\(\int \frac {(a+b \sqrt {x})^{10}}{x^9} \, dx\) [2166]

Optimal result

Integrand size = 15, antiderivative size = 146 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{11}}{8 a x^8}+\frac {b \left (a+b \sqrt {x}\right )^{11}}{24 a^2 x^{15/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{84 a^3 x^7}+\frac {b^3 \left (a+b \sqrt {x}\right )^{11}}{364 a^4 x^{13/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{11}}{2184 a^5 x^6}+\frac {b^5 \left (a+b \sqrt {x}\right )^{11}}{24024 a^6 x^{11/2}} \]

[Out]

-1/8*(a+b*x^(1/2))^11/a/x^8+1/24*b*(a+b*x^(1/2))^11/a^2/x^(15/2)-1/84*b^2*(a+b*x^(1/2))^11/a^3/x^7+1/364*b^3*(
a+b*x^(1/2))^11/a^4/x^(13/2)-1/2184*b^4*(a+b*x^(1/2))^11/a^5/x^6+1/24024*b^5*(a+b*x^(1/2))^11/a^6/x^(11/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 47, 37} \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=\frac {b^5 \left (a+b \sqrt {x}\right )^{11}}{24024 a^6 x^{11/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{11}}{2184 a^5 x^6}+\frac {b^3 \left (a+b \sqrt {x}\right )^{11}}{364 a^4 x^{13/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{84 a^3 x^7}+\frac {b \left (a+b \sqrt {x}\right )^{11}}{24 a^2 x^{15/2}}-\frac {\left (a+b \sqrt {x}\right )^{11}}{8 a x^8} \]

[In]

Int[(a + b*Sqrt[x])^10/x^9,x]

[Out]

-1/8*(a + b*Sqrt[x])^11/(a*x^8) + (b*(a + b*Sqrt[x])^11)/(24*a^2*x^(15/2)) - (b^2*(a + b*Sqrt[x])^11)/(84*a^3*
x^7) + (b^3*(a + b*Sqrt[x])^11)/(364*a^4*x^(13/2)) - (b^4*(a + b*Sqrt[x])^11)/(2184*a^5*x^6) + (b^5*(a + b*Sqr
t[x])^11)/(24024*a^6*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^{17}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \sqrt {x}\right )^{11}}{8 a x^8}-\frac {(5 b) \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{16}} \, dx,x,\sqrt {x}\right )}{8 a} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{11}}{8 a x^8}+\frac {b \left (a+b \sqrt {x}\right )^{11}}{24 a^2 x^{15/2}}+\frac {b^2 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{15}} \, dx,x,\sqrt {x}\right )}{6 a^2} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{11}}{8 a x^8}+\frac {b \left (a+b \sqrt {x}\right )^{11}}{24 a^2 x^{15/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{84 a^3 x^7}-\frac {b^3 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{14}} \, dx,x,\sqrt {x}\right )}{28 a^3} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{11}}{8 a x^8}+\frac {b \left (a+b \sqrt {x}\right )^{11}}{24 a^2 x^{15/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{84 a^3 x^7}+\frac {b^3 \left (a+b \sqrt {x}\right )^{11}}{364 a^4 x^{13/2}}+\frac {b^4 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{13}} \, dx,x,\sqrt {x}\right )}{182 a^4} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{11}}{8 a x^8}+\frac {b \left (a+b \sqrt {x}\right )^{11}}{24 a^2 x^{15/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{84 a^3 x^7}+\frac {b^3 \left (a+b \sqrt {x}\right )^{11}}{364 a^4 x^{13/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{11}}{2184 a^5 x^6}-\frac {b^5 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{12}} \, dx,x,\sqrt {x}\right )}{2184 a^5} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{11}}{8 a x^8}+\frac {b \left (a+b \sqrt {x}\right )^{11}}{24 a^2 x^{15/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{11}}{84 a^3 x^7}+\frac {b^3 \left (a+b \sqrt {x}\right )^{11}}{364 a^4 x^{13/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{11}}{2184 a^5 x^6}+\frac {b^5 \left (a+b \sqrt {x}\right )^{11}}{24024 a^6 x^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=\frac {-3003 a^{10}-32032 a^9 b \sqrt {x}-154440 a^8 b^2 x-443520 a^7 b^3 x^{3/2}-840840 a^6 b^4 x^2-1100736 a^5 b^5 x^{5/2}-1009008 a^4 b^6 x^3-640640 a^3 b^7 x^{7/2}-270270 a^2 b^8 x^4-68640 a b^9 x^{9/2}-8008 b^{10} x^5}{24024 x^8} \]

