3.23.27 \(\int \frac {x}{(a+b \sqrt {x})^8} \, dx\) [2227]

Optimal. Leaf size=78 \[ \frac {2 a^3}{7 b^4 \left (a+b \sqrt {x}\right )^7}-\frac {a^2}{b^4 \left (a+b \sqrt {x}\right )^6}+\frac {6 a}{5 b^4 \left (a+b \sqrt {x}\right )^5}-\frac {1}{2 b^4 \left (a+b \sqrt {x}\right )^4} \]

[Out]

2/7*a^3/b^4/(a+b*x^(1/2))^7-a^2/b^4/(a+b*x^(1/2))^6+6/5*a/b^4/(a+b*x^(1/2))^5-1/2/b^4/(a+b*x^(1/2))^4

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Rubi [A]
time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} \frac {2 a^3}{7 b^4 \left (a+b \sqrt {x}\right )^7}-\frac {a^2}{b^4 \left (a+b \sqrt {x}\right )^6}+\frac {6 a}{5 b^4 \left (a+b \sqrt {x}\right )^5}-\frac {1}{2 b^4 \left (a+b \sqrt {x}\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sqrt[x])^8,x]

[Out]

(2*a^3)/(7*b^4*(a + b*Sqrt[x])^7) - a^2/(b^4*(a + b*Sqrt[x])^6) + (6*a)/(5*b^4*(a + b*Sqrt[x])^5) - 1/(2*b^4*(
a + b*Sqrt[x])^4)

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

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Mathematica [A]
time = 0.04, size = 52, normalized size = 0.67 \begin {gather*} \frac {-a^3-7 a^2 b \sqrt {x}-21 a b^2 x-35 b^3 x^{3/2}}{70 b^4 \left (a+b \sqrt {x}\right )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Sqrt[x])^8,x]

[Out]

(-a^3 - 7*a^2*b*Sqrt[x] - 21*a*b^2*x - 35*b^3*x^(3/2))/(70*b^4*(a + b*Sqrt[x])^7)

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Maple [A]
time = 0.19, size = 65, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {2 a^{3}}{7 b^{4} \left (a +b \sqrt {x}\right )^{7}}-\frac {a^{2}}{b^{4} \left (a +b \sqrt {x}\right )^{6}}+\frac {6 a}{5 b^{4} \left (a +b \sqrt {x}\right )^{5}}-\frac {1}{2 b^{4} \left (a +b \sqrt {x}\right )^{4}}\)	\(65\)
default	\(\frac {2 a^{3}}{7 b^{4} \left (a +b \sqrt {x}\right )^{7}}-\frac {a^{2}}{b^{4} \left (a +b \sqrt {x}\right )^{6}}+\frac {6 a}{5 b^{4} \left (a +b \sqrt {x}\right )^{5}}-\frac {1}{2 b^{4} \left (a +b \sqrt {x}\right )^{4}}\)	\(65\)
trager	\(\frac {\left (x -1\right ) \left (-a^{10} b^{10} x^{6}+7 a^{8} b^{12} x^{6}+14 a^{6} b^{14} x^{6}+630 a^{4} b^{16} x^{6}+595 a^{2} b^{18} x^{6}+35 b^{20} x^{6}+7 a^{12} b^{8} x^{5}-50 a^{10} b^{10} x^{5}-91 a^{8} b^{12} x^{5}-4396 a^{6} b^{14} x^{5}-3535 a^{4} b^{16} x^{5}+350 a^{2} b^{18} x^{5}+35 b^{20} x^{5}+14 a^{14} b^{6} x^{4}-91 a^{12} b^{8} x^{4}+979 a^{10} b^{10} x^{4}+11914 a^{8} b^{12} x^{4}+9324 a^{6} b^{14} x^{4}-3535 a^{4} b^{16} x^{4}+595 a^{2} b^{18} x^{4}+630 a^{16} b^{4} x^{3}-4396 a^{14} b^{6} x^{3}+11914 a^{12} b^{8} x^{3}-41896 a^{10} b^{10} x^{3}+11914 a^{8} b^{12} x^{3}-4396 a^{6} b^{14} x^{3}+630 a^{4} b^{16} x^{3}+595 a^{18} b^{2} x^{2}-3535 a^{16} b^{4} x^{2}+9324 a^{14} b^{6} x^{2}+11914 a^{12} b^{8} x^{2}+979 a^{10} b^{10} x^{2}-91 a^{8} b^{12} x^{2}+14 a^{6} b^{14} x^{2}+35 a^{20} x +350 a^{18} b^{2} x -3535 a^{16} b^{4} x -4396 a^{14} b^{6} x -91 a^{12} b^{8} x -50 a^{10} b^{10} x +7 a^{8} b^{12} x +35 a^{20}+595 a^{18} b^{2}+630 a^{16} b^{4}+14 a^{14} b^{6}+7 b^{8} a^{12}-b^{10} a^{10}\right )}{70 \left (-b^{2} x +a^{2}\right )^{7} \left (a^{14}-7 a^{12} b^{2}+21 a^{10} b^{4}-35 a^{8} b^{6}+35 a^{6} b^{8}-21 a^{4} b^{10}+7 a^{2} b^{12}-b^{14}\right )}-\frac {16 b a \,x^{\frac {5}{2}} \left (7 b^{4} x^{2}+26 a^{2} b^{2} x +7 a^{4}\right )}{35 \left (-b^{2} x +a^{2}\right )^{7}}\)	\(594\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*a^3/b^4/(a+b*x^(1/2))^7-a^2/b^4/(a+b*x^(1/2))^6+6/5*a/b^4/(a+b*x^(1/2))^5-1/2/b^4/(a+b*x^(1/2))^4

