\(\int \frac {x^3}{(a+b \sqrt {x})^8} \, dx\) [2225]

Optimal result

Integrand size = 15, antiderivative size = 157 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 a^7}{7 b^8 \left (a+b \sqrt {x}\right )^7}-\frac {7 a^6}{3 b^8 \left (a+b \sqrt {x}\right )^6}+\frac {42 a^5}{5 b^8 \left (a+b \sqrt {x}\right )^5}-\frac {35 a^4}{2 b^8 \left (a+b \sqrt {x}\right )^4}+\frac {70 a^3}{3 b^8 \left (a+b \sqrt {x}\right )^3}-\frac {21 a^2}{b^8 \left (a+b \sqrt {x}\right )^2}+\frac {14 a}{b^8 \left (a+b \sqrt {x}\right )}+\frac {2 \log \left (a+b \sqrt {x}\right )}{b^8} \]

[Out]

2*ln(a+b*x^(1/2))/b^8+2/7*a^7/b^8/(a+b*x^(1/2))^7-7/3*a^6/b^8/(a+b*x^(1/2))^6+42/5*a^5/b^8/(a+b*x^(1/2))^5-35/
2*a^4/b^8/(a+b*x^(1/2))^4+70/3*a^3/b^8/(a+b*x^(1/2))^3-21*a^2/b^8/(a+b*x^(1/2))^2+14*a/b^8/(a+b*x^(1/2))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 a^7}{7 b^8 \left (a+b \sqrt {x}\right )^7}-\frac {7 a^6}{3 b^8 \left (a+b \sqrt {x}\right )^6}+\frac {42 a^5}{5 b^8 \left (a+b \sqrt {x}\right )^5}-\frac {35 a^4}{2 b^8 \left (a+b \sqrt {x}\right )^4}+\frac {70 a^3}{3 b^8 \left (a+b \sqrt {x}\right )^3}-\frac {21 a^2}{b^8 \left (a+b \sqrt {x}\right )^2}+\frac {14 a}{b^8 \left (a+b \sqrt {x}\right )}+\frac {2 \log \left (a+b \sqrt {x}\right )}{b^8} \]

[In]

Int[x^3/(a + b*Sqrt[x])^8,x]

[Out]

(2*a^7)/(7*b^8*(a + b*Sqrt[x])^7) - (7*a^6)/(3*b^8*(a + b*Sqrt[x])^6) + (42*a^5)/(5*b^8*(a + b*Sqrt[x])^5) - (
35*a^4)/(2*b^8*(a + b*Sqrt[x])^4) + (70*a^3)/(3*b^8*(a + b*Sqrt[x])^3) - (21*a^2)/(b^8*(a + b*Sqrt[x])^2) + (1
4*a)/(b^8*(a + b*Sqrt[x])) + (2*Log[a + b*Sqrt[x]])/b^8

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])

Rule 272

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*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^7}{(a+b x)^8} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {a^7}{b^7 (a+b x)^8}+\frac {7 a^6}{b^7 (a+b x)^7}-\frac {21 a^5}{b^7 (a+b x)^6}+\frac {35 a^4}{b^7 (a+b x)^5}-\frac {35 a^3}{b^7 (a+b x)^4}+\frac {21 a^2}{b^7 (a+b x)^3}-\frac {7 a}{b^7 (a+b x)^2}+\frac {1}{b^7 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 a^7}{7 b^8 \left (a+b \sqrt {x}\right )^7}-\frac {7 a^6}{3 b^8 \left (a+b \sqrt {x}\right )^6}+\frac {42 a^5}{5 b^8 \left (a+b \sqrt {x}\right )^5}-\frac {35 a^4}{2 b^8 \left (a+b \sqrt {x}\right )^4}+\frac {70 a^3}{3 b^8 \left (a+b \sqrt {x}\right )^3}-\frac {21 a^2}{b^8 \left (a+b \sqrt {x}\right )^2}+\frac {14 a}{b^8 \left (a+b \sqrt {x}\right )}+\frac {2 \log \left (a+b \sqrt {x}\right )}{b^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.66 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {a \left (1089 a^6+7203 a^5 b \sqrt {x}+20139 a^4 b^2 x+30625 a^3 b^3 x^{3/2}+26950 a^2 b^4 x^2+13230 a b^5 x^{5/2}+2940 b^6 x^3\right )}{210 b^8 \left (a+b \sqrt {x}\right )^7}+\frac {2 \log \left (a+b \sqrt {x}\right )}{b^8} \]

