3.2232 \(\int \sqrt{a+b \sqrt{x}} x \, dx\)
Optimal. Leaf size=88 \[ -\frac{4 a^3 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^4}+\frac{12 a^2 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^4}+\frac{4 \left (a+b \sqrt{x}\right )^{9/2}}{9 b^4}-\frac{12 a \left (a+b \sqrt{x}\right )^{7/2}}{7 b^4} \]
[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 = 0.0989765, 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
\[ -\frac{4 a^3 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^4}+\frac{12 a^2 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^4}+\frac{4 \left (a+b \sqrt{x}\right )^{9/2}}{9 b^4}-\frac{12 a \left (a+b \sqrt{x}\right )^{7/2}}{7 b^4} \]
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)
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Rubi in Sympy [A] time = 14.387, size = 82, normalized size = 0.93 \[ - \frac{4 a^{3} \left (a + b \sqrt{x}\right )^{\frac{3}{2}}}{3 b^{4}} + \frac{12 a^{2} \left (a + b \sqrt{x}\right )^{\frac{5}{2}}}{5 b^{4}} - \frac{12 a \left (a + b \sqrt{x}\right )^{\frac{7}{2}}}{7 b^{4}} + \frac{4 \left (a + b \sqrt{x}\right )^{\frac{9}{2}}}{9 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(a+b*x**(1/2))**(1/2),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)
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Mathematica [A] time = 0.0280919, size = 65, normalized size = 0.74 \[ \frac{4 \sqrt{a+b \sqrt{x}} \left (-16 a^4+8 a^3 b \sqrt{x}-6 a^2 b^2 x+5 a b^3 x^{3/2}+35 b^4 x^2\right )}{315 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*Sqrt[x]]*x,x]
[Out]
(4*Sqrt[a + b*Sqrt[x]]*(-16*a^4 + 8*a^3*b*Sqrt[x] - 6*a^2*b^2*x + 5*a*b^3*x^(3/2
) + 35*b^4*x^2))/(315*b^4)
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Maple [A] time = 0.003, size = 58, normalized size = 0.7 \[ 4\,{\frac{1/9\, \left ( a+b\sqrt{x} \right ) ^{9/2}-3/7\,a \left ( a+b\sqrt{x} \right ) ^{7/2}+3/5\, \left ( a+b\sqrt{x} \right ) ^{5/2}{a}^{2}-1/3\, \left ( a+b\sqrt{x} \right ) ^{3/2}{a}^{3}}{{b}^{4}}} \]
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+b*x^(1/2))^(5/2)
*a^2-1/3*(a+b*x^(1/2))^(3/2)*a^3)
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Maxima [A] time = 1.44211, size = 86, normalized size = 0.98 \[ \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{9}{2}}}{9 \, b^{4}} - \frac{12 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} a}{7 \, b^{4}} + \frac{12 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{4}} - \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{3}}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(x) + a)*x,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 = 0.243527, size = 74, normalized size = 0.84 \[ \frac{4 \,{\left (35 \, b^{4} x^{2} - 6 \, a^{2} b^{2} x - 16 \, a^{4} +{\left (5 \, a b^{3} x + 8 \, a^{3} b\right )} \sqrt{x}\right )} \sqrt{b \sqrt{x} + a}}{315 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(x) + a)*x,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 [A] time = 8.59073, size = 1987, normalized size = 22.58 \[ \text{result too large to display} \]
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/(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**1
4*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**(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
) + 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**(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**(45/2)*b**2*x**9/(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) - 924
*a**(43/2)*b**3*x**(19/2)*sqrt(1 + b*sqrt(x)/a)/(315*a**20*b**4*x**8 + 1890*a**1
9*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)/(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) - 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 + 630
0*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)/(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**(39/2)*b**5*x**(2
1/2)/(315*a**20*b**4*x**8 + 1890*a**19*b**5*x**(17/2) + 4725*a**18*b**6*x**9 + 6
300*a**17*b**7*x**(19/2) + 4725*a**16*b**8*x**10 + 1890*a**15*b**9*x**(21/2) + 3
15*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**1
0*x**11) + 64*a**(37/2)*b**6*x**11/(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) + 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) + 472
5*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**(2
1/2) + 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 + 630
0*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*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)
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GIAC/XCAS [A] time = 0.259561, size = 77, normalized size = 0.88 \[ \frac{4 \,{\left (35 \,{\left (b \sqrt{x} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{3}\right )}}{315 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(x) + a)*x,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