[In]

Integrate[(a + b*Sqrt[x])^10/x^9,x]

[Out]

(-3003*a^10 - 32032*a^9*b*Sqrt[x] - 154440*a^8*b^2*x - 443520*a^7*b^3*x^(3/2) - 840840*a^6*b^4*x^2 - 1100736*a
^5*b^5*x^(5/2) - 1009008*a^4*b^6*x^3 - 640640*a^3*b^7*x^(7/2) - 270270*a^2*b^8*x^4 - 68640*a*b^9*x^(9/2) - 800
8*b^10*x^5)/(24024*x^8)

Maple [A] (verified)

Time = 3.48 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(-\frac {35 a^{6} b^{4}}{x^{6}}-\frac {504 a^{5} b^{5}}{11 x^{\frac {11}{2}}}-\frac {a^{10}}{8 x^{8}}-\frac {4 a^{9} b}{3 x^{\frac {15}{2}}}-\frac {80 a^{3} b^{7}}{3 x^{\frac {9}{2}}}-\frac {240 a^{7} b^{3}}{13 x^{\frac {13}{2}}}-\frac {b^{10}}{3 x^{3}}-\frac {45 a^{2} b^{8}}{4 x^{4}}-\frac {20 a \,b^{9}}{7 x^{\frac {7}{2}}}-\frac {45 a^{8} b^{2}}{7 x^{7}}-\frac {42 a^{4} b^{6}}{x^{5}}\)	\(113\)
default	\(-\frac {35 a^{6} b^{4}}{x^{6}}-\frac {504 a^{5} b^{5}}{11 x^{\frac {11}{2}}}-\frac {a^{10}}{8 x^{8}}-\frac {4 a^{9} b}{3 x^{\frac {15}{2}}}-\frac {80 a^{3} b^{7}}{3 x^{\frac {9}{2}}}-\frac {240 a^{7} b^{3}}{13 x^{\frac {13}{2}}}-\frac {b^{10}}{3 x^{3}}-\frac {45 a^{2} b^{8}}{4 x^{4}}-\frac {20 a \,b^{9}}{7 x^{\frac {7}{2}}}-\frac {45 a^{8} b^{2}}{7 x^{7}}-\frac {42 a^{4} b^{6}}{x^{5}}\)	\(113\)
trager	\(\frac {\left (-1+x \right ) \left (21 a^{10} x^{7}+1080 a^{8} b^{2} x^{7}+5880 a^{6} b^{4} x^{7}+7056 a^{4} b^{6} x^{7}+1890 a^{2} b^{8} x^{7}+56 b^{10} x^{7}+21 a^{10} x^{6}+1080 a^{8} b^{2} x^{6}+5880 a^{6} b^{4} x^{6}+7056 a^{4} b^{6} x^{6}+1890 a^{2} b^{8} x^{6}+56 b^{10} x^{6}+21 a^{10} x^{5}+1080 a^{8} b^{2} x^{5}+5880 a^{6} b^{4} x^{5}+7056 a^{4} b^{6} x^{5}+1890 a^{2} b^{8} x^{5}+56 b^{10} x^{5}+21 a^{10} x^{4}+1080 a^{8} b^{2} x^{4}+5880 a^{6} b^{4} x^{4}+7056 x^{4} a^{4} b^{6}+1890 a^{2} b^{8} x^{4}+21 a^{10} x^{3}+1080 a^{8} b^{2} x^{3}+5880 a^{6} b^{4} x^{3}+7056 a^{4} b^{6} x^{3}+21 a^{10} x^{2}+1080 a^{8} b^{2} x^{2}+5880 x^{2} a^{6} b^{4}+21 a^{10} x +1080 a^{8} b^{2} x +21 a^{10}\right )}{168 x^{8}}-\frac {4 \left (2145 b^{8} x^{4}+20020 a^{2} b^{6} x^{3}+34398 a^{4} b^{4} x^{2}+13860 a^{6} b^{2} x +1001 a^{8}\right ) a b}{3003 x^{\frac {15}{2}}}\)	\(386\)

