3.351 \(\int \frac{x^4}{(a+b x)^{5/2}} \, dx\)
Optimal. Leaf size=87 \[ -\frac{2 a^4}{3 b^5 (a+b x)^{3/2}}+\frac{8 a^3}{b^5 \sqrt{a+b x}}+\frac{12 a^2 \sqrt{a+b x}}{b^5}-\frac{8 a (a+b x)^{3/2}}{3 b^5}+\frac{2 (a+b x)^{5/2}}{5 b^5} \]
[Out]
(-2*a^4)/(3*b^5*(a + b*x)^(3/2)) + (8*a^3)/(b^5*Sqrt[a + b*x]) + (12*a^2*Sqrt[a + b*x])/b^5 - (8*a*(a + b*x)^(
3/2))/(3*b^5) + (2*(a + b*x)^(5/2))/(5*b^5)
________________________________________________________________________________________
Rubi [A] time = 0.0217943, antiderivative size = 87, 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} \[ -\frac{2 a^4}{3 b^5 (a+b x)^{3/2}}+\frac{8 a^3}{b^5 \sqrt{a+b x}}+\frac{12 a^2 \sqrt{a+b x}}{b^5}-\frac{8 a (a+b x)^{3/2}}{3 b^5}+\frac{2 (a+b x)^{5/2}}{5 b^5} \]
Antiderivative was successfully verified.
[In]
Int[x^4/(a + b*x)^(5/2),x]
[Out]
(-2*a^4)/(3*b^5*(a + b*x)^(3/2)) + (8*a^3)/(b^5*Sqrt[a + b*x]) + (12*a^2*Sqrt[a + b*x])/b^5 - (8*a*(a + b*x)^(
3/2))/(3*b^5) + (2*(a + b*x)^(5/2))/(5*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}{(a+b x)^{5/2}} \, dx &=\int \left (\frac{a^4}{b^4 (a+b x)^{5/2}}-\frac{4 a^3}{b^4 (a+b x)^{3/2}}+\frac{6 a^2}{b^4 \sqrt{a+b x}}-\frac{4 a \sqrt{a+b x}}{b^4}+\frac{(a+b x)^{3/2}}{b^4}\right ) \, dx\\ &=-\frac{2 a^4}{3 b^5 (a+b x)^{3/2}}+\frac{8 a^3}{b^5 \sqrt{a+b x}}+\frac{12 a^2 \sqrt{a+b x}}{b^5}-\frac{8 a (a+b x)^{3/2}}{3 b^5}+\frac{2 (a+b x)^{5/2}}{5 b^5}\\ \end{align*}
Mathematica [A] time = 0.0472755, size = 57, normalized size = 0.66 \[ \frac{2 \left (48 a^2 b^2 x^2+192 a^3 b x+128 a^4-8 a b^3 x^3+3 b^4 x^4\right )}{15 b^5 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^4/(a + b*x)^(5/2),x]
[Out]
(2*(128*a^4 + 192*a^3*b*x + 48*a^2*b^2*x^2 - 8*a*b^3*x^3 + 3*b^4*x^4))/(15*b^5*(a + b*x)^(3/2))
________________________________________________________________________________________
Maple [A] time = 0.005, size = 54, normalized size = 0.6 \begin{align*}{\frac{6\,{x}^{4}{b}^{4}-16\,a{x}^{3}{b}^{3}+96\,{a}^{2}{x}^{2}{b}^{2}+384\,{a}^{3}xb+256\,{a}^{4}}{15\,{b}^{5}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/(b*x+a)^(5/2),x)
[Out]
2/15/(b*x+a)^(3/2)*(3*b^4*x^4-8*a*b^3*x^3+48*a^2*b^2*x^2+192*a^3*b*x+128*a^4)/b^5
________________________________________________________________________________________
Maxima [A] time = 1.06957, size = 96, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, b^{5}} - \frac{8 \,{\left (b x + a\right )}^{\frac{3}{2}} a}{3 \, b^{5}} + \frac{12 \, \sqrt{b x + a} a^{2}}{b^{5}} + \frac{8 \, a^{3}}{\sqrt{b x + a} b^{5}} - \frac{2 \, a^{4}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^(5/2),x, algorithm="maxima")
[Out]
2/5*(b*x + a)^(5/2)/b^5 - 8/3*(b*x + a)^(3/2)*a/b^5 + 12*sqrt(b*x + a)*a^2/b^5 + 8*a^3/(sqrt(b*x + a)*b^5) - 2
/3*a^4/((b*x + a)^(3/2)*b^5)
________________________________________________________________________________________
Fricas [A] time = 1.54716, size = 161, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (3 \, b^{4} x^{4} - 8 \, a b^{3} x^{3} + 48 \, a^{2} b^{2} x^{2} + 192 \, a^{3} b x + 128 \, a^{4}\right )} \sqrt{b x + a}}{15 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^(5/2),x, algorithm="fricas")
[Out]
2/15*(3*b^4*x^4 - 8*a*b^3*x^3 + 48*a^2*b^2*x^2 + 192*a^3*b*x + 128*a^4)*sqrt(b*x + a)/(b^7*x^2 + 2*a*b^6*x + a
^2*b^5)
________________________________________________________________________________________
Sympy [B] time = 5.72088, size = 3456, normalized size = 39.72 \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)**(5/2),x)
[Out]
256*a**(85/2)*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 +
3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**1
3*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 256*a**(85/2)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*
a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6
+ 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) + 2432*a**(83/2)
*b*x*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**
36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 +
150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 2560*a**(83/2)*b*x/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**3
8*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 18
00*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) + 10336*a**(81/2)*b*
*2*x**2*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*
