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

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

[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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Rubi [A]  time = 0.14, antiderivative size = 155, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.19, size = 150, normalized size = 0.97 \[ \frac {-1375 a^9-2980 a^8 b \sqrt {x}+570 a^7 b^2 x+5420 a^6 b^3 x^{3/2}+3875 a^5 b^4 x^2-2520 a^5 \left (a+b \sqrt {x}\right )^4 \log \left (a+b \sqrt {x}\right )+504 a^4 b^5 x^{5/2}-84 a^3 b^6 x^3+24 a^2 b^7 x^{7/2}-9 a b^8 x^4+4 b^9 x^{9/2}}{10 b^{10} \left (a+b \sqrt {x}\right )^4} \]

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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fricas [A]  time = 1.10, size = 245, normalized size = 1.58 \[ -\frac {25 \, a b^{12} x^{6} + 250 \, a^{3} b^{10} x^{5} - 1250 \, a^{5} b^{8} x^{4} - 40 \, a^{7} b^{6} x^{3} + 3840 \, a^{9} b^{4} x^{2} - 4240 \, a^{11} b^{2} x + 1375 \, a^{13} + 2520 \, {\left (a^{5} b^{8} x^{4} - 4 \, a^{7} b^{6} x^{3} + 6 \, a^{9} b^{4} x^{2} - 4 \, a^{11} b^{2} x + a^{13}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (b^{13} x^{6} + 21 \, a^{2} b^{11} x^{5} + 256 \, a^{4} b^{9} x^{4} - 1674 \, a^{6} b^{7} x^{3} + 3066 \, a^{8} b^{5} x^{2} - 2310 \, a^{10} b^{3} x + 630 \, a^{12} b\right )} \sqrt {x}}{10 \, {\left (b^{18} x^{4} - 4 \, a^{2} b^{16} x^{3} + 6 \, a^{4} b^{14} x^{2} - 4 \, a^{6} b^{12} x + a^{8} b^{10}\right )}} \]

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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giac [A]  time = 0.19, size = 121, normalized size = 0.78 \[ -\frac {252 \, a^{5} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{10}} - \frac {336 \, a^{6} b^{3} x^{\frac {3}{2}} + 936 \, a^{7} b^{2} x + 876 \, a^{8} b \sqrt {x} + 275 \, a^{9}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{10}} + \frac {4 \, b^{20} x^{\frac {5}{2}} - 25 \, a b^{19} x^{2} + 100 \, a^{2} b^{18} x^{\frac {3}{2}} - 350 \, a^{3} b^{17} x + 1400 \, a^{4} b^{16} \sqrt {x}}{10 \, b^{25}} \]

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

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maple [A]  time = 0.01, size = 134, normalized size = 0.86 \[ \frac {a^{9}}{2 \left (b \sqrt {x}+a \right )^{4} b^{10}}-\frac {6 a^{8}}{\left (b \sqrt {x}+a \right )^{3} b^{10}}+\frac {2 x^{\frac {5}{2}}}{5 b^{5}}+\frac {36 a^{7}}{\left (b \sqrt {x}+a \right )^{2} b^{10}}-\frac {5 a \,x^{2}}{2 b^{6}}+\frac {10 a^{2} x^{\frac {3}{2}}}{b^{7}}-\frac {168 a^{6}}{\left (b \sqrt {x}+a \right ) b^{10}}-\frac {252 a^{5} \ln \left (b \sqrt {x}+a \right )}{b^{10}}-\frac {35 a^{3} x}{b^{8}}+\frac {140 a^{4} \sqrt {x}}{b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.85, size = 163, normalized size = 1.05 \[ -\frac {252 \, a^{5} \log \left (b \sqrt {x} + a\right )}{b^{10}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{5}}{5 \, b^{10}} - \frac {9 \, {\left (b \sqrt {x} + a\right )}^{4} a}{2 \, b^{10}} + \frac {24 \, {\left (b \sqrt {x} + a\right )}^{3} a^{2}}{b^{10}} - \frac {84 \, {\left (b \sqrt {x} + a\right )}^{2} a^{3}}{b^{10}} + \frac {252 \, {\left (b \sqrt {x} + a\right )} a^{4}}{b^{10}} - \frac {168 \, a^{6}}{{\left (b \sqrt {x} + a\right )} b^{10}} + \frac {36 \, a^{7}}{{\left (b \sqrt {x} + a\right )}^{2} b^{10}} - \frac {6 \, a^{8}}{{\left (b \sqrt {x} + a\right )}^{3} b^{10}} + \frac {a^{9}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{10}} \]

