\(\int \frac {(a+b \sqrt {x})^{15}}{x^{14}} \, dx\) [2187]

Optimal result

Integrand size = 15, antiderivative size = 270 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac {21 b^4 \left (a+b \sqrt {x}\right )^{16}}{16445 a^5 x^{11}}+\frac {6 b^5 \left (a+b \sqrt {x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac {3 b^6 \left (a+b \sqrt {x}\right )^{16}}{32890 a^7 x^{10}}+\frac {6 b^7 \left (a+b \sqrt {x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac {b^8 \left (a+b \sqrt {x}\right )^{16}}{312455 a^9 x^9}+\frac {2 b^9 \left (a+b \sqrt {x}\right )^{16}}{5311735 a^{10} x^{17/2}}-\frac {b^{10} \left (a+b \sqrt {x}\right )^{16}}{42493880 a^{11} x^8} \]

[Out]

-1/13*(a+b*x^(1/2))^16/a/x^13+2/65*b*(a+b*x^(1/2))^16/a^2/x^(25/2)-3/260*b^2*(a+b*x^(1/2))^16/a^3/x^12+6/1495*
b^3*(a+b*x^(1/2))^16/a^4/x^(23/2)-21/16445*b^4*(a+b*x^(1/2))^16/a^5/x^11+6/16445*b^5*(a+b*x^(1/2))^16/a^6/x^(2
1/2)-3/32890*b^6*(a+b*x^(1/2))^16/a^7/x^10+6/312455*b^7*(a+b*x^(1/2))^16/a^8/x^(19/2)-1/312455*b^8*(a+b*x^(1/2
))^16/a^9/x^9+2/5311735*b^9*(a+b*x^(1/2))^16/a^10/x^(17/2)-1/42493880*b^10*(a+b*x^(1/2))^16/a^11/x^8

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 47, 37} \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {b^{10} \left (a+b \sqrt {x}\right )^{16}}{42493880 a^{11} x^8}+\frac {2 b^9 \left (a+b \sqrt {x}\right )^{16}}{5311735 a^{10} x^{17/2}}-\frac {b^8 \left (a+b \sqrt {x}\right )^{16}}{312455 a^9 x^9}+\frac {6 b^7 \left (a+b \sqrt {x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac {3 b^6 \left (a+b \sqrt {x}\right )^{16}}{32890 a^7 x^{10}}+\frac {6 b^5 \left (a+b \sqrt {x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac {21 b^4 \left (a+b \sqrt {x}\right )^{16}}{16445 a^5 x^{11}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}} \]

[In]

Int[(a + b*Sqrt[x])^15/x^14,x]

[Out]

-1/13*(a + b*Sqrt[x])^16/(a*x^13) + (2*b*(a + b*Sqrt[x])^16)/(65*a^2*x^(25/2)) - (3*b^2*(a + b*Sqrt[x])^16)/(2
60*a^3*x^12) + (6*b^3*(a + b*Sqrt[x])^16)/(1495*a^4*x^(23/2)) - (21*b^4*(a + b*Sqrt[x])^16)/(16445*a^5*x^11) +
(6*b^5*(a + b*Sqrt[x])^16)/(16445*a^6*x^(21/2)) - (3*b^6*(a + b*Sqrt[x])^16)/(32890*a^7*x^10) + (6*b^7*(a + b
*Sqrt[x])^16)/(312455*a^8*x^(19/2)) - (b^8*(a + b*Sqrt[x])^16)/(312455*a^9*x^9) + (2*b^9*(a + b*Sqrt[x])^16)/(
5311735*a^10*x^(17/2)) - (b^10*(a + b*Sqrt[x])^16)/(42493880*a^11*x^8)

