3.343 \(\int \frac{x^4}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{2 a^4}{b^5 \sqrt{a+b x}}-\frac{8 a^3 \sqrt{a+b x}}{b^5}+\frac{4 a^2 (a+b x)^{3/2}}{b^5}-\frac{8 a (a+b x)^{5/2}}{5 b^5}+\frac{2 (a+b x)^{7/2}}{7 b^5} \]

[Out]

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

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Rubi [A]  time = 0.060735, antiderivative size = 85, 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 \[ -\frac{2 a^4}{b^5 \sqrt{a+b x}}-\frac{8 a^3 \sqrt{a+b x}}{b^5}+\frac{4 a^2 (a+b x)^{3/2}}{b^5}-\frac{8 a (a+b x)^{5/2}}{5 b^5}+\frac{2 (a+b x)^{7/2}}{7 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.6587, size = 82, normalized size = 0.96 \[ - \frac{2 a^{4}}{b^{5} \sqrt{a + b x}} - \frac{8 a^{3} \sqrt{a + b x}}{b^{5}} + \frac{4 a^{2} \left (a + b x\right )^{\frac{3}{2}}}{b^{5}} - \frac{8 a \left (a + b x\right )^{\frac{5}{2}}}{5 b^{5}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}}}{7 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0322296, size = 57, normalized size = 0.67 \[ \frac{2 \left (-128 a^4-64 a^3 b x+16 a^2 b^2 x^2-8 a b^3 x^3+5 b^4 x^4\right )}{35 b^5 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x)^(3/2),x]

[Out]

(2*(-128*a^4 - 64*a^3*b*x + 16*a^2*b^2*x^2 - 8*a*b^3*x^3 + 5*b^4*x^4))/(35*b^5*S
qrt[a + b*x])

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Maple [A]  time = 0.008, size = 54, normalized size = 0.6 \[ -{\frac{-10\,{x}^{4}{b}^{4}+16\,a{x}^{3}{b}^{3}-32\,{a}^{2}{x}^{2}{b}^{2}+128\,{a}^{3}xb+256\,{a}^{4}}{35\,{b}^{5}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)^(3/2),x)

[Out]

-2/35/(b*x+a)^(1/2)*(-5*b^4*x^4+8*a*b^3*x^3-16*a^2*b^2*x^2+64*a^3*b*x+128*a^4)/b
^5

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Maxima [A]  time = 1.3431, size = 96, normalized size = 1.13 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}}}{7 \, b^{5}} - \frac{8 \,{\left (b x + a\right )}^{\frac{5}{2}} a}{5 \, b^{5}} + \frac{4 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}}{b^{5}} - \frac{8 \, \sqrt{b x + a} a^{3}}{b^{5}} - \frac{2 \, a^{4}}{\sqrt{b x + a} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.210502, size = 72, normalized size = 0.85 \[ \frac{2 \,{\left (5 \, b^{4} x^{4} - 8 \, a b^{3} x^{3} + 16 \, a^{2} b^{2} x^{2} - 64 \, a^{3} b x - 128 \, a^{4}\right )}}{35 \, \sqrt{b x + a} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*b^4*x^4 - 8*a*b^3*x^3 + 16*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 = 7.64992, size = 3606, normalized size = 42.42 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x+a)**(3/2),x)

[Out]

-256*a**(87/2)*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b*
*7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 +
7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**3
1*b**14*x**9 + 35*a**30*b**15*x**10) + 256*a**(87/2)/(35*a**40*b**5 + 350*a**39*
b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 88
20*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32
*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 2432*a**(85/2)*b*x*
sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*
a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**1
1*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 +
35*a**30*b**15*x**10) + 2560*a**(85/2)*b*x/(35*a**40*b**5 + 350*a**39*b**6*x + 1
575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b
**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**
8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 10336*a**(83/2)*b**2*x**2*sqr
t(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**
37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x
**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*
a**30*b**15*x**10) + 11520*a**(83/2)*b**2*x**2/(35*a**40*b**5 + 350*a**39*b**6*x
+ 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**
35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13
*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 25840*a**(81/2)*b**3*x**3
*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200
*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**
11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 +
35*a**30*b**15*x**10) + 30720*a**(81/2)*b**3*x**3/(35*a**40*b**5 + 350*a**39*b*
*6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820
*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b
**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 41990*a**(79/2)*b**4*
x**4*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 +
4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34
*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x*
*9 + 35*a**30*b**15*x**10) + 53760*a**(79/2)*b**4*x**4/(35*a**40*b**5 + 350*a**3
9*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 +
8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**
32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 46182*a**(77/2)*b
**5*x**5*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**
2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a
**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**1
4*x**9 + 35*a**30*b**15*x**10) + 64512*a**(77/2)*b**5*x**5/(35*a**40*b**5 + 350*
a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**
4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575
*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 34584*a**(75/
2)*b**6*x**6*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7
*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 73
50*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*
b**14*x**9 + 35*a**30*b**15*x**10) + 53760*a**(75/2)*b**6*x**6/(35*a**40*b**5 +
350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9
*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 +
1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 17112*a**
(73/2)*b**7*x**7*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*
b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5
+ 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a*
*31*b**14*x**9 + 35*a**30*b**15*x**10) + 30720*a**(73/2)*b**7*x**7/(35*a**40*b**
5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*
b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**
7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 4980*
a**(71/2)*b**8*x**8*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**
38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x*
*5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350
*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 11520*a**(71/2)*b**8*x**8/(35*a**40*
b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**
36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*
x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) - 34
0*a**(69/2)*b**9*x**9*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a
**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*
x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 3
50*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 2560*a**(69/2)*b**9*x**9/(35*a**40
*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a*
*36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12
*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 4
24*a**(67/2)*b**10*x**10*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 157
5*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**
10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8
+ 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 256*a**(67/2)*b**10*x**10/(35*a
**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 735
0*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b
**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10)
+ 248*a**(65/2)*b**11*x**11*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x +
1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35
*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x
**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 74*a**(63/2)*b**12*x**12*sq
rt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a*
*37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*
x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35
*a**30*b**15*x**10) + 10*a**(61/2)*b**13*x**13*sqrt(1 + b*x/a)/(35*a**40*b**5 +
350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9
*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 +
1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10)

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GIAC/XCAS [A]  time = 0.203407, size = 104, normalized size = 1.22 \[ -\frac{2 \, a^{4}}{\sqrt{b x + a} b^{5}} + \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{30} - 28 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{30} + 70 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{30} - 140 \, \sqrt{b x + a} a^{3} b^{30}\right )}}{35 \, b^{35}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^(3/2),x, algorithm="giac")

[Out]

-2*a^4/(sqrt(b*x + a)*b^5) + 2/35*(5*(b*x + a)^(7/2)*b^30 - 28*(b*x + a)^(5/2)*a
*b^30 + 70*(b*x + a)^(3/2)*a^2*b^30 - 140*sqrt(b*x + a)*a^3*b^30)/b^35