3.2238 \(\int \frac{x}{\sqrt{a+b \sqrt{x}}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{4 a^3 \sqrt{a+b \sqrt{x}}}{b^4}+\frac{4 a^2 \left (a+b \sqrt{x}\right )^{3/2}}{b^4}+\frac{4 \left (a+b \sqrt{x}\right )^{7/2}}{7 b^4}-\frac{12 a \left (a+b \sqrt{x}\right )^{5/2}}{5 b^4} \]

[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 = 0.0978998, 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 \[ -\frac{4 a^3 \sqrt{a+b \sqrt{x}}}{b^4}+\frac{4 a^2 \left (a+b \sqrt{x}\right )^{3/2}}{b^4}+\frac{4 \left (a+b \sqrt{x}\right )^{7/2}}{7 b^4}-\frac{12 a \left (a+b \sqrt{x}\right )^{5/2}}{5 b^4} \]

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)

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Rubi in Sympy [A]  time = 14.4778, size = 78, normalized size = 0.93 \[ - \frac{4 a^{3} \sqrt{a + b \sqrt{x}}}{b^{4}} + \frac{4 a^{2} \left (a + b \sqrt{x}\right )^{\frac{3}{2}}}{b^{4}} - \frac{12 a \left (a + b \sqrt{x}\right )^{\frac{5}{2}}}{5 b^{4}} + \frac{4 \left (a + b \sqrt{x}\right )^{\frac{7}{2}}}{7 b^{4}} \]

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*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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Mathematica [A]  time = 0.0279793, size = 54, normalized size = 0.64 \[ \frac{4 \sqrt{a+b \sqrt{x}} \left (-16 a^3+8 a^2 b \sqrt{x}-6 a b^2 x+5 b^3 x^{3/2}\right )}{35 b^4} \]

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 = 0.003, size = 57, normalized size = 0.7 \[ 4\,{\frac{1/7\, \left ( a+b\sqrt{x} \right ) ^{7/2}-3/5\,a \left ( a+b\sqrt{x} \right ) ^{5/2}+ \left ( a+b\sqrt{x} \right ) ^{3/2}{a}^{2}-\sqrt{a+b\sqrt{x}}{a}^{3}}{{b}^{4}}} \]

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*(1/7*(a+b*x^(1/2))^(7/2)-3/5*a*(a+b*x^(1/2))^(5/2)+(a+b*x^(1/2))^(3/2)*a^2
-(a+b*x^(1/2))^(1/2)*a^3)

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Maxima [A]  time = 1.43805, size = 86, normalized size = 1.02 \[ \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}}}{7 \, b^{4}} - \frac{12 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a}{5 \, b^{4}} + \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{2}}{b^{4}} - \frac{4 \, \sqrt{b \sqrt{x} + a} a^{3}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(b*sqrt(x) + a),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 = 0.24437, size = 61, normalized size = 0.73 \[ -\frac{4 \,{\left (6 \, a b^{2} x + 16 \, a^{3} -{\left (5 \, b^{3} x + 8 \, a^{2} b\right )} \sqrt{x}\right )} \sqrt{b \sqrt{x} + a}}{35 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(b*sqrt(x) + a),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 [A]  time = 8.66146, size = 1872, normalized size = 22.29 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-64*a**(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) + 525*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) + 525*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 + 210*a**15*b**9*x**(21/2) + 35*a**14*b**10*x**11) + 9
60*a**(43/2)*b**2*x**9/(35*a**20*b**4*x**8 + 210*a**19*b**5*x**(17/2) + 525*a**1
8*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 + 70
0*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**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) + 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**1
4*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**10 + 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/XCAS [A]  time = 0.3005, size = 77, normalized size = 0.92 \[ \frac{4 \,{\left (5 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} - 21 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b \sqrt{x} + a} a^{3}\right )}}{35 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(b*sqrt(x) + a),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