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)^{15}}{x^{27}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}-\frac {(10 b) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{26}} \, dx,x,\sqrt {x}\right )}{13 a} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}+\frac {\left (18 b^2\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{25}} \, dx,x,\sqrt {x}\right )}{65 a^2} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}-\frac {\left (6 b^3\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{24}} \, dx,x,\sqrt {x}\right )}{65 a^3} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}+\frac {\left (42 b^4\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{23}} \, dx,x,\sqrt {x}\right )}{1495 a^4} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac {21 b^4 \left (a+b \sqrt {x}\right )^{16}}{16445 a^5 x^{11}}-\frac {\left (126 b^5\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{22}} \, dx,x,\sqrt {x}\right )}{16445 a^5} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac {21 b^4 \left (a+b \sqrt {x}\right )^{16}}{16445 a^5 x^{11}}+\frac {6 b^5 \left (a+b \sqrt {x}\right )^{16}}{16445 a^6 x^{21/2}}+\frac {\left (6 b^6\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{21}} \, dx,x,\sqrt {x}\right )}{3289 a^6} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac {21 b^4 \left (a+b \sqrt {x}\right )^{16}}{16445 a^5 x^{11}}+\frac {6 b^5 \left (a+b \sqrt {x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac {3 b^6 \left (a+b \sqrt {x}\right )^{16}}{32890 a^7 x^{10}}-\frac {\left (6 b^7\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{20}} \, dx,x,\sqrt {x}\right )}{16445 a^7} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac {21 b^4 \left (a+b \sqrt {x}\right )^{16}}{16445 a^5 x^{11}}+\frac {6 b^5 \left (a+b \sqrt {x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac {3 b^6 \left (a+b \sqrt {x}\right )^{16}}{32890 a^7 x^{10}}+\frac {6 b^7 \left (a+b \sqrt {x}\right )^{16}}{312455 a^8 x^{19/2}}+\frac {\left (18 b^8\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt {x}\right )}{312455 a^8} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac {21 b^4 \left (a+b \sqrt {x}\right )^{16}}{16445 a^5 x^{11}}+\frac {6 b^5 \left (a+b \sqrt {x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac {3 b^6 \left (a+b \sqrt {x}\right )^{16}}{32890 a^7 x^{10}}+\frac {6 b^7 \left (a+b \sqrt {x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac {b^8 \left (a+b \sqrt {x}\right )^{16}}{312455 a^9 x^9}-\frac {\left (2 b^9\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt {x}\right )}{312455 a^9} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac {21 b^4 \left (a+b \sqrt {x}\right )^{16}}{16445 a^5 x^{11}}+\frac {6 b^5 \left (a+b \sqrt {x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac {3 b^6 \left (a+b \sqrt {x}\right )^{16}}{32890 a^7 x^{10}}+\frac {6 b^7 \left (a+b \sqrt {x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac {b^8 \left (a+b \sqrt {x}\right )^{16}}{312455 a^9 x^9}+\frac {2 b^9 \left (a+b \sqrt {x}\right )^{16}}{5311735 a^{10} x^{17/2}}+\frac {\left (2 b^{10}\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt {x}\right )}{5311735 a^{10}} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac {21 b^4 \left (a+b \sqrt {x}\right )^{16}}{16445 a^5 x^{11}}+\frac {6 b^5 \left (a+b \sqrt {x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac {3 b^6 \left (a+b \sqrt {x}\right )^{16}}{32890 a^7 x^{10}}+\frac {6 b^7 \left (a+b \sqrt {x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac {b^8 \left (a+b \sqrt {x}\right )^{16}}{312455 a^9 x^9}+\frac {2 b^9 \left (a+b \sqrt {x}\right )^{16}}{5311735 a^{10} x^{17/2}}-\frac {b^{10} \left (a+b \sqrt {x}\right )^{16}}{42493880 a^{11} x^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=\frac {-3268760 a^{15}-50992656 a^{14} b \sqrt {x}-371821450 a^{13} b^2 x-1681279600 a^{12} b^3 x^{3/2}-5273104200 a^{11} b^4 x^2-12153249680 a^{10} b^5 x^{5/2}-21268186940 a^9 b^6 x^3-28784012400 a^8 b^7 x^{7/2}-30383124200 a^7 b^8 x^4-25021396400 a^6 b^9 x^{9/2}-15951140205 a^5 b^{10} x^5-7733886160 a^4 b^{11} x^{11/2}-2762102200 a^3 b^{12} x^6-686439600 a^2 b^{13} x^{13/2}-106234700 a b^{14} x^7-7726160 b^{15} x^{15/2}}{42493880 x^{13}} \]

[In]

Integrate[(a + b*Sqrt[x])^15/x^14,x]

[Out]

