3.334 \(\int \frac{x^4}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=89 \[ \frac{2 a^4 \sqrt{a+b x}}{b^5}-\frac{8 a^3 (a+b x)^{3/2}}{3 b^5}+\frac{12 a^2 (a+b x)^{5/2}}{5 b^5}+\frac{2 (a+b x)^{9/2}}{9 b^5}-\frac{8 a (a+b x)^{7/2}}{7 b^5} \]

[Out]

(2*a^4*Sqrt[a + b*x])/b^5 - (8*a^3*(a + b*x)^(3/2))/(3*b^5) + (12*a^2*(a + b*x)^
(5/2))/(5*b^5) - (8*a*(a + b*x)^(7/2))/(7*b^5) + (2*(a + b*x)^(9/2))/(9*b^5)

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Rubi [A]  time = 0.0620837, antiderivative size = 89, 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 \[ \frac{2 a^4 \sqrt{a+b x}}{b^5}-\frac{8 a^3 (a+b x)^{3/2}}{3 b^5}+\frac{12 a^2 (a+b x)^{5/2}}{5 b^5}+\frac{2 (a+b x)^{9/2}}{9 b^5}-\frac{8 a (a+b x)^{7/2}}{7 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[a + b*x],x]

[Out]

(2*a^4*Sqrt[a + b*x])/b^5 - (8*a^3*(a + b*x)^(3/2))/(3*b^5) + (12*a^2*(a + b*x)^
(5/2))/(5*b^5) - (8*a*(a + b*x)^(7/2))/(7*b^5) + (2*(a + b*x)^(9/2))/(9*b^5)

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Rubi in Sympy [A]  time = 13.7062, size = 85, normalized size = 0.96 \[ \frac{2 a^{4} \sqrt{a + b x}}{b^{5}} - \frac{8 a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3 b^{5}} + \frac{12 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5 b^{5}} - \frac{8 a \left (a + b x\right )^{\frac{7}{2}}}{7 b^{5}} + \frac{2 \left (a + b x\right )^{\frac{9}{2}}}{9 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x+a)**(1/2),x)

[Out]

2*a**4*sqrt(a + b*x)/b**5 - 8*a**3*(a + b*x)**(3/2)/(3*b**5) + 12*a**2*(a + b*x)
**(5/2)/(5*b**5) - 8*a*(a + b*x)**(7/2)/(7*b**5) + 2*(a + b*x)**(9/2)/(9*b**5)

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Mathematica [A]  time = 0.0258089, size = 57, normalized size = 0.64 \[ \frac{2 \sqrt{a+b x} \left (128 a^4-64 a^3 b x+48 a^2 b^2 x^2-40 a b^3 x^3+35 b^4 x^4\right )}{315 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(128*a^4 - 64*a^3*b*x + 48*a^2*b^2*x^2 - 40*a*b^3*x^3 + 35*b^4*
x^4))/(315*b^5)

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Maple [A]  time = 0.01, size = 54, normalized size = 0.6 \[{\frac{70\,{x}^{4}{b}^{4}-80\,a{x}^{3}{b}^{3}+96\,{a}^{2}{x}^{2}{b}^{2}-128\,{a}^{3}xb+256\,{a}^{4}}{315\,{b}^{5}}\sqrt{bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)^(1/2),x)

[Out]

2/315*(b*x+a)^(1/2)*(35*b^4*x^4-40*a*b^3*x^3+48*a^2*b^2*x^2-64*a^3*b*x+128*a^4)/
b^5

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Maxima [A]  time = 1.34814, size = 96, normalized size = 1.08 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{9}{2}}}{9 \, b^{5}} - \frac{8 \,{\left (b x + a\right )}^{\frac{7}{2}} a}{7 \, b^{5}} + \frac{12 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{5}} - \frac{8 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}}{3 \, b^{5}} + \frac{2 \, \sqrt{b x + a} a^{4}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(b*x + a),x, algorithm="maxima")

[Out]

2/9*(b*x + a)^(9/2)/b^5 - 8/7*(b*x + a)^(7/2)*a/b^5 + 12/5*(b*x + a)^(5/2)*a^2/b
^5 - 8/3*(b*x + a)^(3/2)*a^3/b^5 + 2*sqrt(b*x + a)*a^4/b^5

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Fricas [A]  time = 0.219371, size = 72, normalized size = 0.81 \[ \frac{2 \,{\left (35 \, b^{4} x^{4} - 40 \, a b^{3} x^{3} + 48 \, a^{2} b^{2} x^{2} - 64 \, a^{3} b x + 128 \, a^{4}\right )} \sqrt{b x + a}}{315 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(b*x + a),x, algorithm="fricas")

[Out]

2/315*(35*b^4*x^4 - 40*a*b^3*x^3 + 48*a^2*b^2*x^2 - 64*a^3*b*x + 128*a^4)*sqrt(b
*x + a)/b^5

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Sympy [A]  time = 7.07812, size = 3755, normalized size = 42.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x+a)**(1/2),x)

[Out]

