\(\int \frac {(a+b \sqrt {x})^{15}}{x^{13}} \, dx\) [2186]

Optimal result

Integrand size = 15, antiderivative size = 220 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}-\frac {7 b^2 \left (a+b \sqrt {x}\right )^{16}}{759 a^3 x^{11}}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{759 a^4 x^{21/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{1518 a^5 x^{10}}+\frac {2 b^5 \left (a+b \sqrt {x}\right )^{16}}{14421 a^6 x^{19/2}}-\frac {b^6 \left (a+b \sqrt {x}\right )^{16}}{43263 a^7 x^9}+\frac {2 b^7 \left (a+b \sqrt {x}\right )^{16}}{735471 a^8 x^{17/2}}-\frac {b^8 \left (a+b \sqrt {x}\right )^{16}}{5883768 a^9 x^8} \]

[Out]

-1/12*(a+b*x^(1/2))^16/a/x^12+2/69*b*(a+b*x^(1/2))^16/a^2/x^(23/2)-7/759*b^2*(a+b*x^(1/2))^16/a^3/x^11+2/759*b
^3*(a+b*x^(1/2))^16/a^4/x^(21/2)-1/1518*b^4*(a+b*x^(1/2))^16/a^5/x^10+2/14421*b^5*(a+b*x^(1/2))^16/a^6/x^(19/2
)-1/43263*b^6*(a+b*x^(1/2))^16/a^7/x^9+2/735471*b^7*(a+b*x^(1/2))^16/a^8/x^(17/2)-1/5883768*b^8*(a+b*x^(1/2))^
16/a^9/x^8

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 47, 37} \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {b^8 \left (a+b \sqrt {x}\right )^{16}}{5883768 a^9 x^8}+\frac {2 b^7 \left (a+b \sqrt {x}\right )^{16}}{735471 a^8 x^{17/2}}-\frac {b^6 \left (a+b \sqrt {x}\right )^{16}}{43263 a^7 x^9}+\frac {2 b^5 \left (a+b \sqrt {x}\right )^{16}}{14421 a^6 x^{19/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{1518 a^5 x^{10}}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{759 a^4 x^{21/2}}-\frac {7 b^2 \left (a+b \sqrt {x}\right )^{16}}{759 a^3 x^{11}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}-\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}} \]

[In]

Int[(a + b*Sqrt[x])^15/x^13,x]

[Out]