[In]

int((a+b*x^(1/2))^10/x^9,x,method=_RETURNVERBOSE)

[Out]

-35*a^6*b^4/x^6-504/11*a^5*b^5/x^(11/2)-1/8*a^10/x^8-4/3*a^9*b/x^(15/2)-80/3*a^3*b^7/x^(9/2)-240/13*a^7*b^3/x^
(13/2)-1/3*b^10/x^3-45/4*a^2*b^8/x^4-20/7*a*b^9/x^(7/2)-45/7*a^8*b^2/x^7-42*a^4*b^6/x^5

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=-\frac {8008 \, b^{10} x^{5} + 270270 \, a^{2} b^{8} x^{4} + 1009008 \, a^{4} b^{6} x^{3} + 840840 \, a^{6} b^{4} x^{2} + 154440 \, a^{8} b^{2} x + 3003 \, a^{10} + 32 \, {\left (2145 \, a b^{9} x^{4} + 20020 \, a^{3} b^{7} x^{3} + 34398 \, a^{5} b^{5} x^{2} + 13860 \, a^{7} b^{3} x + 1001 \, a^{9} b\right )} \sqrt {x}}{24024 \, x^{8}} \]

[In]

integrate((a+b*x^(1/2))^10/x^9,x, algorithm="fricas")

[Out]

-1/24024*(8008*b^10*x^5 + 270270*a^2*b^8*x^4 + 1009008*a^4*b^6*x^3 + 840840*a^6*b^4*x^2 + 154440*a^8*b^2*x + 3
003*a^10 + 32*(2145*a*b^9*x^4 + 20020*a^3*b^7*x^3 + 34398*a^5*b^5*x^2 + 13860*a^7*b^3*x + 1001*a^9*b)*sqrt(x))
/x^8

Sympy [A] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=- \frac {a^{10}}{8 x^{8}} - \frac {4 a^{9} b}{3 x^{\frac {15}{2}}} - \frac {45 a^{8} b^{2}}{7 x^{7}} - \frac {240 a^{7} b^{3}}{13 x^{\frac {13}{2}}} - \frac {35 a^{6} b^{4}}{x^{6}} - \frac {504 a^{5} b^{5}}{11 x^{\frac {11}{2}}} - \frac {42 a^{4} b^{6}}{x^{5}} - \frac {80 a^{3} b^{7}}{3 x^{\frac {9}{2}}} - \frac {45 a^{2} b^{8}}{4 x^{4}} - \frac {20 a b^{9}}{7 x^{\frac {7}{2}}} - \frac {b^{10}}{3 x^{3}} \]

[In]

integrate((a+b*x**(1/2))**10/x**9,x)

[Out]