[In]

Integrate[x^3/(a + b*Sqrt[x])^8,x]

[Out]

(a*(1089*a^6 + 7203*a^5*b*Sqrt[x] + 20139*a^4*b^2*x + 30625*a^3*b^3*x^(3/2) + 26950*a^2*b^4*x^2 + 13230*a*b^5*
x^(5/2) + 2940*b^6*x^3))/(210*b^8*(a + b*Sqrt[x])^7) + (2*Log[a + b*Sqrt[x]])/b^8

Maple [A] (verified)

Time = 5.99 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {2 \ln \left (a +b \sqrt {x}\right )}{b^{8}}+\frac {2 a^{7}}{7 b^{8} \left (a +b \sqrt {x}\right )^{7}}-\frac {7 a^{6}}{3 b^{8} \left (a +b \sqrt {x}\right )^{6}}+\frac {42 a^{5}}{5 b^{8} \left (a +b \sqrt {x}\right )^{5}}-\frac {35 a^{4}}{2 b^{8} \left (a +b \sqrt {x}\right )^{4}}+\frac {70 a^{3}}{3 b^{8} \left (a +b \sqrt {x}\right )^{3}}-\frac {21 a^{2}}{b^{8} \left (a +b \sqrt {x}\right )^{2}}+\frac {14 a}{b^{8} \left (a +b \sqrt {x}\right )}\)	\(132\)
default	\(\frac {2 \ln \left (a +b \sqrt {x}\right )}{b^{8}}+\frac {2 a^{7}}{7 b^{8} \left (a +b \sqrt {x}\right )^{7}}-\frac {7 a^{6}}{3 b^{8} \left (a +b \sqrt {x}\right )^{6}}+\frac {42 a^{5}}{5 b^{8} \left (a +b \sqrt {x}\right )^{5}}-\frac {35 a^{4}}{2 b^{8} \left (a +b \sqrt {x}\right )^{4}}+\frac {70 a^{3}}{3 b^{8} \left (a +b \sqrt {x}\right )^{3}}-\frac {21 a^{2}}{b^{8} \left (a +b \sqrt {x}\right )^{2}}+\frac {14 a}{b^{8} \left (a +b \sqrt {x}\right )}\)	\(132\)

[In]

int(x^3/(a+b*x^(1/2))^8,x,method=_RETURNVERBOSE)

[Out]

2*ln(a+b*x^(1/2))/b^8+2/7*a^7/b^8/(a+b*x^(1/2))^7-7/3*a^6/b^8/(a+b*x^(1/2))^6+42/5*a^5/b^8/(a+b*x^(1/2))^5-35/
2*a^4/b^8/(a+b*x^(1/2))^4+70/3*a^3/b^8/(a+b*x^(1/2))^3-21*a^2/b^8/(a+b*x^(1/2))^2+14*a/b^8/(a+b*x^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (131) = 262\).

