\(\int \frac {(a+b \sqrt {x})^{10}}{x^8} \, dx\) [2165]

Optimal result

Integrand size = 15, antiderivative size = 96 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{11}}{7 a x^7}+\frac {3 b \left (a+b \sqrt {x}\right )^{11}}{91 a^2 x^{13/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{182 a^3 x^6}+\frac {b^3 \left (a+b \sqrt {x}\right )^{11}}{2002 a^4 x^{11/2}} \]

[Out]

-1/7*(a+b*x^(1/2))^11/a/x^7+3/91*b*(a+b*x^(1/2))^11/a^2/x^(13/2)-1/182*b^2*(a+b*x^(1/2))^11/a^3/x^6+1/2002*b^3
*(a+b*x^(1/2))^11/a^4/x^(11/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 47, 37} \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=\frac {b^3 \left (a+b \sqrt {x}\right )^{11}}{2002 a^4 x^{11/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{182 a^3 x^6}+\frac {3 b \left (a+b \sqrt {x}\right )^{11}}{91 a^2 x^{13/2}}-\frac {\left (a+b \sqrt {x}\right )^{11}}{7 a x^7} \]

[In]

Int[(a + b*Sqrt[x])^10/x^8,x]

[Out]

-1/7*(a + b*Sqrt[x])^11/(a*x^7) + (3*b*(a + b*Sqrt[x])^11)/(91*a^2*x^(13/2)) - (b^2*(a + b*Sqrt[x])^11)/(182*a
^3*x^6) + (b^3*(a + b*Sqrt[x])^11)/(2002*a^4*x^(11/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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 {(a+b x)^{10}}{x^{15}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \sqrt {x}\right )^{11}}{7 a x^7}-\frac {(3 b) \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{14}} \, dx,x,\sqrt {x}\right )}{7 a} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{11}}{7 a x^7}+\frac {3 b \left (a+b \sqrt {x}\right )^{11}}{91 a^2 x^{13/2}}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{13}} \, dx,x,\sqrt {x}\right )}{91 a^2} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{11}}{7 a x^7}+\frac {3 b \left (a+b \sqrt {x}\right )^{11}}{91 a^2 x^{13/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{182 a^3 x^6}-\frac {b^3 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{12}} \, dx,x,\sqrt {x}\right )}{182 a^3} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{11}}{7 a x^7}+\frac {3 b \left (a+b \sqrt {x}\right )^{11}}{91 a^2 x^{13/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{182 a^3 x^6}+\frac {b^3 \left (a+b \sqrt {x}\right )^{11}}{2002 a^4 x^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=\frac {-286 a^{10}-3080 a^9 b \sqrt {x}-15015 a^8 b^2 x-43680 a^7 b^3 x^{3/2}-84084 a^6 b^4 x^2-112112 a^5 b^5 x^{5/2}-105105 a^4 b^6 x^3-68640 a^3 b^7 x^{7/2}-30030 a^2 b^8 x^4-8008 a b^9 x^{9/2}-1001 b^{10} x^5}{2002 x^7} \]

[In]

Integrate[(a + b*Sqrt[x])^10/x^8,x]

[Out]

(-286*a^10 - 3080*a^9*b*Sqrt[x] - 15015*a^8*b^2*x - 43680*a^7*b^3*x^(3/2) - 84084*a^6*b^4*x^2 - 112112*a^5*b^5
*x^(5/2) - 105105*a^4*b^6*x^3 - 68640*a^3*b^7*x^(7/2) - 30030*a^2*b^8*x^4 - 8008*a*b^9*x^(9/2) - 1001*b^10*x^5
)/(2002*x^7)

Maple [A] (verified)

