2} b^{2} x - 16 \, a^{4} +{\left (5 \, a b^{3} x + 8 \, a^{3} b\right )} \sqrt{x}\right )} \sqrt{b \sqrt{x} + a}}{315 \, 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]
4/315*(35*b^4*x^2 - 6*a^2*b^2*x - 16*a^4 + (5*a*b^3*x + 8*a^3*b)*sqrt(x))*sqrt(b*sqrt(x) + a)/b^4
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Sympy [B] time = 3.45847, size = 1987, normalized size = 22.58 \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**(49/2)*x**8*sqrt(1 + b*sqrt(x)/a)/(315*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x*
*9 + 6300*a**17*b**7*x**(19/2) + 4725*a**16*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) +
64*a**(49/2)*x**8/(315*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x*
*(19/2) + 4725*a**16*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) - 352*a**(47/2)*b*x**(17/
2)*sqrt(1 + b*sqrt(x)/a)/(315*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*
b**7*x**(19/2) + 4725*a**16*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 384*a**(47/2)*b*
x**(17/2)/(315*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2)
+ 4725*a**16*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) - 792*a**(45/2)*b**2*x**9*sqrt(1
+ b*sqrt(x)/a)/(315*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(1
9/2) + 4725*a**16*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 960*a**(45/2)*b**2*x**9/(3
15*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*a**16
*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) - 924*a**(43/2)*b**3*x**(19/2)*sqrt(1 + b*sqr
t(x)/a)/(315*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) +
4725*a**16*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 1280*a**(43/2)*b**3*x**(19/2)/(31
5*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*a**16*
b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) - 420*a**(41/2)*b**4*x**10*sqrt(1 + b*sqrt(x)/
a)/(315*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*
a**16*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 960*a**(41/2)*b**4*x**10/(315*a**20*b*
*4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*a**16*b**8*x**10
+ 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 756*a**(39/2)*b**5*x**(21/2)*sqrt(1 + b*sqrt(x)/a)/(31
5*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*a**16*
b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 384*a**(39/2)*b**5*x**(21/2)/(315*a**20*b**4
*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*a**16*b**8*x**10 +
1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 2268*a**(37/2)*b**6*x**11*sqrt(1 + b*sqrt(x)/a)/(315*a**
20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*a**16*b**8*
x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 64*a**(37/2)*b**6*x**11/(315*a**20*b**4*x**8 + 18
90*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*a**16*b**8*x**10 + 1890*a**1
5*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 2988*a**(35/2)*b**7*x**(23/2)*sqrt(1 + b*sqrt(x)/a)/(315*a**20*b**
4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*a**16*b**8*x**10
+ 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 2196*a**(33/2)*b**8*x**12*sqrt(1 + b*sqrt(x)/a)/(315*a*
*20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*a**16*b**8
*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 860*a**(31/2)*b**9*x**(25/2)*sqrt(1 + b*sqrt(x)/
a)/(315*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2) + 4725*
a**16*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11) + 140*a**(29/2)*b**10*x**13*sqrt(1 + b*s
qrt(x)/a)/(315*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6300*a**17*b**7*x**(19/2)
+ 4725*a**16*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 315*a**14*b**10*x**11)
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Giac [A] time = 1.10387, size = 77, normalized size = 0.88 \begin{align*} \frac{4 \,{\left (35 \,{\left (b \sqrt{x} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{3}\right )}}{315 \, 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]
4/315*(35*(b*sqrt(x) + a)^(9/2) - 135*(b*sqrt(x) + a)^(7/2)*a + 189*(b*sqrt(x) + a)^(5/2)*a^2 - 105*(b*sqrt(x)
+ a)^(3/2)*a^3)/b^4