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Maxima [A]
time = 0.30, size = 64, normalized size = 0.82 \begin {gather*} -\frac {1}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{4}} + \frac {6 \, a}{5 \, {\left (b \sqrt {x} + a\right )}^{5} b^{4}} - \frac {a^{2}}{{\left (b \sqrt {x} + a\right )}^{6} b^{4}} + \frac {2 \, a^{3}}{7 \, {\left (b \sqrt {x} + a\right )}^{7} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((b*sqrt(x) + a)^4*b^4) + 6/5*a/((b*sqrt(x) + a)^5*b^4) - a^2/((b*sqrt(x) + a)^6*b^4) + 2/7*a^3/((b*sqrt(
x) + a)^7*b^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (64) = 128\).
time = 0.36, size = 177, normalized size = 2.27 \begin {gather*} -\frac {35 \, b^{10} x^{5} + 595 \, a^{2} b^{8} x^{4} + 630 \, a^{4} b^{6} x^{3} + 14 \, a^{6} b^{4} x^{2} + 7 \, a^{8} b^{2} x - a^{10} - 32 \, {\left (7 \, a b^{9} x^{4} + 26 \, a^{3} b^{7} x^{3} + 7 \, a^{5} b^{5} x^{2}\right )} \sqrt {x}}{70 \, {\left (b^{18} x^{7} - 7 \, a^{2} b^{16} x^{6} + 21 \, a^{4} b^{14} x^{5} - 35 \, a^{6} b^{12} x^{4} + 35 \, a^{8} b^{10} x^{3} - 21 \, a^{10} b^{8} x^{2} + 7 \, a^{12} b^{6} x - a^{14} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/70*(35*b^10*x^5 + 595*a^2*b^8*x^4 + 630*a^4*b^6*x^3 + 14*a^6*b^4*x^2 + 7*a^8*b^2*x - a^10 - 32*(7*a*b^9*x^4
+ 26*a^3*b^7*x^3 + 7*a^5*b^5*x^2)*sqrt(x))/(b^18*x^7 - 7*a^2*b^16*x^6 + 21*a^4*b^14*x^5 - 35*a^6*b^12*x^4 + 3
5*a^8*b^10*x^3 - 21*a^10*b^8*x^2 + 7*a^12*b^6*x - a^14*b^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (71) = 142\).
time = 1.35, size = 410, normalized size = 5.26 \begin {gather*} \begin {cases} - \frac {a^{3}}{70 a^{7} b^{4} + 490 a^{6} b^{5} \sqrt {x} + 1470 a^{5} b^{6} x + 2450 a^{4} b^{7} x^{\frac {3}{2}} + 2450 a^{3} b^{8} x^{2} + 1470 a^{2} b^{9} x^{\frac {5}{2}} + 490 a b^{10} x^{3} + 70 b^{11} x^{\frac {7}{2}}} - \frac {7 a^{2} b \sqrt {x}}{70 a^{7} b^{4} + 490 a^{6} b^{5} \sqrt {x} + 1470 a^{5} b^{6} x + 2450 a^{4} b^{7} x^{\frac {3}{2}} + 2450 a^{3} b^{8} x^{2} + 1470 a^{2} b^{9} x^{\frac {5}{2}} + 490 a b^{10} x^{3} + 70 b^{11} x^{\frac {7}{2}}} - \frac {21 a b^{2} x}{70 a^{7} b^{4} + 490 a^{6} b^{5} \sqrt {x} + 1470 a^{5} b^{6} x + 2450 a^{4} b^{7} x^{\frac {3}{2}} + 