3.335 \(\int \frac{x^3}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{2 a^3 \sqrt{a+b x}}{b^4}+\frac{2 a^2 (a+b x)^{3/2}}{b^4}+\frac{2 (a+b x)^{7/2}}{7 b^4}-\frac{6 a (a+b x)^{5/2}}{5 b^4} \]

[Out]

(-2*a^3*Sqrt[a + b*x])/b^4 + (2*a^2*(a + b*x)^(3/2))/b^4 - (6*a*(a + b*x)^(5/2))
/(5*b^4) + (2*(a + b*x)^(7/2))/(7*b^4)

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Rubi [A]  time = 0.0507547, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 a^3 \sqrt{a+b x}}{b^4}+\frac{2 a^2 (a+b x)^{3/2}}{b^4}+\frac{2 (a+b x)^{7/2}}{7 b^4}-\frac{6 a (a+b x)^{5/2}}{5 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a + b*x],x]

[Out]

(-2*a^3*Sqrt[a + b*x])/b^4 + (2*a^2*(a + b*x)^(3/2))/b^4 - (6*a*(a + b*x)^(5/2))
/(5*b^4) + (2*(a + b*x)^(7/2))/(7*b^4)

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Rubi in Sympy [A]  time = 10.9907, size = 65, normalized size = 0.96 \[ - \frac{2 a^{3} \sqrt{a + b x}}{b^{4}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{3}{2}}}{b^{4}} - \frac{6 a \left (a + b x\right )^{\frac{5}{2}}}{5 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}}}{7 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x+a)**(1/2),x)

[Out]

-2*a**3*sqrt(a + b*x)/b**4 + 2*a**2*(a + b*x)**(3/2)/b**4 - 6*a*(a + b*x)**(5/2)
/(5*b**4) + 2*(a + b*x)**(7/2)/(7*b**4)

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Mathematica [A]  time = 0.0211253, size = 46, normalized size = 0.68 \[ \frac{2 \sqrt{a+b x} \left (-16 a^3+8 a^2 b x-6 a b^2 x^2+5 b^3 x^3\right )}{35 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(-16*a^3 + 8*a^2*b*x - 6*a*b^2*x^2 + 5*b^3*x^3))/(35*b^4)

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Maple [A]  time = 0.009, size = 43, normalized size = 0.6 \[ -{\frac{-10\,{b}^{3}{x}^{3}+12\,a{b}^{2}{x}^{2}-16\,{a}^{2}bx+32\,{a}^{3}}{35\,{b}^{4}}\sqrt{bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)^(1/2),x)

[Out]

-2/35*(b*x+a)^(1/2)*(-5*b^3*x^3+6*a*b^2*x^2-8*a^2*b*x+16*a^3)/b^4

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Maxima [A]  time = 1.35091, size = 76, normalized size = 1.12 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}}}{7 \, b^{4}} - \frac{6 \,{\left (b x + a\right )}^{\frac{5}{2}} a}{5 \, b^{4}} + \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}}{b^{4}} - \frac{2 \, \sqrt{b x + a} a^{3}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x + a),x, algorithm="maxima")

[Out]

2/7*(b*x + a)^(7/2)/b^4 - 6/5*(b*x + a)^(5/2)*a/b^4 + 2*(b*x + a)^(3/2)*a^2/b^4
- 2*sqrt(b*x + a)*a^3/b^4

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Fricas [A]  time = 0.231927, size = 57, normalized size = 0.84 \[ \frac{2 \,{\left (5 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 8 \, a^{2} b x - 16 \, a^{3}\right )} \sqrt{b x + a}}{35 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x + a),x, algorithm="fricas")

[Out]

2/35*(5*b^3*x^3 - 6*a*b^2*x^2 + 8*a^2*b*x - 16*a^3)*sqrt(b*x + a)/b^4

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Sympy [A]  time = 4.21008, size = 1640, normalized size = 24.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x+a)**(1/2),x)

[Out]

-32*a**(47/2)*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6
*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**
14*b**10*x**6) + 32*a**(47/2)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6
*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**
14*b**10*x**6) - 176*a**(45/2)*b*x*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b*
*5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**
15*b**9*x**5 + 35*a**14*b**10*x**6) + 192*a**(45/2)*b*x/(35*a**20*b**4 + 210*a**
19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 21
0*a**15*b**9*x**5 + 35*a**14*b**10*x**6) - 396*a**(43/2)*b**2*x**2*sqrt(1 + b*x/
a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3
+ 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 480*a**(43
/2)*b**2*x**2/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**1
7*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) -
462*a**(41/2)*b**3*x**3*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525
*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x*
*5 + 35*a**14*b**10*x**6) + 640*a**(41/2)*b**3*x**3/(35*a**20*b**4 + 210*a**19*b
**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a*
*15*b**9*x**5 + 35*a**14*b**10*x**6) - 280*a**(39/2)*b**4*x**4*sqrt(1 + b*x/a)/(
35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 5
25*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 480*a**(39/2)*
b**4*x**4/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b*
*7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) - 42*
a**(37/2)*b**5*x**5*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**1
8*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 +
35*a**14*b**10*x**6) + 192*a**(37/2)*b**5*x**5/(35*a**20*b**4 + 210*a**19*b**5*x
+ 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b
**9*x**5 + 35*a**14*b**10*x**6) + 84*a**(35/2)*b**6*x**6*sqrt(1 + b*x/a)/(35*a**
20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**
16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 32*a**(35/2)*b**6*x*
*6/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3
+ 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 94*a**(33/
2)*b**7*x**7*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*
x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**1
4*b**10*x**6) + 48*a**(31/2)*b**8*x**8*sqrt(1 + b*x/a)/(35*a**20*b**4 + 210*a**1
9*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 + 525*a**16*b**8*x**4 + 210
*a**15*b**9*x**5 + 35*a**14*b**10*x**6) + 10*a**(29/2)*b**9*x**9*sqrt(1 + b*x/a)
/(35*a**20*b**4 + 210*a**19*b**5*x + 525*a**18*b**6*x**2 + 700*a**17*b**7*x**3 +
525*a**16*b**8*x**4 + 210*a**15*b**9*x**5 + 35*a**14*b**10*x**6)

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GIAC/XCAS [A]  time = 0.20902, size = 82, normalized size = 1.21 \[ \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{18} - 21 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{18} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{18} - 35 \, \sqrt{b x + a} a^{3} b^{18}\right )}}{35 \, b^{22}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x + a),x, algorithm="giac")

[Out]

2/35*(5*(b*x + a)^(7/2)*b^18 - 21*(b*x + a)^(5/2)*a*b^18 + 35*(b*x + a)^(3/2)*a^
2*b^18 - 35*sqrt(b*x + a)*a^3*b^18)/b^22