(-3268760*a^15 - 50992656*a^14*b*Sqrt[x] - 371821450*a^13*b^2*x - 1681279600*a^12*b^3*x^(3/2) - 5273104200*a^1
1*b^4*x^2 - 12153249680*a^10*b^5*x^(5/2) - 21268186940*a^9*b^6*x^3 - 28784012400*a^8*b^7*x^(7/2) - 30383124200
*a^7*b^8*x^4 - 25021396400*a^6*b^9*x^(9/2) - 15951140205*a^5*b^10*x^5 - 7733886160*a^4*b^11*x^(11/2) - 2762102
200*a^3*b^12*x^6 - 686439600*a^2*b^13*x^(13/2) - 106234700*a*b^14*x^7 - 7726160*b^15*x^(15/2))/(42493880*x^13)

Maple [A] (verified)

Time = 3.58 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.62

method	result	size
derivativedivides	\(-\frac {910 a^{12} b^{3}}{23 x^{\frac {23}{2}}}-\frac {2 b^{15}}{11 x^{\frac {11}{2}}}-\frac {65 a^{3} b^{12}}{x^{7}}-\frac {6 a^{14} b}{5 x^{\frac {25}{2}}}-\frac {5 a \,b^{14}}{2 x^{6}}-\frac {1001 a^{9} b^{6}}{2 x^{10}}-\frac {3003 a^{5} b^{10}}{8 x^{8}}-\frac {1365 a^{11} b^{4}}{11 x^{11}}-\frac {a^{15}}{13 x^{13}}-\frac {10010 a^{6} b^{9}}{17 x^{\frac {17}{2}}}-\frac {35 a^{13} b^{2}}{4 x^{12}}-\frac {715 a^{7} b^{8}}{x^{9}}-\frac {210 a^{2} b^{13}}{13 x^{\frac {13}{2}}}-\frac {12870 a^{8} b^{7}}{19 x^{\frac {19}{2}}}-\frac {286 a^{10} b^{5}}{x^{\frac {21}{2}}}-\frac {182 a^{4} b^{11}}{x^{\frac {15}{2}}}\)	\(168\)
default	\(-\frac {910 a^{12} b^{3}}{23 x^{\frac {23}{2}}}-\frac {2 b^{15}}{11 x^{\frac {11}{2}}}-\frac {65 a^{3} b^{12}}{x^{7}}-\frac {6 a^{14} b}{5 x^{\frac {25}{2}}}-\frac {5 a \,b^{14}}{2 x^{6}}-\frac {1001 a^{9} b^{6}}{2 x^{10}}-\frac {3003 a^{5} b^{10}}{8 x^{8}}-\frac {1365 a^{11} b^{4}}{11 x^{11}}-\frac {a^{15}}{13 x^{13}}-\frac {10010 a^{6} b^{9}}{17 x^{\frac {17}{2}}}-\frac {35 a^{13} b^{2}}{4 x^{12}}-\frac {715 a^{7} b^{8}}{x^{9}}-\frac {210 a^{2} b^{13}}{13 x^{\frac {13}{2}}}-\frac {12870 a^{8} b^{7}}{19 x^{\frac {19}{2}}}-\frac {286 a^{10} b^{5}}{x^{\frac {21}{2}}}-\frac {182 a^{4} b^{11}}{x^{\frac {15}{2}}}\)	\(168\)
trager	\(\frac {\left (-1+x \right ) \left (88 a^{14} x^{12}+10010 a^{12} b^{2} x^{12}+141960 a^{10} b^{4} x^{12}+572572 a^{8} b^{6} x^{12}+817960 a^{6} b^{8} x^{12}+429429 a^{4} b^{10} x^{12}+74360 a^{2} b^{12} x^{12}+2860 b^{14} x^{12}+88 a^{14} x^{11}+10010 a^{12} b^{2} x^{11}+141960 a^{10} b^{4} x^{11}+572572 a^{8} b^{6} x^{11}+817960 a^{6} b^{8} x^{11}+429429 a^{4} b^{10} x^{11}+74360 a^{2} b^{12} x^{11}+2860 b^{14} x^{11}+88 a^{14} x^{10}+10010 a^{12} b^{2} x^{10}+141960 a^{10} b^{4} x^{10}+572572 a^{8} b^{6} x^{10}+817960 a^{6} b^{8} x^{10}+429429 a^{4} b^{10} x^{10}+74360 a^{2} b^{12} x^{10}+2860 b^{14} x^{10}+88 a^{14} x^{9}+10010 a^{12} b^{2} x^{9}+141960 a^{10} b^{4} x^{9}+572572 a^{8} b^{6} x^{9}+817960 a^{6} b^{8} x^{9}+429429 a^{4} b^{10} x^{9}+74360 a^{2} b^{12} x^{9}+2860 b^{14} x^{9}+88 a^{14} x^{8}+10010 a^{12} b^{2} x^{8}+141960 a^{10} b^{4} x^{8}+572572 a^{8} b^{6} x^{8}+817960 a^{6} b^{8} x^{8}+429429 a^{4} b^{10} x^{8}+74360 a^{2} b^{12} x^{8}+2860 b^{14} x^{8}+88 a^{14} x^{7}+10010 a^{12} b^{2} x^{7}+141960 a^{10} b^{4} x^{7}+572572 a^{8} b^{6} x^{7}+817960 a^{6} b^{8} x^{7}+429429 a^{4} b^{10} x^{7}+74360 a^{2} b^{12} x^{7}+2860 x^{7} b^{14}+88 a^{14} x^{6}+10010 a^{12} b^{2} x^{6}+141960 a^{10} b^{4} x^{6}+572572 a^{8} b^{6} x^{6}+817960 a^{6} b^{8} x^{6}+429429 a^{4} b^{10} x^{6}+74360 a^{2} b^{12} x^{6}+88 a^{14} x^{5}+10010 a^{12} b^{2} x^{5}+141960 a^{10} b^{4} x^{5}+572572 a^{8} b^{6} x^{5}+817960 a^{6} b^{8} x^{5}+429429 a^{4} b^{10} x^{5}+88 a^{14} x^{4}+10010 x^{4} a^{12} b^{2}+141960 x^{4} a^{10} b^{4}+572572 a^{8} b^{6} x^{4}+817960 a^{6} b^{8} x^{4}+88 a^{14} x^{3}+10010 a^{12} b^{2} x^{3}+141960 a^{10} b^{4} x^{3}+572572 a^{8} b^{6} x^{3}+88 x^{2} a^{14}+10010 a^{12} b^{2} x^{2}+141960 a^{10} b^{4} x^{2}+88 a^{14} x +10010 a^{12} b^{2} x +88 a^{14}\right ) a}{1144 x^{13}}-\frac {2 \left (482885 x^{7} b^{14}[phone] a^{2} b^{12} x^{6}[phone] a^{4} b^{10} x^{5}[phone] a^{6} b^{8} x^{4}[phone] a^{8} b^{6} x^{3}[phone] a^{10} b^{4} x^{2}[phone] a^{12} b^{2} x +3187041 a^{14}\right ) b}{5311735 x^{\frac {25}{2}}}\)	\(868\)