Time = 0.30 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.01 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=-\frac {7350 \, a^{2} b^{12} x^{6} - 16905 \, a^{4} b^{10} x^{5} + 32585 \, a^{6} b^{8} x^{4} - 34370 \, a^{8} b^{6} x^{3} + 21504 \, a^{10} b^{4} x^{2} - 7413 \, a^{12} b^{2} x + 1089 \, a^{14} - 420 \, {\left (b^{14} x^{7} - 7 \, a^{2} b^{12} x^{6} + 21 \, a^{4} b^{10} x^{5} - 35 \, a^{6} b^{8} x^{4} + 35 \, a^{8} b^{6} x^{3} - 21 \, a^{10} b^{4} x^{2} + 7 \, a^{12} b^{2} x - a^{14}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (735 \, a b^{13} x^{6} - 980 \, a^{3} b^{11} x^{5} + 2891 \, a^{5} b^{9} x^{4} - 3072 \, a^{7} b^{7} x^{3} + 1981 \, a^{9} b^{5} x^{2} - 700 \, a^{11} b^{3} x + 105 \, a^{13} b\right )} \sqrt {x}}{210 \, {\left (b^{22} x^{7} - 7 \, a^{2} b^{20} x^{6} + 21 \, a^{4} b^{18} x^{5} - 35 \, a^{6} b^{16} x^{4} + 35 \, a^{8} b^{14} x^{3} - 21 \, a^{10} b^{12} x^{2} + 7 \, a^{12} b^{10} x - a^{14} b^{8}\right )}} \]

[In]

integrate(x^3/(a+b*x^(1/2))^8,x, algorithm="fricas")

[Out]

-1/210*(7350*a^2*b^12*x^6 - 16905*a^4*b^10*x^5 + 32585*a^6*b^8*x^4 - 34370*a^8*b^6*x^3 + 21504*a^10*b^4*x^2 -
7413*a^12*b^2*x + 1089*a^14 - 420*(b^14*x^7 - 7*a^2*b^12*x^6 + 21*a^4*b^10*x^5 - 35*a^6*b^8*x^4 + 35*a^8*b^6*x
^3 - 21*a^10*b^4*x^2 + 7*a^12*b^2*x - a^14)*log(b*sqrt(x) + a) - 4*(735*a*b^13*x^6 - 980*a^3*b^11*x^5 + 2891*a
^5*b^9*x^4 - 3072*a^7*b^7*x^3 + 1981*a^9*b^5*x^2 - 700*a^11*b^3*x + 105*a^13*b)*sqrt(x))/(b^22*x^7 - 7*a^2*b^2
0*x^6 + 21*a^4*b^18*x^5 - 35*a^6*b^16*x^4 + 35*a^8*b^14*x^3 - 21*a^10*b^12*x^2 + 7*a^12*b^10*x - a^14*b^8)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1629 vs. \(2 (150) = 300\).

Time = 1.61 (sec) , antiderivative size = 1629, normalized size of antiderivative = 10.38 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\text {Too large to display} \]

[In]

integrate(x**3/(a+b*x**(1/2))**8,x)

[Out]

Piecewise((420*a**7*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4
*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) +
1089*a**7/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b
**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 2940*a**6*b*sqrt(x)*log(a/b +
sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b
**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 7203*a**6*b*sqrt(x)/(210*a**7
*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a*
*2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 8820*a**5*b**2*x*log(a/b + sqrt(x))/(210*a**7*b*
*8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*
b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 20139*a**5*b**2*x/(210*a**7*b**8 + 1470*a**6*b**9*s
qrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470
*a*b**14*x**3 + 210*b**15*x**(7/2)) + 14700*a**4*b**3*x**(3/2)*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b
**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) +
1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 30625*a**4*b**3*x**(3/2)/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) +
4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*
x**3 + 210*b**15*x**(7/2)) + 14700*a**3*b**4*x**2*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) +
4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14
*x**3 + 210*b**15*x**(7/2)) + 26950*a**3*b**4*x**2/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x
+ 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*
x**(7/2)) + 8820*a**2*b**5*x**(5/2)*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**
10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b*
*15*x**(7/2)) + 13230*a**2*b**5*x**(5/2)/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a*
*4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2))
+ 2940*a*b**6*x**3*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*
b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 2
940*a*b**6*x**3/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*
a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 420*b**7*x**(7/2)*log(a
/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a*
*3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)), Ne(b, 0)), (x**4/(4*a**8),
True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.83 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 \, \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {14 \, a}{{\left (b \sqrt {x} + a\right )} b^{8}} - \frac {21 \, a^{2}}{{\left (b \sqrt {x} + a\right )}^{2} b^{8}} + \frac {70 \, a^{3}}{3 \, {\left (b \sqrt {x} + a\right )}^{3} b^{8}} - \frac {35 \, a^{4}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{8}} + \frac {42 \, a^{5}}{5 \, {\left (b \sqrt {x} + a\right )}^{5} b^{8}} - \frac {7 \, a^{6}}{3 \, {\left (b \sqrt {x} + a\right )}^{6} b^{8}} + \frac {2 \, a^{7}}{7 \, {\left (b \sqrt {x} + a\right )}^{7} b^{8}} \]

