3.2224 \(\int \frac{x^4}{(a+b \sqrt{x})^8} \, dx\)
Optimal. Leaf size=172 \[ \frac{2 a^9}{7 b^{10} \left (a+b \sqrt{x}\right )^7}-\frac{3 a^8}{b^{10} \left (a+b \sqrt{x}\right )^6}+\frac{72 a^7}{5 b^{10} \left (a+b \sqrt{x}\right )^5}-\frac{42 a^6}{b^{10} \left (a+b \sqrt{x}\right )^4}+\frac{84 a^5}{b^{10} \left (a+b \sqrt{x}\right )^3}-\frac{126 a^4}{b^{10} \left (a+b \sqrt{x}\right )^2}+\frac{168 a^3}{b^{10} \left (a+b \sqrt{x}\right )}+\frac{72 a^2 \log \left (a+b \sqrt{x}\right )}{b^{10}}-\frac{16 a \sqrt{x}}{b^9}+\frac{x}{b^8} \]
[Out]
(2*a^9)/(7*b^10*(a + b*Sqrt[x])^7) - (3*a^8)/(b^10*(a + b*Sqrt[x])^6) + (72*a^7)/(5*b^10*(a + b*Sqrt[x])^5) -
(42*a^6)/(b^10*(a + b*Sqrt[x])^4) + (84*a^5)/(b^10*(a + b*Sqrt[x])^3) - (126*a^4)/(b^10*(a + b*Sqrt[x])^2) + (
168*a^3)/(b^10*(a + b*Sqrt[x])) - (16*a*Sqrt[x])/b^9 + x/b^8 + (72*a^2*Log[a + b*Sqrt[x]])/b^10
________________________________________________________________________________________
Rubi [A] time = 0.157361, antiderivative size = 172, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used =
{266, 43} \[ \frac{2 a^9}{7 b^{10} \left (a+b \sqrt{x}\right )^7}-\frac{3 a^8}{b^{10} \left (a+b \sqrt{x}\right )^6}+\frac{72 a^7}{5 b^{10} \left (a+b \sqrt{x}\right )^5}-\frac{42 a^6}{b^{10} \left (a+b \sqrt{x}\right )^4}+\frac{84 a^5}{b^{10} \left (a+b \sqrt{x}\right )^3}-\frac{126 a^4}{b^{10} \left (a+b \sqrt{x}\right )^2}+\frac{168 a^3}{b^{10} \left (a+b \sqrt{x}\right )}+\frac{72 a^2 \log \left (a+b \sqrt{x}\right )}{b^{10}}-\frac{16 a \sqrt{x}}{b^9}+\frac{x}{b^8} \]
Antiderivative was successfully verified.
[In]
Int[x^4/(a + b*Sqrt[x])^8,x]
[Out]
(2*a^9)/(7*b^10*(a + b*Sqrt[x])^7) - (3*a^8)/(b^10*(a + b*Sqrt[x])^6) + (72*a^7)/(5*b^10*(a + b*Sqrt[x])^5) -
(42*a^6)/(b^10*(a + b*Sqrt[x])^4) + (84*a^5)/(b^10*(a + b*Sqrt[x])^3) - (126*a^4)/(b^10*(a + b*Sqrt[x])^2) + (
168*a^3)/(b^10*(a + b*Sqrt[x])) - (16*a*Sqrt[x])/b^9 + x/b^8 + (72*a^2*Log[a + b*Sqrt[x]])/b^10
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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}{\left (a+b \sqrt{x}\right )^8} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^9}{(a+b x)^8} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{8 a}{b^9}+\frac{x}{b^8}-\frac{a^9}{b^9 (a+b x)^8}+\frac{9 a^8}{b^9 (a+b x)^7}-\frac{36 a^7}{b^9 (a+b x)^6}+\frac{84 a^6}{b^9 (a+b x)^5}-\frac{126 a^5}{b^9 (a+b x)^4}+\frac{126 a^4}{b^9 (a+b x)^3}-\frac{84 a^3}{b^9 (a+b x)^2}+\frac{36 a^2}{b^9 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^9}{7 b^{10} \left (a+b \sqrt{x}\right )^7}-\frac{3 a^8}{b^{10} \left (a+b \sqrt{x}\right )^6}+\frac{72 a^7}{5 b^{10} \left (a+b \sqrt{x}\right )^5}-\frac{42 a^6}{b^{10} \left (a+b \sqrt{x}\right )^4}+\frac{84 a^5}{b^{10} \left (a+b \sqrt{x}\right )^3}-\frac{126 a^4}{b^{10} \left (a+b \sqrt{x}\right )^2}+\frac{168 a^3}{b^{10} \left (a+b \sqrt{x}\right )}-\frac{16 a \sqrt{x}}{b^9}+\frac{x}{b^8}+\frac{72 a^2 \log \left (a+b \sqrt{x}\right )}{b^{10}}\\ \end{align*}
