3.2228 \(\int \frac {1}{(a+b \sqrt {x})^8} \, dx\)

Optimal. Leaf size=38 \[ \frac {2 a}{7 b^2 \left (a+b \sqrt {x}\right )^7}-\frac {1}{3 b^2 \left (a+b \sqrt {x}\right )^6} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac {2 a}{7 b^2 \left (a+b \sqrt {x}\right )^7}-\frac {1}{3 b^2 \left (a+b \sqrt {x}\right )^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^(-8),x]

[Out]

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

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 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 28, normalized size = 0.74 \[ -\frac {a+7 b \sqrt {x}}{21 b^2 \left (a+b \sqrt {x}\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^(-8),x]

[Out]

-1/21*(a + 7*b*Sqrt[x])/(b^2*(a + b*Sqrt[x])^7)

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fricas [B]  time = 0.94, size = 164, normalized size = 4.32 \[ -\frac {7 \, b^{8} x^{4} + 140 \, a^{2} b^{6} x^{3} + 210 \, a^{4} b^{4} x^{2} + 28 \, a^{6} b^{2} x - a^{8} - 16 \, {\left (3 \, a b^{7} x^{3} + 14 \, a^{3} b^{5} x^{2} + 7 \, a^{5} b^{3} x\right )} \sqrt {x}}{21 \, {\left (b^{16} x^{7} - 7 \, a^{2} b^{14} x^{6} + 21 \, a^{4} b^{12} x^{5} - 35 \, a^{6} b^{10} x^{4} + 35 \, a^{8} b^{8} x^{3} - 21 \, a^{10} b^{6} x^{2} + 7 \, a^{12} b^{4} x - a^{14} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(7*b^8*x^4 + 140*a^2*b^6*x^3 + 210*a^4*b^4*x^2 + 28*a^6*b^2*x - a^8 - 16*(3*a*b^7*x^3 + 14*a^3*b^5*x^2 +
7*a^5*b^3*x)*sqrt(x))/(b^16*x^7 - 7*a^2*b^14*x^6 + 21*a^4*b^12*x^5 - 35*a^6*b^10*x^4 + 35*a^8*b^8*x^3 - 21*a^
10*b^6*x^2 + 7*a^12*b^4*x - a^14*b^2)

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giac [A]  time = 0.18, size = 22, normalized size = 0.58 \[ -\frac {7 \, b \sqrt {x} + a}{21 \, {\left (b \sqrt {x} + a\right )}^{7} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(7*b*sqrt(x) + a)/((b*sqrt(x) + a)^7*b^2)

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maple [B]  time = 0.07, size = 399, normalized size = 10.50 \[ 28 \left (-\frac {a^{2}}{7 \left (b^{2} x -a^{2}\right )^{7} b^{4}}-\frac {1}{6 \left (b^{2} x -a^{2}\right )^{6} b^{4}}\right ) a^{6} b^{2}+70 \left (-\frac {a^{4}}{7 \left (b^{2} x -a^{2}\right )^{7} b^{6}}-\frac {a^{2}}{3 \left (b^{2} x -a^{2}\right )^{6} b^{6}}-\frac {1}{5 \left (b^{2} x -a^{2}\right )^{5} b^{6}}\right ) a^{4} b^{4}+28 \left (-\frac {a^{6}}{7 \left (b^{2} x -a^{2}\right )^{7} b^{8}}-\frac {a^{4}}{2 \left (b^{2} x -a^{2}\right )^{6} b^{8}}-\frac {3 a^{2}}{5 \left (b^{2} x -a^{2}\right )^{5} b^{8}}-\frac {1}{4 \left (b^{2} x -a^{2}\right )^{4} b^{8}}\right ) a^{2} b^{6}+\left (-\frac {a^{8}}{7 \left (b^{2} x -a^{2}\right )^{7} b^{10}}-\frac {2 a^{6}}{3 \left (b^{2} x -a^{2}\right )^{6} b^{10}}-\frac {6 a^{4}}{5 \left (b^{2} x -a^{2}\right )^{5} b^{10}}-\frac {a^{2}}{\left (b^{2} x -a^{2}\right )^{4} b^{10}}-\frac {1}{3 \left (b^{2} x -a^{2}\right )^{3} b^{10}}\right ) b^{8}-\frac {a^{8}}{7 \left (b^{2} x -a^{2}\right )^{7} b^{2}}+\frac {a}{7 \left (b \sqrt {x}+a \right )^{7} b^{2}}+\frac {a}{7 \left (b \sqrt {x}-a \right )^{7} b^{2}}-\frac {1}{6 \left (b \sqrt {x}+a \right )^{6} b^{2}}+\frac {1}{6 \left (b \sqrt {x}-a \right )^{6} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^8/(b^2*x-a^2)^7/b^2+b^8*(-a^2/b^10/(b^2*x-a^2)^4-1/3/b^10/(b^2*x-a^2)^3-6/5*a^4/b^10/(b^2*x-a^2)^5-1/7*
a^8/b^10/(b^2*x-a^2)^7-2/3*a^6/b^10/(b^2*x-a^2)^6)+1/7*a/b^2/(b*x^(1/2)+a)^7+1/7*a/b^2/(b*x^(1/2)-a)^7-1/6/b^2
/(b*x^(1/2)+a)^6+1/6/b^2/(b*x^(1/2)-a)^6+28*b^6*a^2*(-1/4/(b^2*x-a^2)^4/b^8-3/5*a^2/b^8/(b^2*x-a^2)^5-1/7*a^6/
b^8/(b^2*x-a^2)^7-1/2*a^4/b^8/(b^2*x-a^2)^6)+28*b^2*a^6*(-1/7*a^2/b^4/(b^2*x-a^2)^7-1/6/b^4/(b^2*x-a^2)^6)+70*
a^4*b^4*(-1/5/b^6/(b^2*x-a^2)^5-1/7*a^4/b^6/(b^2*x-a^2)^7-1/3*a^2/b^6/(b^2*x-a^2)^6)