[In]

integrate(x^3/(a+b*x^(1/2))^8,x, algorithm="maxima")

[Out]

2*log(b*sqrt(x) + a)/b^8 + 14*a/((b*sqrt(x) + a)*b^8) - 21*a^2/((b*sqrt(x) + a)^2*b^8) + 70/3*a^3/((b*sqrt(x)
+ a)^3*b^8) - 35/2*a^4/((b*sqrt(x) + a)^4*b^8) + 42/5*a^5/((b*sqrt(x) + a)^5*b^8) - 7/3*a^6/((b*sqrt(x) + a)^6
*b^8) + 2/7*a^7/((b*sqrt(x) + a)^7*b^8)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.61 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 \, \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} + \frac {2940 \, a b^{5} x^{3} + 13230 \, a^{2} b^{4} x^{\frac {5}{2}} + 26950 \, a^{3} b^{3} x^{2} + 30625 \, a^{4} b^{2} x^{\frac {3}{2}} + 20139 \, a^{5} b x + 7203 \, a^{6} \sqrt {x} + \frac {1089 \, a^{7}}{b}}{210 \, {\left (b \sqrt {x} + a\right )}^{7} b^{7}} \]

[In]

integrate(x^3/(a+b*x^(1/2))^8,x, algorithm="giac")

[Out]

2*log(abs(b*sqrt(x) + a))/b^8 + 1/210*(2940*a*b^5*x^3 + 13230*a^2*b^4*x^(5/2) + 26950*a^3*b^3*x^2 + 30625*a^4*
b^2*x^(3/2) + 20139*a^5*b*x + 7203*a^6*sqrt(x) + 1089*a^7/b)/((b*sqrt(x) + a)^7*b^7)

Mupad [B] (verification not implemented)

Time = 5.98 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {\frac {363\,a^7}{70\,b^8}+\frac {14\,a\,x^3}{b^2}+\frac {959\,a^5\,x}{10\,b^6}+\frac {385\,a^3\,x^2}{3\,b^4}+\frac {63\,a^2\,x^{5/2}}{b^3}+\frac {875\,a^4\,x^{3/2}}{6\,b^5}+\frac {343\,a^6\,\sqrt {x}}{10\,b^7}}{a^7+b^7\,x^{7/2}+21\,a^5\,b^2\,x+7\,a\,b^6\,x^3+7\,a^6\,b\,\sqrt {x}+35\,a^3\,b^4\,x^2+35\,a^4\,b^3\,x^{3/2}+21\,a^2\,b^5\,x^{5/2}}+\frac {2\,\ln \left (a+b\,\sqrt {x}\right )}{b^8} \]

[In]

int(x^3/(a + b*x^(1/2))^8,x)

[Out]

((363*a^7)/(70*b^8) + (14*a*x^3)/b^2 + (959*a^5*x)/(10*b^6) + (385*a^3*x^2)/(3*b^4) + (63*a^2*x^(5/2))/b^3 + (
875*a^4*x^(3/2))/(6*b^5) + (343*a^6*x^(1/2))/(10*b^7))/(a^7 + b^7*x^(7/2) + 21*a^5*b^2*x + 7*a*b^6*x^3 + 7*a^6
*b*x^(1/2) + 35*a^3*b^4*x^2 + 35*a^4*b^3*x^(3/2) + 21*a^2*b^5*x^(5/2)) + (2*log(a + b*x^(1/2)))/b^8