[In]

int((a+b*x^(1/2))^15/x^14,x,method=_RETURNVERBOSE)

[Out]

-910/23*a^12*b^3/x^(23/2)-2/11*b^15/x^(11/2)-65*a^3*b^12/x^7-6/5*a^14*b/x^(25/2)-5/2*a*b^14/x^6-1001/2*a^9*b^6
/x^10-3003/8*a^5*b^10/x^8-1365/11*a^11*b^4/x^11-1/13*a^15/x^13-10010/17*a^6*b^9/x^(17/2)-35/4*a^13*b^2/x^12-71
5*a^7*b^8/x^9-210/13*a^2*b^13/x^(13/2)-12870/19*a^8*b^7/x^(19/2)-286*a^10*b^5/x^(21/2)-182*a^4*b^11/x^(15/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {106234700 \, a b^{14} x^{7} + 2762102200 \, a^{3} b^{12} x^{6} + 15951140205 \, a^{5} b^{10} x^{5} + 30383124200 \, a^{7} b^{8} x^{4} + 21268186940 \, a^{9} b^{6} x^{3} + 5273104200 \, a^{11} b^{4} x^{2} + 371821450 \, a^{13} b^{2} x + 3268760 \, a^{15} + 16 \, {\left (482885 \, b^{15} x^{7} + 42902475 \, a^{2} b^{13} x^{6} + 483367885 \, a^{4} b^{11} x^{5} + 1563837275 \, a^{6} b^{9} x^{4} + 1799000775 \, a^{8} b^{7} x^{3} + 759578105 \, a^{10} b^{5} x^{2} + 105079975 \, a^{12} b^{3} x + 3187041 \, a^{14} b\right )} \sqrt {x}}{42493880 \, x^{13}} \]

[In]

integrate((a+b*x^(1/2))^15/x^14,x, algorithm="fricas")

[Out]