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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mupad [B]  time = 0.06, size = 148, normalized size = 0.95 \[ \frac {2\,x^{5/2}}{5\,b^5}-\frac {\frac {275\,a^9}{2\,b}+438\,a^8\,\sqrt {x}+168\,a^6\,b^2\,x^{3/2}+468\,a^7\,b\,x}{a^4\,b^9+b^{13}\,x^2+6\,a^2\,b^{11}\,x+4\,a\,b^{12}\,x^{3/2}+4\,a^3\,b^{10}\,\sqrt {x}}-\frac {5\,a\,x^2}{2\,b^6}-\frac {35\,a^3\,x}{b^8}-\frac {252\,a^5\,\ln \left (a+b\,\sqrt {x}\right )}{b^{10}}+\frac {10\,a^2\,x^{3/2}}{b^7}+\frac {140\,a^4\,\sqrt {x}}{b^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b*x^(1/2))^5,x)

[Out]

(2*x^(5/2))/(5*b^5) - ((275*a^9)/(2*b) + 438*a^8*x^(1/2) + 168*a^6*b^2*x^(3/2) + 468*a^7*b*x)/(a^4*b^9 + b^13*
x^2 + 6*a^2*b^11*x + 4*a*b^12*x^(3/2) + 4*a^3*b^10*x^(1/2)) - (5*a*x^2)/(2*b^6) - (35*a^3*x)/b^8 - (252*a^5*lo
g(a + b*x^(1/2)))/b^10 + (10*a^2*x^(3/2))/b^7 + (140*a^4*x^(1/2))/b^9

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sympy [A]  time = 5.95, size = 949, normalized size = 6.12 \[ \begin {cases} - \frac {2520 a^{9} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{4} b^{10} + 40 a^{3} b^{11} \sqrt {x} + 60 a^{2} b^{12} x + 40 a b^{13} x^{\frac {3}{2}} + 10 b^{14} x^{2}} - \frac {5250 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^{\frac {3}{2}} + 10 b^{14} x^{2}} - \frac {10080 a^{8} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{4} b^{10} + 40 a^{3} b^{11} \sqrt {x} + 60 a^{2} b^{12} x + 40 a b^{13} x^{\frac {3}{2}} + 10 b^{14} x^{2}} - \frac {18480 a^{8} 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^{\frac {3}{2}} + 10 b^{14} x^{2}} - \frac {15120 a^{7} b^{2} x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{4} b^{10} + 40 a^{3} b^{11} \sqrt {x} + 60 a^{2} b^{12} x + 40 a b^{13} x^{\frac {3}{2}} + 10 b^{14} x^{2}} - \frac {22680 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^{\frac {3}{2}} + 10 b^{14} x^{2}} - \frac {10080 a^{6} b^{3} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{4} b^{10} + 40 a^{3} b^{11} \sqrt {x} + 60 a^{2} b^{12} x + 40 a b^{13} x^{\frac {3}{2}} + 10 b^{14} x^{2}} - \frac {10080 a^{6} b^{3} x^{\frac {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^{\frac {3}{2}} + 10 b^{14} x^{2}} - \frac {2520 a^{5} b^{4} x^{2} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{4} b^{10} + 40 a^{3} b^{11} \sqrt {x} + 60 a^{2} b^{12} x + 40 a b^{13} x^{\frac {3}{2}} + 10 b^{14} x^{2}} + \frac {504 a^{4} b^{5} x^{\frac {5}{2}}}{10 a^{4} b^{10} + 40 a^{3} b^{11} \sqrt {x} + 60 a^{2} b^{12} x + 40 a b^{13} x^{\frac {3}{2}} + 10 b^{14} x^{2}} - \frac {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^{\frac {3}{2}} + 10 b^{14} x^{2}} + \frac {24 a^{2} b^{7} x^{\frac {7}{2}}}{10 a^{4} b^{10} + 40 a^{3} b^{11} \sqrt {x} + 60 a^{2} b^{12} x + 40 a b^{13} x^{\frac {3}{2}} + 10 b^{14} x^{2}} - \frac {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^{\frac {3}{2}} + 10 b^{14} x^{2}} + \frac {4 b^{9} x^{\frac {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^{\frac {3}{2}} + 10 b^{14} x^{2}} & \text {for}\: b \neq 0 \\\frac {x^{5}}{5 a^{5}} & \text {otherwise} \end {cases} \]

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) - 5250*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) - 18480*a**8*b*sqrt(x)/(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) - 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) - 22680*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)*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) - 10080*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**1
3*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**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
*sqrt(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
+ 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) - 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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