-a**10/(8*x**8) - 4*a**9*b/(3*x**(15/2)) - 45*a**8*b**2/(7*x**7) - 240*a**7*b**3/(13*x**(13/2)) - 35*a**6*b**4
/x**6 - 504*a**5*b**5/(11*x**(11/2)) - 42*a**4*b**6/x**5 - 80*a**3*b**7/(3*x**(9/2)) - 45*a**2*b**8/(4*x**4) -
20*a*b**9/(7*x**(7/2)) - b**10/(3*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=-\frac {8008 \, b^{10} x^{5} + 68640 \, a b^{9} x^{\frac {9}{2}} + 270270 \, a^{2} b^{8} x^{4} + 640640 \, a^{3} b^{7} x^{\frac {7}{2}} + 1009008 \, a^{4} b^{6} x^{3} + 1100736 \, a^{5} b^{5} x^{\frac {5}{2}} + 840840 \, a^{6} b^{4} x^{2} + 443520 \, a^{7} b^{3} x^{\frac {3}{2}} + 154440 \, a^{8} b^{2} x + 32032 \, a^{9} b \sqrt {x} + 3003 \, a^{10}}{24024 \, x^{8}} \]

[In]

integrate((a+b*x^(1/2))^10/x^9,x, algorithm="maxima")

[Out]

-1/24024*(8008*b^10*x^5 + 68640*a*b^9*x^(9/2) + 270270*a^2*b^8*x^4 + 640640*a^3*b^7*x^(7/2) + 1009008*a^4*b^6*
x^3 + 1100736*a^5*b^5*x^(5/2) + 840840*a^6*b^4*x^2 + 443520*a^7*b^3*x^(3/2) + 154440*a^8*b^2*x + 32032*a^9*b*s
qrt(x) + 3003*a^10)/x^8

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=-\frac {8008 \, b^{10} x^{5} + 68640 \, a b^{9} x^{\frac {9}{2}} + 270270 \, a^{2} b^{8} x^{4} + 640640 \, a^{3} b^{7} x^{\frac {7}{2}} + 1009008 \, a^{4} b^{6} x^{3} + 1100736 \, a^{5} b^{5} x^{\frac {5}{2}} + 840840 \, a^{6} b^{4} x^{2} + 443520 \, a^{7} b^{3} x^{\frac {3}{2}} + 154440 \, a^{8} b^{2} x + 32032 \, a^{9} b \sqrt {x} + 3003 \, a^{10}}{24024 \, x^{8}} \]

[In]

integrate((a+b*x^(1/2))^10/x^9,x, algorithm="giac")

[Out]

-1/24024*(8008*b^10*x^5 + 68640*a*b^9*x^(9/2) + 270270*a^2*b^8*x^4 + 640640*a^3*b^7*x^(7/2) + 1009008*a^4*b^6*
x^3 + 1100736*a^5*b^5*x^(5/2) + 840840*a^6*b^4*x^2 + 443520*a^7*b^3*x^(3/2) + 154440*a^8*b^2*x + 32032*a^9*b*s
qrt(x) + 3003*a^10)/x^8

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^9} \, dx=-\frac {\frac {a^{10}}{8}+\frac {b^{10}\,x^5}{3}+\frac {45\,a^8\,b^2\,x}{7}+\frac {4\,a^9\,b\,\sqrt {x}}{3}+\frac {20\,a\,b^9\,x^{9/2}}{7}+35\,a^6\,b^4\,x^2+42\,a^4\,b^6\,x^3+\frac {45\,a^2\,b^8\,x^4}{4}+\frac {240\,a^7\,b^3\,x^{3/2}}{13}+\frac {504\,a^5\,b^5\,x^{5/2}}{11}+\frac {80\,a^3\,b^7\,x^{7/2}}{3}}{x^8} \]

[In]

int((a + b*x^(1/2))^10/x^9,x)

[Out]

-(a^10/8 + (b^10*x^5)/3 + (45*a^8*b^2*x)/7 + (4*a^9*b*x^(1/2))/3 + (20*a*b^9*x^(9/2))/7 + 35*a^6*b^4*x^2 + 42*
a^4*b^6*x^3 + (45*a^2*b^8*x^4)/4 + (240*a^7*b^3*x^(3/2))/13 + (504*a^5*b^5*x^(5/2))/11 + (80*a^3*b^7*x^(7/2))/
3)/x^8