3.2215 \(\int \frac{x^4}{(a+b \sqrt{x})^5} \, dx\)

Optimal. Leaf size=155 \[ \frac{10 a^2 x^{3/2}}{b^7}+\frac{a^9}{2 b^{10} \left (a+b \sqrt{x}\right )^4}-\frac{6 a^8}{b^{10} \left (a+b \sqrt{x}\right )^3}+\frac{36 a^7}{b^{10} \left (a+b \sqrt{x}\right )^2}-\frac{168 a^6}{b^{10} \left (a+b \sqrt{x}\right )}+\frac{140 a^4 \sqrt{x}}{b^9}-\frac{35 a^3 x}{b^8}-\frac{252 a^5 \log \left (a+b \sqrt{x}\right )}{b^{10}}-\frac{5 a x^2}{2 b^6}+\frac{2 x^{5/2}}{5 b^5} \]

[Out]

a^9/(2*b^10*(a + b*Sqrt[x])^4) - (6*a^8)/(b^10*(a + b*Sqrt[x])^3) + (36*a^7)/(b^10*(a + b*Sqrt[x])^2) - (168*a
^6)/(b^10*(a + b*Sqrt[x])) + (140*a^4*Sqrt[x])/b^9 - (35*a^3*x)/b^8 + (10*a^2*x^(3/2))/b^7 - (5*a*x^2)/(2*b^6)
+ (2*x^(5/2))/(5*b^5) - (252*a^5*Log[a + b*Sqrt[x]])/b^10

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Rubi [A]  time = 0.139889, antiderivative size = 155, 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} \[ \frac{10 a^2 x^{3/2}}{b^7}+\frac{a^9}{2 b^{10} \left (a+b \sqrt{x}\right )^4}-\frac{6 a^8}{b^{10} \left (a+b \sqrt{x}\right )^3}+\frac{36 a^7}{b^{10} \left (a+b \sqrt{x}\right )^2}-\frac{168 a^6}{b^{10} \left (a+b \sqrt{x}\right )}+\frac{140 a^4 \sqrt{x}}{b^9}-\frac{35 a^3 x}{b^8}-\frac{252 a^5 \log \left (a+b \sqrt{x}\right )}{b^{10}}-\frac{5 a x^2}{2 b^6}+\frac{2 x^{5/2}}{5 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*Sqrt[x])^5,x]

[Out]

a^9/(2*b^10*(a + b*Sqrt[x])^4) - (6*a^8)/(b^10*(a + b*Sqrt[x])^3) + (36*a^7)/(b^10*(a + b*Sqrt[x])^2) - (168*a
^6)/(b^10*(a + b*Sqrt[x])) + (140*a^4*Sqrt[x])/b^9 - (35*a^3*x)/b^8 + (10*a^2*x^(3/2))/b^7 - (5*a*x^2)/(2*b^6)
+ (2*x^(5/2))/(5*b^5) - (252*a^5*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 )^5} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^9}{(a+b x)^5} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{70 a^4}{b^9}-\frac{35 a^3 x}{b^8}+\frac{15 a^2 x^2}{b^7}-\frac{5 a x^3}{b^6}+\frac{x^4}{b^5}-\frac{a^9}{b^9 (a+b x)^5}+\frac{9 a^8}{b^9 (a+b x)^4}-\frac{36 a^7}{b^9 (a+b x)^3}+\frac{84 a^6}{b^9 (a+b x)^2}-\frac{126 a^5}{b^9 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^9}{2 b^{10} \left (a+b \sqrt{x}\right )^4}-\frac{6 a^8}{b^{10} \left (a+b \sqrt{x}\right )^3}+\frac{36 a^7}{b^{10} \left (a+b \sqrt{x}\right )^2}-\frac{168 a^6}{b^{10} \left (a+b \sqrt{x}\right )}+\frac{140 a^4 \sqrt{x}}{b^9}-\frac{35 a^3 x}{b^8}+\frac{10 a^2 x^{3/2}}{b^7}-\frac{5 a x^2}{2 b^6}+\frac{2 x^{5/2}}{5 b^5}-\frac{252 a^5 \log \left (a+b \sqrt{x}\right )}{b^{10}}\\ \end{align*}

Mathematica [A]  time = 0.127373, size = 150, normalized size = 0.97 \[ \frac{5420 a^6 b^3 x^{3/2}+3875 a^5 b^4 x^2+504 a^4 b^5 x^{5/2}-84 a^3 b^6 x^3+24 a^2 b^7 x^{7/2}+570 a^7 b^2 x-2980 a^8 b \sqrt{x}-2520 a^5 \left (a+b \sqrt{x}\right )^4 \log \left (a+b \sqrt{x}\right )-1375 a^9-9 a b^8 x^4+4 b^9 x^{9/2}}{10 b^{10} \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*Sqrt[x])^5,x]