Mathematica [A] time = 0.132702, size = 150, normalized size = 0.87 \[ \frac{72275 a^6 b^3 x^{3/2}+50225 a^5 b^4 x^2+12495 a^4 b^5 x^{5/2}-4655 a^3 b^6 x^3-3185 a^2 b^7 x^{7/2}+53949 a^7 b^2 x+20923 a^8 b \sqrt{x}+2520 a^2 \left (a+b \sqrt{x}\right )^7 \log \left (a+b \sqrt{x}\right )+3349 a^9-315 a b^8 x^4+35 b^9 x^{9/2}}{35 b^{10} \left (a+b \sqrt{x}\right )^7} \]
Antiderivative was successfully verified.
[In]
Integrate[x^4/(a + b*Sqrt[x])^8,x]
[Out]
(3349*a^9 + 20923*a^8*b*Sqrt[x] + 53949*a^7*b^2*x + 72275*a^6*b^3*x^(3/2) + 50225*a^5*b^4*x^2 + 12495*a^4*b^5*
x^(5/2) - 4655*a^3*b^6*x^3 - 3185*a^2*b^7*x^(7/2) - 315*a*b^8*x^4 + 35*b^9*x^(9/2) + 2520*a^2*(a + b*Sqrt[x])^
7*Log[a + b*Sqrt[x]])/(35*b^10*(a + b*Sqrt[x])^7)
________________________________________________________________________________________
Maple [A] time = 0.012, size = 151, normalized size = 0.9 \begin{align*}{\frac{x}{{b}^{8}}}+72\,{\frac{{a}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{10}}}-16\,{\frac{a\sqrt{x}}{{b}^{9}}}+{\frac{2\,{a}^{9}}{7\,{b}^{10}} \left ( a+b\sqrt{x} \right ) ^{-7}}-3\,{\frac{{a}^{8}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{6}}}+{\frac{72\,{a}^{7}}{5\,{b}^{10}} \left ( a+b\sqrt{x} \right ) ^{-5}}-42\,{\frac{{a}^{6}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{4}}}+84\,{\frac{{a}^{5}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{3}}}-126\,{\frac{{a}^{4}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{2}}}+168\,{\frac{{a}^{3}}{{b}^{10} \left ( a+b\sqrt{x} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/(a+b*x^(1/2))^8,x)
[Out]
x/b^8+72*a^2*ln(a+b*x^(1/2))/b^10-16*a*x^(1/2)/b^9+2/7*a^9/b^10/(a+b*x^(1/2))^7-3*a^8/b^10/(a+b*x^(1/2))^6+72/
5*a^7/b^10/(a+b*x^(1/2))^5-42*a^6/b^10/(a+b*x^(1/2))^4+84*a^5/b^10/(a+b*x^(1/2))^3-126*a^4/b^10/(a+b*x^(1/2))^
2+168*a^3/b^10/(a+b*x^(1/2))
________________________________________________________________________________________
Maxima [A] time = 0.968824, size = 219, normalized size = 1.27 \begin{align*} \frac{72 \, a^{2} \log \left (b \sqrt{x} + a\right )}{b^{10}} + \frac{{\left (b \sqrt{x} + a\right )}^{2}}{b^{10}} - \frac{18 \,{\left (b \sqrt{x} + a\right )} a}{b^{10}} + \frac{168 \, a^{3}}{{\left (b \sqrt{x} + a\right )} b^{10}} - \frac{126 \, a^{4}}{{\left (b \sqrt{x} + a\right )}^{2} b^{10}} + \frac{84 \, a^{5}}{{\left (b \sqrt{x} + a\right )}^{3} b^{10}} - \frac{42 \, a^{6}}{{\left (b \sqrt{x} + a\right )}^{4} b^{10}} + \frac{72 \, a^{7}}{5 \,{\left (b \sqrt{x} + a\right )}^{5} b^{10}} - \frac{3 \, a^{8}}{{\left (b \sqrt{x} + a\right )}^{6} b^{10}} + \frac{2 \, a^{9}}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(a+b*x^(1/2))^8,x, algorithm="maxima")
[Out]
72*a^2*log(b*sqrt(x) + a)/b^10 + (b*sqrt(x) + a)^2/b^10 - 18*(b*sqrt(x) + a)*a/b^10 + 168*a^3/((b*sqrt(x) + a)