Time = 3.51 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(-\frac {b^{10}}{2 x^{2}}-\frac {20 a^{9} b}{13 x^{\frac {13}{2}}}-\frac {a^{10}}{7 x^{7}}-\frac {105 a^{4} b^{6}}{2 x^{4}}-\frac {240 a^{3} b^{7}}{7 x^{\frac {7}{2}}}-\frac {240 a^{7} b^{3}}{11 x^{\frac {11}{2}}}-\frac {42 a^{6} b^{4}}{x^{5}}-\frac {4 a \,b^{9}}{x^{\frac {5}{2}}}-\frac {15 a^{8} b^{2}}{2 x^{6}}-\frac {15 a^{2} b^{8}}{x^{3}}-\frac {56 a^{5} b^{5}}{x^{\frac {9}{2}}}\)	\(113\)
default	\(-\frac {b^{10}}{2 x^{2}}-\frac {20 a^{9} b}{13 x^{\frac {13}{2}}}-\frac {a^{10}}{7 x^{7}}-\frac {105 a^{4} b^{6}}{2 x^{4}}-\frac {240 a^{3} b^{7}}{7 x^{\frac {7}{2}}}-\frac {240 a^{7} b^{3}}{11 x^{\frac {11}{2}}}-\frac {42 a^{6} b^{4}}{x^{5}}-\frac {4 a \,b^{9}}{x^{\frac {5}{2}}}-\frac {15 a^{8} b^{2}}{2 x^{6}}-\frac {15 a^{2} b^{8}}{x^{3}}-\frac {56 a^{5} b^{5}}{x^{\frac {9}{2}}}\)	\(113\)
trager	\(\frac {\left (-1+x \right ) \left (2 a^{10} x^{6}+105 a^{8} b^{2} x^{6}+588 a^{6} b^{4} x^{6}+735 a^{4} b^{6} x^{6}+210 a^{2} b^{8} x^{6}+7 b^{10} x^{6}+2 a^{10} x^{5}+105 a^{8} b^{2} x^{5}+588 a^{6} b^{4} x^{5}+735 a^{4} b^{6} x^{5}+210 a^{2} b^{8} x^{5}+7 b^{10} x^{5}+2 a^{10} x^{4}+105 a^{8} b^{2} x^{4}+588 a^{6} b^{4} x^{4}+735 x^{4} a^{4} b^{6}+210 a^{2} b^{8} x^{4}+2 a^{10} x^{3}+105 a^{8} b^{2} x^{3}+588 a^{6} b^{4} x^{3}+735 a^{4} b^{6} x^{3}+2 a^{10} x^{2}+105 a^{8} b^{2} x^{2}+588 x^{2} a^{6} b^{4}+2 a^{10} x +105 a^{8} b^{2} x +2 a^{10}\right )}{14 x^{7}}-\frac {4 \left (1001 b^{8} x^{4}+8580 a^{2} b^{6} x^{3}+14014 a^{4} b^{4} x^{2}+5460 a^{6} b^{2} x +385 a^{8}\right ) a b}{1001 x^{\frac {13}{2}}}\)	\(326\)

[In]

int((a+b*x^(1/2))^10/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/2*b^10/x^2-20/13*a^9*b/x^(13/2)-1/7*a^10/x^7-105/2*a^4*b^6/x^4-240/7*a^3*b^7/x^(7/2)-240/11*a^7*b^3/x^(11/2
)-42*a^6*b^4/x^5-4*a*b^9/x^(5/2)-15/2*a^8*b^2/x^6-15*a^2*b^8/x^3-56*a^5*b^5/x^(9/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=-\frac {1001 \, b^{10} x^{5} + 30030 \, a^{2} b^{8} x^{4} + 105105 \, a^{4} b^{6} x^{3} + 84084 \, a^{6} b^{4} x^{2} + 15015 \, a^{8} b^{2} x + 286 \, a^{10} + 8 \, {\left (1001 \, a b^{9} x^{4} + 8580 \, a^{3} b^{7} x^{3} + 14014 \, a^{5} b^{5} x^{2} + 5460 \, a^{7} b^{3} x + 385 \, a^{9} b\right )} \sqrt {x}}{2002 \, x^{7}} \]

[In]

integrate((a+b*x^(1/2))^10/x^8,x, algorithm="fricas")

[Out]

-1/2002*(1001*b^10*x^5 + 30030*a^2*b^8*x^4 + 105105*a^4*b^6*x^3 + 84084*a^6*b^4*x^2 + 15015*a^8*b^2*x + 286*a^
10 + 8*(1001*a*b^9*x^4 + 8580*a^3*b^7*x^3 + 14014*a^5*b^5*x^2 + 5460*a^7*b^3*x + 385*a^9*b)*sqrt(x))/x^7

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=- \frac {a^{10}}{7 x^{7}} - \frac {20 a^{9} b}{13 x^{\frac {13}{2}}} - \frac {15 a^{8} b^{2}}{2 x^{6}} - \frac {240 a^{7} b^{3}}{11 x^{\frac {11}{2}}} - \frac {42 a^{6} b^{4}}{x^{5}} - \frac {56 a^{5} b^{5}}{x^{\frac {9}{2}}} - \frac {105 a^{4} b^{6}}{2 x^{4}} - \frac {240 a^{3} b^{7}}{7 x^{\frac {7}{2}}} - \frac {15 a^{2} b^{8}}{x^{3}} - \frac {4 a b^{9}}{x^{\frac {5}{2}}} - \frac {b^{10}}{2 x^{2}} \]