[Out]

(-1375*a^9 - 2980*a^8*b*Sqrt[x] + 570*a^7*b^2*x + 5420*a^6*b^3*x^(3/2) + 3875*a^5*b^4*x^2 + 504*a^4*b^5*x^(5/2
) - 84*a^3*b^6*x^3 + 24*a^2*b^7*x^(7/2) - 9*a*b^8*x^4 + 4*b^9*x^(9/2) - 2520*a^5*(a + b*Sqrt[x])^4*Log[a + b*S
qrt[x]])/(10*b^10*(a + b*Sqrt[x])^4)

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Maple [A]  time = 0.007, size = 134, normalized size = 0.9 \begin{align*} -35\,{\frac{{a}^{3}x}{{b}^{8}}}+10\,{\frac{{a}^{2}{x}^{3/2}}{{b}^{7}}}-{\frac{5\,a{x}^{2}}{2\,{b}^{6}}}+{\frac{2}{5\,{b}^{5}}{x}^{{\frac{5}{2}}}}-252\,{\frac{{a}^{5}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{10}}}+140\,{\frac{{a}^{4}\sqrt{x}}{{b}^{9}}}+{\frac{{a}^{9}}{2\,{b}^{10}} \left ( a+b\sqrt{x} \right ) ^{-4}}-6\,{\frac{{a}^{8}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{3}}}+36\,{\frac{{a}^{7}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{2}}}-168\,{\frac{{a}^{6}}{{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))^5,x)

[Out]

-35*a^3*x/b^8+10*a^2*x^(3/2)/b^7-5/2*a*x^2/b^6+2/5*x^(5/2)/b^5-252*a^5*ln(a+b*x^(1/2))/b^10+140*a^4*x^(1/2)/b^
9+1/2*a^9/b^10/(a+b*x^(1/2))^4-6*a^8/b^10/(a+b*x^(1/2))^3+36*a^7/b^10/(a+b*x^(1/2))^2-168*a^6/b^10/(a+b*x^(1/2
))

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Maxima [A]  time = 0.965948, size = 220, normalized size = 1.42 \begin{align*} -\frac{252 \, a^{5} \log \left (b \sqrt{x} + a\right )}{b^{10}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{5}}{5 \, b^{10}} - \frac{9 \,{\left (b \sqrt{x} + a\right )}^{4} a}{2 \, b^{10}} + \frac{24 \,{\left (b \sqrt{x} + a\right )}^{3} a^{2}}{b^{10}} - \frac{84 \,{\left (b \sqrt{x} + a\right )}^{2} a^{3}}{b^{10}} + \frac{252 \,{\left (b \sqrt{x} + a\right )} a^{4}}{b^{10}} - \frac{168 \, a^{6}}{{\left (b \sqrt{x} + a\right )} b^{10}} + \frac{36 \, a^{7}}{{\left (b \sqrt{x} + a\right )}^{2} b^{10}} - \frac{6 \, a^{8}}{{\left (b \sqrt{x} + a\right )}^{3} b^{10}} + \frac{a^{9}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

-252*a^5*log(b*sqrt(x) + a)/b^10 + 2/5*(b*sqrt(x) + a)^5/b^10 - 9/2*(b*sqrt(x) + a)^4*a/b^10 + 24*(b*sqrt(x) +
a)^3*a^2/b^10 - 84*(b*sqrt(x) + a)^2*a^3/b^10 + 252*(b*sqrt(x) + a)*a^4/b^10 - 168*a^6/((b*sqrt(x) + a)*b^10)
+ 36*a^7/((b*sqrt(x) + a)^2*b^10) - 6*a^8/((b*sqrt(x) + a)^3*b^10) + 1/2*a^9/((b*sqrt(x) + a)^4*b^10)