*b^10) - 126*a^4/((b*sqrt(x) + a)^2*b^10) + 84*a^5/((b*sqrt(x) + a)^3*b^10) - 42*a^6/((b*sqrt(x) + a)^4*b^10)
+ 72/5*a^7/((b*sqrt(x) + a)^5*b^10) - 3*a^8/((b*sqrt(x) + a)^6*b^10) + 2/7*a^9/((b*sqrt(x) + a)^7*b^10)
________________________________________________________________________________________
Fricas [B] time = 1.34013, size = 824, normalized size = 4.79 \begin{align*} \frac{35 \, b^{16} x^{8} - 245 \, a^{2} b^{14} x^{7} - 9555 \, a^{4} b^{12} x^{6} + 41405 \, a^{6} b^{10} x^{5} - 83720 \, a^{8} b^{8} x^{4} + 94745 \, a^{10} b^{6} x^{3} - 62139 \, a^{12} b^{4} x^{2} + 22183 \, a^{14} b^{2} x - 3349 \, a^{16} + 2520 \,{\left (a^{2} b^{14} x^{7} - 7 \, a^{4} b^{12} x^{6} + 21 \, a^{6} b^{10} x^{5} - 35 \, a^{8} b^{8} x^{4} + 35 \, a^{10} b^{6} x^{3} - 21 \, a^{12} b^{4} x^{2} + 7 \, a^{14} b^{2} x - a^{16}\right )} \log \left (b \sqrt{x} + a\right ) - 8 \,{\left (70 \, a b^{15} x^{7} - 1225 \, a^{3} b^{13} x^{6} + 4410 \, a^{5} b^{11} x^{5} - 8393 \, a^{7} b^{9} x^{4} + 9216 \, a^{9} b^{7} x^{3} - 5943 \, a^{11} b^{5} x^{2} + 2100 \, a^{13} b^{3} x - 315 \, a^{15} b\right )} \sqrt{x}}{35 \,{\left (b^{24} x^{7} - 7 \, a^{2} b^{22} x^{6} + 21 \, a^{4} b^{20} x^{5} - 35 \, a^{6} b^{18} x^{4} + 35 \, a^{8} b^{16} x^{3} - 21 \, a^{10} b^{14} x^{2} + 7 \, a^{12} b^{12} x - a^{14} b^{10}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(a+b*x^(1/2))^8,x, algorithm="fricas")
[Out]
1/35*(35*b^16*x^8 - 245*a^2*b^14*x^7 - 9555*a^4*b^12*x^6 + 41405*a^6*b^10*x^5 - 83720*a^8*b^8*x^4 + 94745*a^10
*b^6*x^3 - 62139*a^12*b^4*x^2 + 22183*a^14*b^2*x - 3349*a^16 + 2520*(a^2*b^14*x^7 - 7*a^4*b^12*x^6 + 21*a^6*b^
10*x^5 - 35*a^8*b^8*x^4 + 35*a^10*b^6*x^3 - 21*a^12*b^4*x^2 + 7*a^14*b^2*x - a^16)*log(b*sqrt(x) + a) - 8*(70*
a*b^15*x^7 - 1225*a^3*b^13*x^6 + 4410*a^5*b^11*x^5 - 8393*a^7*b^9*x^4 + 9216*a^9*b^7*x^3 - 5943*a^11*b^5*x^2 +
2100*a^13*b^3*x - 315*a^15*b)*sqrt(x))/(b^24*x^7 - 7*a^2*b^22*x^6 + 21*a^4*b^20*x^5 - 35*a^6*b^18*x^4 + 35*a^
8*b^16*x^3 - 21*a^10*b^14*x^2 + 7*a^12*b^12*x - a^14*b^10)
________________________________________________________________________________________
Sympy [A] time = 10.6282, size = 1841, normalized size = 10.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**4/(a+b*x**(1/2))**8,x)
[Out]
Piecewise((2520*a**9*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4
*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 360
*a**9/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*
x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 17640*a**8*b*sqrt(x)*log(a/b + sqrt(x
))/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**
2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 52920*a**7*b**2*x*log(a/b + sqrt(x))/(35
*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 73
5*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) - 8820*a**7*b**2*x/(35*a**7*b**10 + 245*a**6*b**