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maxima [A]  time = 0.86, size = 30, normalized size = 0.79 \[ -\frac {1}{3 \, {\left (b \sqrt {x} + a\right )}^{6} b^{2}} + \frac {2 \, a}{7 \, {\left (b \sqrt {x} + a\right )}^{7} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.22, size = 90, normalized size = 2.37 \[ -\frac {\frac {a}{21\,b^2}+\frac {\sqrt {x}}{3\,b}}{a^7+b^7\,x^{7/2}+21\,a^5\,b^2\,x+7\,a\,b^6\,x^3+7\,a^6\,b\,\sqrt {x}+35\,a^3\,b^4\,x^2+35\,a^4\,b^3\,x^{3/2}+21\,a^2\,b^5\,x^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^(1/2))^8,x)

[Out]

-(a/(21*b^2) + x^(1/2)/(3*b))/(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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sympy [A]  time = 5.67, size = 199, normalized size = 5.24 \[ \begin {cases} - \frac {a}{21 a^{7} b^{2} + 147 a^{6} b^{3} \sqrt {x} + 441 a^{5} b^{4} x + 735 a^{4} b^{5} x^{\frac {3}{2}} + 735 a^{3} b^{6} x^{2} + 441 a^{2} b^{7} x^{\frac {5}{2}} + 147 a b^{8} x^{3} + 21 b^{9} x^{\frac {7}{2}}} - \frac {7 b \sqrt {x}}{21 a^{7} b^{2} + 147 a^{6} b^{3} \sqrt {x} + 441 a^{5} b^{4} x + 735 a^{4} b^{5} x^{\frac {3}{2}} + 735 a^{3} b^{6} x^{2} + 441 a^{2} b^{7} x^{\frac {5}{2}} + 147 a b^{8} x^{3} + 21 b^{9} x^{\frac {7}{2}}} & \text {for}\: b \neq 0 \\\frac {x}{a^{8}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a/(21*a**7*b**2 + 147*a**6*b**3*sqrt(x) + 441*a**5*b**4*x + 735*a**4*b**5*x**(3/2) + 735*a**3*b**6
*x**2 + 441*a**2*b**7*x**(5/2) + 147*a*b**8*x**3 + 21*b**9*x**(7/2)) - 7*b*sqrt(x)/(21*a**7*b**2 + 147*a**6*b*
*3*sqrt(x) + 441*a**5*b**4*x + 735*a**4*b**5*x**(3/2) + 735*a**3*b**6*x**2 + 441*a**2*b**7*x**(5/2) + 147*a*b*
*8*x**3 + 21*b**9*x**(7/2)), Ne(b, 0)), (x/a**8, True))

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