-1/12*(a + b*Sqrt[x])^16/(a*x^12) + (2*b*(a + b*Sqrt[x])^16)/(69*a^2*x^(23/2)) - (7*b^2*(a + b*Sqrt[x])^16)/(7
59*a^3*x^11) + (2*b^3*(a + b*Sqrt[x])^16)/(759*a^4*x^(21/2)) - (b^4*(a + b*Sqrt[x])^16)/(1518*a^5*x^10) + (2*b
^5*(a + b*Sqrt[x])^16)/(14421*a^6*x^(19/2)) - (b^6*(a + b*Sqrt[x])^16)/(43263*a^7*x^9) + (2*b^7*(a + b*Sqrt[x]
)^16)/(735471*a^8*x^(17/2)) - (b^8*(a + b*Sqrt[x])^16)/(5883768*a^9*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^{25}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}-\frac {(2 b) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{24}} \, dx,x,\sqrt {x}\right )}{3 a} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}+\frac {\left (14 b^2\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{23}} \, dx,x,\sqrt {x}\right )}{69 a^2} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}-\frac {7 b^2 \left (a+b \sqrt {x}\right )^{16}}{759 a^3 x^{11}}-\frac {\left (14 b^3\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{22}} \, dx,x,\sqrt {x}\right )}{253 a^3} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}-\frac {7 b^2 \left (a+b \sqrt {x}\right )^{16}}{759 a^3 x^{11}}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{759 a^4 x^{21/2}}+\frac {\left (10 b^4\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{21}} \, dx,x,\sqrt {x}\right )}{759 a^4} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}-\frac {7 b^2 \left (a+b \sqrt {x}\right )^{16}}{759 a^3 x^{11}}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{759 a^4 x^{21/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{1518 a^5 x^{10}}-\frac {\left (2 b^5\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{20}} \, dx,x,\sqrt {x}\right )}{759 a^5} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}-\frac {7 b^2 \left (a+b \sqrt {x}\right )^{16}}{759 a^3 x^{11}}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{759 a^4 x^{21/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{1518 a^5 x^{10}}+\frac {2 b^5 \left (a+b \sqrt {x}\right )^{16}}{14421 a^6 x^{19/2}}+\frac {\left (2 b^6\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt {x}\right )}{4807 a^6} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}-\frac {7 b^2 \left (a+b \sqrt {x}\right )^{16}}{759 a^3 x^{11}}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{759 a^4 x^{21/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{1518 a^5 x^{10}}+\frac {2 b^5 \left (a+b \sqrt {x}\right )^{16}}{14421 a^6 x^{19/2}}-\frac {b^6 \left (a+b \sqrt {x}\right )^{16}}{43263 a^7 x^9}-\frac {\left (2 b^7\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt {x}\right )}{43263 a^7} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}-\frac {7 b^2 \left (a+b \sqrt {x}\right )^{16}}{759 a^3 x^{11}}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{759 a^4 x^{21/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{1518 a^5 x^{10}}+\frac {2 b^5 \left (a+b \sqrt {x}\right )^{16}}{14421 a^6 x^{19/2}}-\frac {b^6 \left (a+b \sqrt {x}\right )^{16}}{43263 a^7 x^9}+\frac {2 b^7 \left (a+b \sqrt {x}\right )^{16}}{735471 a^8 x^{17/2}}+\frac {\left (2 b^8\right ) \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt {x}\right )}{735471 a^8} \\ & = -\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}-\frac {7 b^2 \left (a+b \sqrt {x}\right )^{16}}{759 a^3 x^{11}}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{759 a^4 x^{21/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{1518 a^5 x^{10}}+\frac {2 b^5 \left (a+b \sqrt {x}\right )^{16}}{14421 a^6 x^{19/2}}-\frac {b^6 \left (a+b \sqrt {x}\right )^{16}}{43263 a^7 x^9}+\frac {2 b^7 \left (a+b \sqrt {x}\right )^{16}}{735471 a^8 x^{17/2}}-\frac {b^8 \left (a+b \sqrt {x}\right )^{16}}{5883768 a^9 x^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=\frac {-490314 a^{15}-7674480 a^{14} b \sqrt {x}-56163240 a^{13} b^2 x-254963280 a^{12} b^3 x^{3/2}-803134332 a^{11} b^4 x^2-1859890032 a^{10} b^5 x^{5/2}-3272028760 a^9 b^6 x^3-4454358480 a^8 b^7 x^{7/2}-4732755885 a^7 b^8 x^4-3926434512 a^6 b^9 x^{9/2}-2524136472 a^5 b^{10} x^5-1235591280 a^4 b^{11} x^{11/2}-446185740 a^3 b^{12} x^6-112326480 a^2 b^{13} x^{13/2}-17651304 a b^{14} x^7-1307504 b^{15} x^{15/2}}{5883768 x^{12}} \]

[In]

Integrate[(a + b*Sqrt[x])^15/x^13,x]

[Out]

(-490314*a^15 - 7674480*a^14*b*Sqrt[x] - 56163240*a^13*b^2*x - 254963280*a^12*b^3*x^(3/2) - 803134332*a^11*b^4
*x^2 - 1859890032*a^10*b^5*x^(5/2) - 3272028760*a^9*b^6*x^3 - 4454358480*a^8*b^7*x^(7/2) - 4732755885*a^7*b^8*
x^4 - 3926434512*a^6*b^9*x^(9/2) - 2524136472*a^5*b^10*x^5 - 1235591280*a^4*b^11*x^(11/2) - 446185740*a^3*b^12
*x^6 - 112326480*a^2*b^13*x^(13/2) - 17651304*a*b^14*x^7 - 1307504*b^15*x^(15/2))/(5883768*x^12)

Maple [A] (verified)

