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 = 0.0235158, 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, Rules used =
{43} \[ -\frac{2 a^4}{b^5 \sqrt{a+b x}}-\frac{8 a^3 \sqrt{a+b x}}{b^5}+\frac{4 a^2 (a+b x)^{3/2}}{b^5}-\frac{8 a (a+b x)^{5/2}}{5 b^5}+\frac{2 (a+b x)^{7/2}}{7 b^5} \]
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)
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)^{3/2}} \, dx &=\int \left (\frac{a^4}{b^4 (a+b x)^{3/2}}-\frac{4 a^3}{b^4 \sqrt{a+b x}}+\frac{6 a^2 \sqrt{a+b x}}{b^4}-\frac{4 a (a+b x)^{3/2}}{b^4}+\frac{(a+b x)^{5/2}}{b^4}\right ) \, dx\\ &=-\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}\\ \end{align*}
Mathematica [A] time = 0.0424042, size = 57, normalized size = 0.67 \[ \frac{2 \left (16 a^2 b^2 x^2-64 a^3 b x-128 a^4-8 a b^3 x^3+5 b^4 x^4\right )}{35 b^5 \sqrt{a+b x}} \]
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*Sqrt[a + b*x])
________________________________________________________________________________________
Maple [A] time = 0.005, size = 54, normalized size = 0.6 \begin{align*} -{\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}}}} \end{align*}
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
________________________________________________________________________________________
Maxima [A] time = 1.05825, size = 96, normalized size = 1.13 \begin{align*} \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}} \end{align*}
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)
________________________________________________________________________________________
Fricas [A] time = 1.53654, size = 138, normalized size = 1.62 \begin{align*} \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 )} \sqrt{b x + a}}{35 \,{\left (b^{6} x + a b^{5}\right )}} \end{align*}
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^6*x + a*b^5)
________________________________________________________________________________________
Sympy [B] time = 5.73927, size = 3606, normalized size = 42.42 \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)**(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**31*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 + 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) - 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 + 73
50*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) + 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*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**(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**1
0) - 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**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) - 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**14*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**3
6*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) - 340*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 + 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) + 424*a**(67/
2)*b**10*x**10*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) + 256*a**(67/2)*b**10*x**10/(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)
+ 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*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 + 8
820*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 + 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)
________________________________________________________________________________________
Giac [A] time = 1.14232, size = 104, normalized size = 1.22 \begin{align*} -\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}} \end{align*}
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