-1/42493880*(106234700*a*b^14*x^7 + 2762102200*a^3*b^12*x^6 + 15951140205*a^5*b^10*x^5 + 30383124200*a^7*b^8*x
^4 + 21268186940*a^9*b^6*x^3 + 5273104200*a^11*b^4*x^2 + 371821450*a^13*b^2*x + 3268760*a^15 + 16*(482885*b^15
*x^7 + 42902475*a^2*b^13*x^6 + 483367885*a^4*b^11*x^5 + 1563837275*a^6*b^9*x^4 + 1799000775*a^8*b^7*x^3 + 7595
78105*a^10*b^5*x^2 + 105079975*a^12*b^3*x + 3187041*a^14*b)*sqrt(x))/x^13

Sympy [A] (verification not implemented)

Time = 1.52 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=- \frac {a^{15}}{13 x^{13}} - \frac {6 a^{14} b}{5 x^{\frac {25}{2}}} - \frac {35 a^{13} b^{2}}{4 x^{12}} - \frac {910 a^{12} b^{3}}{23 x^{\frac {23}{2}}} - \frac {1365 a^{11} b^{4}}{11 x^{11}} - \frac {286 a^{10} b^{5}}{x^{\frac {21}{2}}} - \frac {1001 a^{9} b^{6}}{2 x^{10}} - \frac {12870 a^{8} b^{7}}{19 x^{\frac {19}{2}}} - \frac {715 a^{7} b^{8}}{x^{9}} - \frac {10010 a^{6} b^{9}}{17 x^{\frac {17}{2}}} - \frac {3003 a^{5} b^{10}}{8 x^{8}} - \frac {182 a^{4} b^{11}}{x^{\frac {15}{2}}} - \frac {65 a^{3} b^{12}}{x^{7}} - \frac {210 a^{2} b^{13}}{13 x^{\frac {13}{2}}} - \frac {5 a b^{14}}{2 x^{6}} - \frac {2 b^{15}}{11 x^{\frac {11}{2}}} \]

[In]

integrate((a+b*x**(1/2))**15/x**14,x)

[Out]

-a**15/(13*x**13) - 6*a**14*b/(5*x**(25/2)) - 35*a**13*b**2/(4*x**12) - 910*a**12*b**3/(23*x**(23/2)) - 1365*a
**11*b**4/(11*x**11) - 286*a**10*b**5/x**(21/2) - 1001*a**9*b**6/(2*x**10) - 12870*a**8*b**7/(19*x**(19/2)) -
715*a**7*b**8/x**9 - 10010*a**6*b**9/(17*x**(17/2)) - 3003*a**5*b**10/(8*x**8) - 182*a**4*b**11/x**(15/2) - 65
*a**3*b**12/x**7 - 210*a**2*b**13/(13*x**(13/2)) - 5*a*b**14/(2*x**6) - 2*b**15/(11*x**(11/2))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {7726160 \, b^{15} x^{\frac {15}{2}} + 106234700 \, a b^{14} x^{7} + 686439600 \, a^{2} b^{13} x^{\frac {13}{2}} + 2762102200 \, a^{3} b^{12} x^{6} + 7733886160 \, a^{4} b^{11} x^{\frac {11}{2}} + 15951140205 \, a^{5} b^{10} x^{5} + 25021396400 \, a^{6} b^{9} x^{\frac {9}{2}} + 30383124200 \, a^{7} b^{8} x^{4} + 28784012400 \, a^{8} b^{7} x^{\frac {7}{2}} + 21268186940 \, a^{9} b^{6} x^{3} + 12153249680 \, a^{10} b^{5} x^{\frac {5}{2}} + 5273104200 \, a^{11} b^{4} x^{2} + 1681279600 \, a^{12} b^{3} x^{\frac {3}{2}} + 371821450 \, a^{13} b^{2} x + 50992656 \, a^{14} b \sqrt {x} + 3268760 \, a^{15}}{42493880 \, x^{13}} \]

[In]

integrate((a+b*x^(1/2))^15/x^14,x, algorithm="maxima")

[Out]