2450 a^{3} b^{8} x^{2} + 1470 a^{2} b^{9} x^{\frac {5}{2}} + 490 a b^{10} x^{3} + 70 b^{11} x^{\frac {7}{2}}} - \frac {35 b^{3} x^{\frac {3}{2}}}{70 a^{7} b^{4} + 490 a^{6} b^{5} \sqrt {x} + 1470 a^{5} b^{6} x + 2450 a^{4} b^{7} x^{\frac {3}{2}} + 2450 a^{3} b^{8} x^{2} + 1470 a^{2} b^{9} x^{\frac {5}{2}} + 490 a b^{10} x^{3} + 70 b^{11} x^{\frac {7}{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{8}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**3/(70*a**7*b**4 + 490*a**6*b**5*sqrt(x) + 1470*a**5*b**6*x + 2450*a**4*b**7*x**(3/2) + 2450*a**
3*b**8*x**2 + 1470*a**2*b**9*x**(5/2) + 490*a*b**10*x**3 + 70*b**11*x**(7/2)) - 7*a**2*b*sqrt(x)/(70*a**7*b**4
+ 490*a**6*b**5*sqrt(x) + 1470*a**5*b**6*x + 2450*a**4*b**7*x**(3/2) + 2450*a**3*b**8*x**2 + 1470*a**2*b**9*x
**(5/2) + 490*a*b**10*x**3 + 70*b**11*x**(7/2)) - 21*a*b**2*x/(70*a**7*b**4 + 490*a**6*b**5*sqrt(x) + 1470*a**
5*b**6*x + 2450*a**4*b**7*x**(3/2) + 2450*a**3*b**8*x**2 + 1470*a**2*b**9*x**(5/2) + 490*a*b**10*x**3 + 70*b**
11*x**(7/2)) - 35*b**3*x**(3/2)/(70*a**7*b**4 + 490*a**6*b**5*sqrt(x) + 1470*a**5*b**6*x + 2450*a**4*b**7*x**(
3/2) + 2450*a**3*b**8*x**2 + 1470*a**2*b**9*x**(5/2) + 490*a*b**10*x**3 + 70*b**11*x**(7/2)), Ne(b, 0)), (x**2
/(2*a**8), True))

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Giac [A]
time = 1.13, size = 42, normalized size = 0.54 \begin {gather*} -\frac {35 \, b^{3} x^{\frac {3}{2}} + 21 \, a b^{2} x + 7 \, a^{2} b \sqrt {x} + a^{3}}{70 \, {\left (b \sqrt {x} + a\right )}^{7} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/70*(35*b^3*x^(3/2) + 21*a*b^2*x + 7*a^2*b*sqrt(x) + a^3)/((b*sqrt(x) + a)^7*b^4)

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Mupad [B]
time = 1.21, size = 110, normalized size = 1.41 \begin {gather*} -\frac {\frac {a^3}{70\,b^4}+\frac {x^{3/2}}{2\,b}+\frac {a^2\,\sqrt {x}}{10\,b^3}+\frac {3\,a\,x}{10\,b^2}}{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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^(1/2))^8,x)

[Out]

-(a^3/(70*b^4) + x^(3/2)/(2*b) + (a^2*x^(1/2))/(10*b^3) + (3*a*x)/(10*b^2))/(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))

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