a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8
+ 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 11520*a**(81/2)*b**2*x**2/(15*a**40*b**5 + 150*a**39*b**6*x
+ 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11
*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) + 25840*a*
*(79/2)*b**3*x**3*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x*
*3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*
b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 30720*a**(79/2)*b**3*x**3/(15*a**40*b**5 + 150*a**
39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a
**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10)
+ 41990*a**(77/2)*b**4*x**4*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**
37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 +
675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 53760*a**(77/2)*b**4*x**4/(15*a**40*b**5
+ 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**
5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**
15*x**10) + 46192*a**(75/2)*b**5*x**5*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2
+ 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**
12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 64512*a**(75/2)*b**5*x**5/(15*
a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35
*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15
*a**30*b**15*x**10) + 34664*a**(73/2)*b**6*x**6*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*
b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800
*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 53760*a**(73/2)*b**6
*x**6/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 +
3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14
*x**9 + 15*a**30*b**15*x**10) + 17392*a**(71/2)*b**7*x**7*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x +
675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x
**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 30720*a**(
71/2)*b**7*x**7/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b*
*9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a
**31*b**14*x**9 + 15*a**30*b**15*x**10) + 5540*a**(69/2)*b**8*x**8*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*
b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**3
4*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 1
1520*a**(69/2)*b**8*x**8/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150
*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**
8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) + 1040*a**(67/2)*b**9*x**9*sqrt(1 + b*x/a)/(15*a**40*b**5 + 1
50*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 +
3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x
**10) - 2560*a**(67/2)*b**9*x**9/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**
3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b
**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) + 136*a**(65/2)*b**10*x**10*sqrt(1 + b*x/a)/(15*a**40
*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**1
0*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**3
0*b**15*x**10) - 256*a**(65/2)*b**10*x**10/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**3
7*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 6
75*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) + 32*a**(63/2)*b**11*x**11*sqrt(1 + b*x/a)/
(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a
**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9
+ 15*a**30*b**15*x**10) + 6*a**(61/2)*b**12*x**12*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**3
8*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 18
00*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10)
________________________________________________________________________________________
Giac [A] time = 1.1844, size = 101, normalized size = 1.16 \begin{align*} \frac{2 \,{\left (12 \,{\left (b x + a\right )} a^{3} - a^{4}\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{5}} + \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{20} - 20 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{20} + 90 \, \sqrt{b x + a} a^{2} b^{20}\right )}}{15 \, b^{25}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^(5/2),x, algorithm="giac")
[Out]
2/3*(12*(b*x + a)*a^3 - a^4)/((b*x + a)^(3/2)*b^5) + 2/15*(3*(b*x + a)^(5/2)*b^20 - 20*(b*x + a)^(3/2)*a*b^20
+ 90*sqrt(b*x + a)*a^2*b^20)/b^25