-1/42493880*(7726160*b^15*x^(15/2) + 106234700*a*b^14*x^7 + 686439600*a^2*b^13*x^(13/2) + 2762102200*a^3*b^12*
x^6 + 7733886160*a^4*b^11*x^(11/2) + 15951140205*a^5*b^10*x^5 + 25021396400*a^6*b^9*x^(9/2) + 30383124200*a^7*
b^8*x^4 + 28784012400*a^8*b^7*x^(7/2) + 21268186940*a^9*b^6*x^3 + 12153249680*a^10*b^5*x^(5/2) + 5273104200*a^
11*b^4*x^2 + 1681279600*a^12*b^3*x^(3/2) + 371821450*a^13*b^2*x + 50992656*a^14*b*sqrt(x) + 3268760*a^15)/x^13

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {7726160 \, b^{15} x^{\frac {15}{2}} + 106234700 \, a b^{14} x^{7} + 686439600 \, a^{2} b^{13} x^{\frac {13}{2}} + 2762102200 \, a^{3} b^{12} x^{6} + 7733886160 \, a^{4} b^{11} x^{\frac {11}{2}} + 15951140205 \, a^{5} b^{10} x^{5} + 25021396400 \, a^{6} b^{9} x^{\frac {9}{2}} + 30383124200 \, a^{7} b^{8} x^{4} + 28784012400 \, a^{8} b^{7} x^{\frac {7}{2}} + 21268186940 \, a^{9} b^{6} x^{3} + 12153249680 \, a^{10} b^{5} x^{\frac {5}{2}} + 5273104200 \, a^{11} b^{4} x^{2} + 1681279600 \, a^{12} b^{3} x^{\frac {3}{2}} + 371821450 \, a^{13} b^{2} x + 50992656 \, a^{14} b \sqrt {x} + 3268760 \, a^{15}}{42493880 \, x^{13}} \]

[In]

integrate((a+b*x^(1/2))^15/x^14,x, algorithm="giac")

[Out]

-1/42493880*(7726160*b^15*x^(15/2) + 106234700*a*b^14*x^7 + 686439600*a^2*b^13*x^(13/2) + 2762102200*a^3*b^12*
x^6 + 7733886160*a^4*b^11*x^(11/2) + 15951140205*a^5*b^10*x^5 + 25021396400*a^6*b^9*x^(9/2) + 30383124200*a^7*
b^8*x^4 + 28784012400*a^8*b^7*x^(7/2) + 21268186940*a^9*b^6*x^3 + 12153249680*a^10*b^5*x^(5/2) + 5273104200*a^
11*b^4*x^2 + 1681279600*a^12*b^3*x^(3/2) + 371821450*a^13*b^2*x + 50992656*a^14*b*sqrt(x) + 3268760*a^15)/x^13

Mupad [B] (verification not implemented)

Time = 5.96 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {\frac {a^{15}}{13}+\frac {2\,b^{15}\,x^{15/2}}{11}+\frac {35\,a^{13}\,b^2\,x}{4}+\frac {6\,a^{14}\,b\,\sqrt {x}}{5}+\frac {5\,a\,b^{14}\,x^7}{2}+\frac {1365\,a^{11}\,b^4\,x^2}{11}+\frac {1001\,a^9\,b^6\,x^3}{2}+715\,a^7\,b^8\,x^4+\frac {3003\,a^5\,b^{10}\,x^5}{8}+\frac {910\,a^{12}\,b^3\,x^{3/2}}{23}+65\,a^3\,b^{12}\,x^6+286\,a^{10}\,b^5\,x^{5/2}+\frac {12870\,a^8\,b^7\,x^{7/2}}{19}+\frac {10010\,a^6\,b^9\,x^{9/2}}{17}+182\,a^4\,b^{11}\,x^{11/2}+\frac {210\,a^2\,b^{13}\,x^{13/2}}{13}}{x^{13}} \]

[In]

int((a + b*x^(1/2))^15/x^14,x)

[Out]

-(a^15/13 + (2*b^15*x^(15/2))/11 + (35*a^13*b^2*x)/4 + (6*a^14*b*x^(1/2))/5 + (5*a*b^14*x^7)/2 + (1365*a^11*b^
4*x^2)/11 + (1001*a^9*b^6*x^3)/2 + 715*a^7*b^8*x^4 + (3003*a^5*b^10*x^5)/8 + (910*a^12*b^3*x^(3/2))/23 + 65*a^
3*b^12*x^6 + 286*a^10*b^5*x^(5/2) + (12870*a^8*b^7*x^(7/2))/19 + (10010*a^6*b^9*x^(9/2))/17 + 182*a^4*b^11*x^(
11/2) + (210*a^2*b^13*x^(13/2))/13)/x^13