Time = 3.38 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(-\frac {455 a^{3} b^{12}}{6 x^{6}}-\frac {3 a \,b^{14}}{x^{5}}-\frac {2002 a^{6} b^{9}}{3 x^{\frac {15}{2}}}-\frac {130 a^{12} b^{3}}{3 x^{\frac {21}{2}}}-\frac {273 a^{11} b^{4}}{2 x^{10}}-\frac {429 a^{5} b^{10}}{x^{7}}-\frac {210 a^{4} b^{11}}{x^{\frac {13}{2}}}-\frac {210 a^{2} b^{13}}{11 x^{\frac {11}{2}}}-\frac {6006 a^{10} b^{5}}{19 x^{\frac {19}{2}}}-\frac {30 a^{14} b}{23 x^{\frac {23}{2}}}-\frac {a^{15}}{12 x^{12}}-\frac {6435 a^{7} b^{8}}{8 x^{8}}-\frac {105 a^{13} b^{2}}{11 x^{11}}-\frac {12870 a^{8} b^{7}}{17 x^{\frac {17}{2}}}-\frac {2 b^{15}}{9 x^{\frac {9}{2}}}-\frac {5005 a^{9} b^{6}}{9 x^{9}}\)	\(168\)
default	\(-\frac {455 a^{3} b^{12}}{6 x^{6}}-\frac {3 a \,b^{14}}{x^{5}}-\frac {2002 a^{6} b^{9}}{3 x^{\frac {15}{2}}}-\frac {130 a^{12} b^{3}}{3 x^{\frac {21}{2}}}-\frac {273 a^{11} b^{4}}{2 x^{10}}-\frac {429 a^{5} b^{10}}{x^{7}}-\frac {210 a^{4} b^{11}}{x^{\frac {13}{2}}}-\frac {210 a^{2} b^{13}}{11 x^{\frac {11}{2}}}-\frac {6006 a^{10} b^{5}}{19 x^{\frac {19}{2}}}-\frac {30 a^{14} b}{23 x^{\frac {23}{2}}}-\frac {a^{15}}{12 x^{12}}-\frac {6435 a^{7} b^{8}}{8 x^{8}}-\frac {105 a^{13} b^{2}}{11 x^{11}}-\frac {12870 a^{8} b^{7}}{17 x^{\frac {17}{2}}}-\frac {2 b^{15}}{9 x^{\frac {9}{2}}}-\frac {5005 a^{9} b^{6}}{9 x^{9}}\)	\(168\)
trager	\(\frac {\left (-1+x \right ) \left (66 a^{14} x^{11}+7560 a^{12} b^{2} x^{11}+108108 a^{10} b^{4} x^{11}+440440 a^{8} b^{6} x^{11}+637065 a^{6} b^{8} x^{11}+339768 a^{4} b^{10} x^{11}+60060 a^{2} b^{12} x^{11}+2376 b^{14} x^{11}+66 a^{14} x^{10}+7560 a^{12} b^{2} x^{10}+108108 a^{10} b^{4} x^{10}+440440 a^{8} b^{6} x^{10}+637065 a^{6} b^{8} x^{10}+339768 a^{4} b^{10} x^{10}+60060 a^{2} b^{12} x^{10}+2376 b^{14} x^{10}+66 a^{14} x^{9}+7560 a^{12} b^{2} x^{9}+108108 a^{10} b^{4} x^{9}+440440 a^{8} b^{6} x^{9}+637065 a^{6} b^{8} x^{9}+339768 a^{4} b^{10} x^{9}+60060 a^{2} b^{12} x^{9}+2376 b^{14} x^{9}+66 a^{14} x^{8}+7560 a^{12} b^{2} x^{8}+108108 a^{10} b^{4} x^{8}+440440 a^{8} b^{6} x^{8}+637065 a^{6} b^{8} x^{8}+339768 a^{4} b^{10} x^{8}+60060 a^{2} b^{12} x^{8}+2376 b^{14} x^{8}+66 a^{14} x^{7}+7560 a^{12} b^{2} x^{7}+108108 a^{10} b^{4} x^{7}+440440 a^{8} b^{6} x^{7}+637065 a^{6} b^{8} x^{7}+339768 a^{4} b^{10} x^{7}+60060 a^{2} b^{12} x^{7}+2376 x^{7} b^{14}+66 a^{14} x^{6}+7560 a^{12} b^{2} x^{6}+108108 a^{10} b^{4} x^{6}+440440 a^{8} b^{6} x^{6}+637065 a^{6} b^{8} x^{6}+339768 a^{4} b^{10} x^{6}+60060 a^{2} b^{12} x^{6}+66 a^{14} x^{5}+7560 a^{12} b^{2} x^{5}+108108 a^{10} b^{4} x^{5}+440440 a^{8} b^{6} x^{5}+637065 a^{6} b^{8} x^{5}+339768 a^{4} b^{10} x^{5}+66 a^{14} x^{4}+7560 x^{4} a^{12} b^{2}+108108 x^{4} a^{10} b^{4}+440440 a^{8} b^{6} x^{4}+637065 a^{6} b^{8} x^{4}+66 a^{14} x^{3}+7560 a^{12} b^{2} x^{3}+108108 a^{10} b^{4} x^{3}+440440 a^{8} b^{6} x^{3}+66 x^{2} a^{14}+7560 a^{12} b^{2} x^{2}+108108 a^{10} b^{4} x^{2}+66 a^{14} x +7560 a^{12} b^{2} x +66 a^{14}\right ) a}{792 x^{12}}-\frac {2 \left (81719 x^{7} b^{14}+7020405 a^{2} b^{12} x^{6}[phone] a^{4} b^{10} x^{5}[phone] a^{6} b^{8} x^{4}[phone] a^{8} b^{6} x^{3}[phone] a^{10} b^{4} x^{2}[phone] a^{12} b^{2} x +479655 a^{14}\right ) b}{735471 x^{\frac {23}{2}}}\)	\(786\)

[In]

int((a+b*x^(1/2))^15/x^13,x,method=_RETURNVERBOSE)

[Out]

