3.334 \(\int \frac{x^4}{\sqrt{a+b x}} \, dx\)
Optimal. Leaf size=89 \[ \frac{12 a^2 (a+b x)^{5/2}}{5 b^5}-\frac{8 a^3 (a+b x)^{3/2}}{3 b^5}+\frac{2 a^4 \sqrt{a+b x}}{b^5}+\frac{2 (a+b x)^{9/2}}{9 b^5}-\frac{8 a (a+b x)^{7/2}}{7 b^5} \]
[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)
________________________________________________________________________________________
Rubi [A] time = 0.0222366, 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, Rules used =
{43} \[ \frac{12 a^2 (a+b x)^{5/2}}{5 b^5}-\frac{8 a^3 (a+b x)^{3/2}}{3 b^5}+\frac{2 a^4 \sqrt{a+b x}}{b^5}+\frac{2 (a+b x)^{9/2}}{9 b^5}-\frac{8 a (a+b x)^{7/2}}{7 b^5} \]
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)
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt{a+b x}} \, dx &=\int \left (\frac{a^4}{b^4 \sqrt{a+b x}}-\frac{4 a^3 \sqrt{a+b x}}{b^4}+\frac{6 a^2 (a+b x)^{3/2}}{b^4}-\frac{4 a (a+b x)^{5/2}}{b^4}+\frac{(a+b x)^{7/2}}{b^4}\right ) \, dx\\ &=\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{8 a (a+b x)^{7/2}}{7 b^5}+\frac{2 (a+b x)^{9/2}}{9 b^5}\\ \end{align*}
Mathematica [A] time = 0.0437078, size = 57, normalized size = 0.64 \[ \frac{2 \sqrt{a+b x} \left (48 a^2 b^2 x^2-64 a^3 b x+128 a^4-40 a b^3 x^3+35 b^4 x^4\right )}{315 b^5} \]
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)
________________________________________________________________________________________
Maple [A] time = 0.004, size = 54, normalized size = 0.6 \begin{align*}{\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}} \end{align*}
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
________________________________________________________________________________________
Maxima [A] time = 1.07997, size = 96, normalized size = 1.08 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^(1/2),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
________________________________________________________________________________________
Fricas [A] time = 1.43878, size = 126, normalized size = 1.42 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^(1/2),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
________________________________________________________________________________________
Sympy [B] time = 6.16229, size = 3755, normalized size = 42.19 \begin{align*} \text{result too large to display} \end{align*}
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**3
9*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) + 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 + 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) - 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**1
4*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**1
0) - 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 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 1417
5*a**32*b**13*x**8 + 3150*a**31*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 + 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) + 4199
0*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**3
7*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**(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 + 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) + 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 + 14175*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**32*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 + 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) + 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 + 14175*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 + 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) + 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 + 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) - 2560*a**(71/2)*b**9*x**9/(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) + 8396*a**(6
9/2)*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 + 66150*a**34*b**11*x**6 + 37800*a**33*b**12*x**7 + 14
175*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 + 14
175*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) + 2
446*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**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) + 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 + 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)
________________________________________________________________________________________
Giac [A] time = 1.23519, size = 82, normalized size = 0.92 \begin{align*} \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 180 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 378 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{b x + a} a^{4}\right )}}{315 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^(1/2),x, algorithm="giac")
[Out]
2/315*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sq
rt(b*x + a)*a^4)/b^5