3.4.51 \(\int \frac {x^4}{(a+b x)^{5/2}} \, dx\) [351]
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]
-2/3*a^4/b^5/(b*x+a)^(3/2)-8/3*a*(b*x+a)^(3/2)/b^5+2/5*(b*x+a)^(5/2)/b^5+8*a^3/b^5/(b*x+a)^(1/2)+12*a^2*(b*x+a
)^(1/2)/b^5
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Rubi [A]
time = 0.02, antiderivative size = 87, normalized size of antiderivative = 1.00, 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 = {45}
\begin {gather*} -\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 {gather*}
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 45
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*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.66 \begin {gather*} \frac {2 \left (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\right )}{15 b^5 (a+b x)^{3/2}} \end {gather*}
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))
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Maple [A]
time = 0.12, size = 62, normalized size = 0.71
| | |
method |
result |
size |
| | |
gosper |
\(\frac {\frac {2}{5} b^{4} x^{4}-\frac {16}{15} a \,b^{3} x^{3}+\frac {32}{5} a^{2} b^{2} x^{2}+\frac {128}{5} a^{3} b x +\frac {256}{15} a^{4}}{\left (b x +a \right )^{\frac {3}{2}} b^{5}}\) |
\(54\) |
trager |
\(\frac {\frac {2}{5} b^{4} x^{4}-\frac {16}{15} a \,b^{3} x^{3}+\frac {32}{5} a^{2} b^{2} x^{2}+\frac {128}{5} a^{3} b x +\frac {256}{15} a^{4}}{\left (b x +a \right )^{\frac {3}{2}} b^{5}}\) |
\(54\) |
risch |
\(\frac {2 \left (3 x^{2} b^{2}-14 a b x +73 a^{2}\right ) \sqrt {b x +a}}{15 b^{5}}+\frac {2 a^{3} \left (12 b x +11 a \right )}{3 b^{5} \left (b x +a \right )^{\frac {3}{2}}}\) |
\(56\) |
derivativedivides |
\(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {8 a \left (b x +a \right )^{\frac {3}{2}}}{3}+12 a^{2} \sqrt {b x +a}-\frac {2 a^{4}}{3 \left (b x +a \right )^{\frac {3}{2}}}+\frac {8 a^{3}}{\sqrt {b x +a}}}{b^{5}}\) |
\(62\) |
default |
\(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {8 a \left (b x +a \right )^{\frac {3}{2}}}{3}+12 a^{2} \sqrt {b x +a}-\frac {2 a^{4}}{3 \left (b x +a \right )^{\frac {3}{2}}}+\frac {8 a^{3}}{\sqrt {b x +a}}}{b^{5}}\) |
\(62\) |
| | |
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/(b*x+a)^(5/2),x,method=_RETURNVERBOSE)
[Out]
2/b^5*(1/5*(b*x+a)^(5/2)-4/3*a*(b*x+a)^(3/2)+6*a^2*(b*x+a)^(1/2)-1/3*a^4/(b*x+a)^(3/2)+4*a^3/(b*x+a)^(1/2))
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Maxima [A]
time = 0.28, size = 71, normalized size = 0.82 \begin {gather*} \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 {gather*}
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)
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Fricas [A]
time = 0.42, size = 74, normalized size = 0.85 \begin {gather*} \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 {gather*}
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)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3456 vs.
\(2 (83) = 166\).
time = 2.02, size = 3456, normalized size = 39.72 \begin {gather*} \text {Too large to display} \end {gather*}
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...
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Giac [A]
time = 1.88, size = 75, normalized size = 0.86 \begin {gather*} \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 {gather*}
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
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Mupad [B]
time = 0.05, size = 68, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{5/2}}{5\,b^5}+\frac {8\,a^3\,\left (a+b\,x\right )-\frac {2\,a^4}{3}}{b^5\,{\left (a+b\,x\right )}^{3/2}}+\frac {12\,a^2\,\sqrt {a+b\,x}}{b^5}-\frac {8\,a\,{\left (a+b\,x\right )}^{3/2}}{3\,b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/(a + b*x)^(5/2),x)
[Out]
(2*(a + b*x)^(5/2))/(5*b^5) + (8*a^3*(a + b*x) - (2*a^4)/3)/(b^5*(a + b*x)^(3/2)) + (12*a^2*(a + b*x)^(1/2))/b
^5 - (8*a*(a + b*x)^(3/2))/(3*b^5)
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