[In]

integrate((a+b*x**(1/2))**10/x**8,x)

[Out]

-a**10/(7*x**7) - 20*a**9*b/(13*x**(13/2)) - 15*a**8*b**2/(2*x**6) - 240*a**7*b**3/(11*x**(11/2)) - 42*a**6*b*
*4/x**5 - 56*a**5*b**5/x**(9/2) - 105*a**4*b**6/(2*x**4) - 240*a**3*b**7/(7*x**(7/2)) - 15*a**2*b**8/x**3 - 4*
a*b**9/x**(5/2) - b**10/(2*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=-\frac {1001 \, b^{10} x^{5} + 8008 \, a b^{9} x^{\frac {9}{2}} + 30030 \, a^{2} b^{8} x^{4} + 68640 \, a^{3} b^{7} x^{\frac {7}{2}} + 105105 \, a^{4} b^{6} x^{3} + 112112 \, a^{5} b^{5} x^{\frac {5}{2}} + 84084 \, a^{6} b^{4} x^{2} + 43680 \, a^{7} b^{3} x^{\frac {3}{2}} + 15015 \, a^{8} b^{2} x + 3080 \, a^{9} b \sqrt {x} + 286 \, a^{10}}{2002 \, x^{7}} \]

[In]

integrate((a+b*x^(1/2))^10/x^8,x, algorithm="maxima")

[Out]

-1/2002*(1001*b^10*x^5 + 8008*a*b^9*x^(9/2) + 30030*a^2*b^8*x^4 + 68640*a^3*b^7*x^(7/2) + 105105*a^4*b^6*x^3 +
112112*a^5*b^5*x^(5/2) + 84084*a^6*b^4*x^2 + 43680*a^7*b^3*x^(3/2) + 15015*a^8*b^2*x + 3080*a^9*b*sqrt(x) + 2
86*a^10)/x^7

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=-\frac {1001 \, b^{10} x^{5} + 8008 \, a b^{9} x^{\frac {9}{2}} + 30030 \, a^{2} b^{8} x^{4} + 68640 \, a^{3} b^{7} x^{\frac {7}{2}} + 105105 \, a^{4} b^{6} x^{3} + 112112 \, a^{5} b^{5} x^{\frac {5}{2}} + 84084 \, a^{6} b^{4} x^{2} + 43680 \, a^{7} b^{3} x^{\frac {3}{2}} + 15015 \, a^{8} b^{2} x + 3080 \, a^{9} b \sqrt {x} + 286 \, a^{10}}{2002 \, x^{7}} \]

[In]

integrate((a+b*x^(1/2))^10/x^8,x, algorithm="giac")

[Out]

-1/2002*(1001*b^10*x^5 + 8008*a*b^9*x^(9/2) + 30030*a^2*b^8*x^4 + 68640*a^3*b^7*x^(7/2) + 105105*a^4*b^6*x^3 +
112112*a^5*b^5*x^(5/2) + 84084*a^6*b^4*x^2 + 43680*a^7*b^3*x^(3/2) + 15015*a^8*b^2*x + 3080*a^9*b*sqrt(x) + 2
86*a^10)/x^7

Mupad [B] (verification not implemented)

Time = 5.82 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^8} \, dx=-\frac {\frac {a^{10}}{7}+\frac {b^{10}\,x^5}{2}+\frac {15\,a^8\,b^2\,x}{2}+\frac {20\,a^9\,b\,\sqrt {x}}{13}+4\,a\,b^9\,x^{9/2}+42\,a^6\,b^4\,x^2+\frac {105\,a^4\,b^6\,x^3}{2}+15\,a^2\,b^8\,x^4+\frac {240\,a^7\,b^3\,x^{3/2}}{11}+56\,a^5\,b^5\,x^{5/2}+\frac {240\,a^3\,b^7\,x^{7/2}}{7}}{x^7} \]

[In]

int((a + b*x^(1/2))^10/x^8,x)

[Out]

-(a^10/7 + (b^10*x^5)/2 + (15*a^8*b^2*x)/2 + (20*a^9*b*x^(1/2))/13 + 4*a*b^9*x^(9/2) + 42*a^6*b^4*x^2 + (105*a
^4*b^6*x^3)/2 + 15*a^2*b^8*x^4 + (240*a^7*b^3*x^(3/2))/11 + 56*a^5*b^5*x^(5/2) + (240*a^3*b^7*x^(7/2))/7)/x^7