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Fricas [A]  time = 1.33678, size = 567, normalized size = 3.66 \begin{align*} -\frac{25 \, a b^{12} x^{6} + 250 \, a^{3} b^{10} x^{5} - 1250 \, a^{5} b^{8} x^{4} - 40 \, a^{7} b^{6} x^{3} + 3840 \, a^{9} b^{4} x^{2} - 4240 \, a^{11} b^{2} x + 1375 \, a^{13} + 2520 \,{\left (a^{5} b^{8} x^{4} - 4 \, a^{7} b^{6} x^{3} + 6 \, a^{9} b^{4} x^{2} - 4 \, a^{11} b^{2} x + a^{13}\right )} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (b^{13} x^{6} + 21 \, a^{2} b^{11} x^{5} + 256 \, a^{4} b^{9} x^{4} - 1674 \, a^{6} b^{7} x^{3} + 3066 \, a^{8} b^{5} x^{2} - 2310 \, a^{10} b^{3} x + 630 \, a^{12} b\right )} \sqrt{x}}{10 \,{\left (b^{18} x^{4} - 4 \, a^{2} b^{16} x^{3} + 6 \, a^{4} b^{14} x^{2} - 4 \, a^{6} b^{12} x + a^{8} b^{10}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

-1/10*(25*a*b^12*x^6 + 250*a^3*b^10*x^5 - 1250*a^5*b^8*x^4 - 40*a^7*b^6*x^3 + 3840*a^9*b^4*x^2 - 4240*a^11*b^2
*x + 1375*a^13 + 2520*(a^5*b^8*x^4 - 4*a^7*b^6*x^3 + 6*a^9*b^4*x^2 - 4*a^11*b^2*x + a^13)*log(b*sqrt(x) + a) -
4*(b^13*x^6 + 21*a^2*b^11*x^5 + 256*a^4*b^9*x^4 - 1674*a^6*b^7*x^3 + 3066*a^8*b^5*x^2 - 2310*a^10*b^3*x + 630
*a^12*b)*sqrt(x))/(b^18*x^4 - 4*a^2*b^16*x^3 + 6*a^4*b^14*x^2 - 4*a^6*b^12*x + a^8*b^10)

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Sympy [A]  time = 8.37872, size = 949, normalized size = 6.12 \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))**5,x)

[Out]

Piecewise((-2520*a**9*log(a/b + sqrt(x))/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13
*x**(3/2) + 10*b**14*x**2) - 630*a**9/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x*
*(3/2) + 10*b**14*x**2) - 10080*a**8*b*sqrt(x)*log(a/b + sqrt(x))/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*
a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 15120*a**7*b**2*x*log(a/b + sqrt(x))/(10*a**4*b**10 + 40
*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) + 5040*a**7*b**2*x/(10*a**4*b**10
+ 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 10080*a**6*b**3*x**(3/2)*l
og(a/b + sqrt(x))/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x*
*2) + 8400*a**6*b**3*x**(3/2)/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) +
10*b**14*x**2) - 2520*a**5*b**4*x**2*log(a/b + sqrt(x))/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**1
2*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) + 4620*a**5*b**4*x**2/(10*a**4*b**10 + 40*a**3*b**11*sqrt(x) + 60*a
**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) + 504*a**4*b**5*x**(5/2)/(10*a**4*b**10 + 40*a**3*b**11*sqr
t(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 84*a**3*b**6*x**3/(10*a**4*b**10 + 40*a**3*b**
11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) + 24*a**2*b**7*x**(7/2)/(10*a**4*b**10 + 4
0*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) - 9*a*b**8*x**4/(10*a**4*b**10 +
40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2) + 4*b**9*x**(9/2)/(10*a**4*b**
10 + 40*a**3*b**11*sqrt(x) + 60*a**2*b**12*x + 40*a*b**13*x**(3/2) + 10*b**14*x**2), Ne(b, 0)), (x**5/(5*a**5)
, True))

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Giac [A]  time = 1.09864, size = 163, normalized size = 1.05 \begin{align*} -\frac{252 \, a^{5} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{10}} - \frac{336 \, a^{6} b^{3} x^{\frac{3}{2}} + 936 \, a^{7} b^{2} x + 876 \, a^{8} b \sqrt{x} + 275 \, a^{9}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{10}} + \frac{4 \, b^{20} x^{\frac{5}{2}} - 25 \, a b^{19} x^{2} + 100 \, a^{2} b^{18} x^{\frac{3}{2}} - 350 \, a^{3} b^{17} x + 1400 \, a^{4} b^{16} \sqrt{x}}{10 \, b^{25}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

-252*a^5*log(abs(b*sqrt(x) + a))/b^10 - 1/2*(336*a^6*b^3*x^(3/2) + 936*a^7*b^2*x + 876*a^8*b*sqrt(x) + 275*a^9
)/((b*sqrt(x) + a)^4*b^10) + 1/10*(4*b^20*x^(5/2) - 25*a*b^19*x^2 + 100*a^2*b^18*x^(3/2) - 350*a^3*b^17*x + 14
00*a^4*b^16*sqrt(x))/b^25