256*a**(89/2)*sqrt(1 + b*x/a)/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*
b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x*
*5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 +
3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10) - 256*a**(89/2)/(315*a**40*b**5 +
3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36
*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12
*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10)
+ 2432*a**(87/2)*b*x*sqrt(1 + b*x/a)/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175
*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b
**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b**13*
x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10) - 2560*a**(87/2)*b*x/(315*
a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 +
66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800
*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b
**15*x**10) + 10336*a**(85/2)*b**2*x**2*sqrt(1 + b*x/a)/(315*a**40*b**5 + 3150*a
**39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x
**4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 +
14175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10) - 11520
*a**(85/2)*b**2*x**2/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2
+ 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 6615
0*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**3
1*b**14*x**9 + 315*a**30*b**15*x**10) + 25840*a**(83/2)*b**3*x**3*sqrt(1 + b*x/a
)/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8
*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6
+ 37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*
a**30*b**15*x**10) - 30720*a**(83/2)*b**3*x**3/(315*a**40*b**5 + 3150*a**39*b**6
*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 793
80*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a*
*32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10) + 41990*a**(81/2
)*b**4*x**4*sqrt(1 + b*x/a)/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b*
*7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5
+ 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 31
50*a**31*b**14*x**9 + 315*a**30*b**15*x**10) - 53760*a**(81/2)*b**4*x**4/(315*a*
*40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 6
6150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a
**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**
15*x**10) + 46252*a**(79/2)*b**5*x**5*sqrt(1 + b*x/a)/(315*a**40*b**5 + 3150*a**
39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**
4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 1
4175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10) - 64512*a
**(79/2)*b**5*x**5/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 +
37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*
a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*
b**14*x**9 + 315*a**30*b**15*x**10) + 35214*a**(77/2)*b**6*x**6*sqrt(1 + b*x/a)/
(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x
**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 +
37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a*
*30*b**15*x**10) - 53760*a**(77/2)*b**6*x**6/(315*a**40*b**5 + 3150*a**39*b**6*x
+ 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380
*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**3
2*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10) + 19632*a**(75/2)*
b**7*x**7*sqrt(1 + b*x/a)/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7
*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 +
66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150
*a**31*b**14*x**9 + 315*a**30*b**15*x**10) - 30720*a**(75/2)*b**7*x**7/(315*a**4
0*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 661
50*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**
33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15
*x**10) + 10860*a**(73/2)*b**8*x**8*sqrt(1 + b*x/a)/(315*a**40*b**5 + 3150*a**39
*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4
+ 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 141
75*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10) - 11520*a**
(73/2)*b**8*x**8/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 3
7800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a*
*34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b*
*14*x**9 + 315*a**30*b**15*x**10) + 9160*a**(71/2)*b**9*x**9*sqrt(1 + b*x/a)/(31
5*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3
+ 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 378
00*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30
*b**15*x**10) - 2560*a**(71/2)*b**9*x**9/(315*a**40*b**5 + 3150*a**39*b**6*x + 1
4175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**
35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b*
*13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10) + 8396*a**(69/2)*b**10
*x**10*sqrt(1 + b*x/a)/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x*
*2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66
150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a*
*31*b**14*x**9 + 315*a**30*b**15*x**10) - 256*a**(69/2)*b**10*x**10/(315*a**40*b
**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*
a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*
b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15*x*
*10) + 5632*a**(67/2)*b**11*x**11*sqrt(1 + b*x/a)/(315*a**40*b**5 + 3150*a**39*b
**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 +
79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175
*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10) + 2446*a**(65
/2)*b**12*x**12*sqrt(1 + b*x/a)/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**3
8*b**7*x**2 + 37800*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*
x**5 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8
+ 3150*a**31*b**14*x**9 + 315*a**30*b**15*x**10) + 620*a**(63/2)*b**13*x**13*sqr
t(1 + b*x/a)/(315*a**40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 37800
*a**37*b**8*x**3 + 66150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a**34*
b**11*x**6 + 37800*a**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b**14*
x**9 + 315*a**30*b**15*x**10) + 70*a**(61/2)*b**14*x**14*sqrt(1 + b*x/a)/(315*a*
*40*b**5 + 3150*a**39*b**6*x + 14175*a**38*b**7*x**2 + 37800*a**37*b**8*x**3 + 6
6150*a**36*b**9*x**4 + 79380*a**35*b**10*x**5 + 66150*a**34*b**11*x**6 + 37800*a
**33*b**12*x**7 + 14175*a**32*b**13*x**8 + 3150*a**31*b**14*x**9 + 315*a**30*b**
15*x**10)

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GIAC/XCAS [A]  time = 0.208835, size = 103, normalized size = 1.16 \[ \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{32} - 180 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{32} + 378 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{32} - 420 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{32} + 315 \, \sqrt{b x + a} a^{4} b^{32}\right )}}{315 \, b^{37}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(b*x + a),x, algorithm="giac")

[Out]

2/315*(35*(b*x + a)^(9/2)*b^32 - 180*(b*x + a)^(7/2)*a*b^32 + 378*(b*x + a)^(5/2
)*a^2*b^32 - 420*(b*x + a)^(3/2)*a^3*b^32 + 315*sqrt(b*x + a)*a^4*b^32)/b^37