11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 24
5*a*b**16*x**3 + 35*b**17*x**(7/2)) + 88200*a**6*b**3*x**(3/2)*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b*
*11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 2
45*a*b**16*x**3 + 35*b**17*x**(7/2)) - 32340*a**6*b**3*x**(3/2)/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*
a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 +
35*b**17*x**(7/2)) + 88200*a**5*b**4*x**2*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**
5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*
b**17*x**(7/2)) - 54390*a**5*b**4*x**2/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*
b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 5292
0*a**4*b**5*x**(5/2)*log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4
*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) - 502
74*a**4*b**5*x**(5/2)/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) +
1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 17640*a**3*b**6*x**3*
log(a/b + sqrt(x))/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 122
5*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) - 25578*a**3*b**6*x**3/(35
*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 73
5*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) + 2520*a**2*b**7*x**(7/2)*log(a/b + sqrt(x))/(35
*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 73
5*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) - 6174*a**2*b**7*x**(7/2)/(35*a**7*b**10 + 245*a
**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/
2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)) - 315*a*b**8*x**4/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**
5*b**12*x + 1225*a**4*b**13*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*
b**17*x**(7/2)) + 35*b**9*x**(9/2)/(35*a**7*b**10 + 245*a**6*b**11*sqrt(x) + 735*a**5*b**12*x + 1225*a**4*b**1
3*x**(3/2) + 1225*a**3*b**14*x**2 + 735*a**2*b**15*x**(5/2) + 245*a*b**16*x**3 + 35*b**17*x**(7/2)), Ne(b, 0))
, (x**5/(5*a**8), True))
________________________________________________________________________________________
Giac [A] time = 1.16156, size = 161, normalized size = 0.94 \begin{align*} \frac{72 \, a^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{10}} + \frac{b^{8} x - 16 \, a b^{7} \sqrt{x}}{b^{16}} + \frac{5880 \, a^{3} b^{6} x^{3} + 30870 \, a^{4} b^{5} x^{\frac{5}{2}} + 69090 \, a^{5} b^{4} x^{2} + 83790 \, a^{6} b^{3} x^{\frac{3}{2}} + 57834 \, a^{7} b^{2} x + 21483 \, a^{8} b \sqrt{x} + 3349 \, a^{9}}{35 \,{\left (b \sqrt{x} + a\right )}^{7} b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(a+b*x^(1/2))^8,x, algorithm="giac")
[Out]
72*a^2*log(abs(b*sqrt(x) + a))/b^10 + (b^8*x - 16*a*b^7*sqrt(x))/b^16 + 1/35*(5880*a^3*b^6*x^3 + 30870*a^4*b^5
*x^(5/2) + 69090*a^5*b^4*x^2 + 83790*a^6*b^3*x^(3/2) + 57834*a^7*b^2*x + 21483*a^8*b*sqrt(x) + 3349*a^9)/((b*s
qrt(x) + a)^7*b^10)