-455/6*a^3*b^12/x^6-3*a*b^14/x^5-2002/3*a^6*b^9/x^(15/2)-130/3*a^12*b^3/x^(21/2)-273/2*a^11*b^4/x^10-429*a^5*b
^10/x^7-210*a^4*b^11/x^(13/2)-210/11*a^2*b^13/x^(11/2)-6006/19*a^10*b^5/x^(19/2)-30/23*a^14*b/x^(23/2)-1/12*a^
15/x^12-6435/8*a^7*b^8/x^8-105/11*a^13*b^2/x^11-12870/17*a^8*b^7/x^(17/2)-2/9*b^15/x^(9/2)-5005/9*a^9*b^6/x^9

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {17651304 \, a b^{14} x^{7} + 446185740 \, a^{3} b^{12} x^{6} + 2524136472 \, a^{5} b^{10} x^{5} + 4732755885 \, a^{7} b^{8} x^{4} + 3272028760 \, a^{9} b^{6} x^{3} + 803134332 \, a^{11} b^{4} x^{2} + 56163240 \, a^{13} b^{2} x + 490314 \, a^{15} + 16 \, {\left (81719 \, b^{15} x^{7} + 7020405 \, a^{2} b^{13} x^{6} + 77224455 \, a^{4} b^{11} x^{5} + 245402157 \, a^{6} b^{9} x^{4} + 278397405 \, a^{8} b^{7} x^{3} + 116243127 \, a^{10} b^{5} x^{2} + 15935205 \, a^{12} b^{3} x + 479655 \, a^{14} b\right )} \sqrt {x}}{5883768 \, x^{12}} \]

[In]

integrate((a+b*x^(1/2))^15/x^13,x, algorithm="fricas")

[Out]

-1/5883768*(17651304*a*b^14*x^7 + 446185740*a^3*b^12*x^6 + 2524136472*a^5*b^10*x^5 + 4732755885*a^7*b^8*x^4 +
3272028760*a^9*b^6*x^3 + 803134332*a^11*b^4*x^2 + 56163240*a^13*b^2*x + 490314*a^15 + 16*(81719*b^15*x^7 + 702
0405*a^2*b^13*x^6 + 77224455*a^4*b^11*x^5 + 245402157*a^6*b^9*x^4 + 278397405*a^8*b^7*x^3 + 116243127*a^10*b^5
*x^2 + 15935205*a^12*b^3*x + 479655*a^14*b)*sqrt(x))/x^12

Sympy [A] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=- \frac {a^{15}}{12 x^{12}} - \frac {30 a^{14} b}{23 x^{\frac {23}{2}}} - \frac {105 a^{13} b^{2}}{11 x^{11}} - \frac {130 a^{12} b^{3}}{3 x^{\frac {21}{2}}} - \frac {273 a^{11} b^{4}}{2 x^{10}} - \frac {6006 a^{10} b^{5}}{19 x^{\frac {19}{2}}} - \frac {5005 a^{9} b^{6}}{9 x^{9}} - \frac {12870 a^{8} b^{7}}{17 x^{\frac {17}{2}}} - \frac {6435 a^{7} b^{8}}{8 x^{8}} - \frac {2002 a^{6} b^{9}}{3 x^{\frac {15}{2}}} - \frac {429 a^{5} b^{10}}{x^{7}} - \frac {210 a^{4} b^{11}}{x^{\frac {13}{2}}} - \frac {455 a^{3} b^{12}}{6 x^{6}} - \frac {210 a^{2} b^{13}}{11 x^{\frac {11}{2}}} - \frac {3 a b^{14}}{x^{5}} - \frac {2 b^{15}}{9 x^{\frac {9}{2}}} \]

[In]

integrate((a+b*x**(1/2))**15/x**13,x)

[Out]

-a**15/(12*x**12) - 30*a**14*b/(23*x**(23/2)) - 105*a**13*b**2/(11*x**11) - 130*a**12*b**3/(3*x**(21/2)) - 273
*a**11*b**4/(2*x**10) - 6006*a**10*b**5/(19*x**(19/2)) - 5005*a**9*b**6/(9*x**9) - 12870*a**8*b**7/(17*x**(17/
2)) - 6435*a**7*b**8/(8*x**8) - 2002*a**6*b**9/(3*x**(15/2)) - 429*a**5*b**10/x**7 - 210*a**4*b**11/x**(13/2)
- 455*a**3*b**12/(6*x**6) - 210*a**2*b**13/(11*x**(11/2)) - 3*a*b**14/x**5 - 2*b**15/(9*x**(9/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {1307504 \, b^{15} x^{\frac {15}{2}} + 17651304 \, a b^{14} x^{7} + 112326480 \, a^{2} b^{13} x^{\frac {13}{2}} + 446185740 \, a^{3} b^{12} x^{6} + 1235591280 \, a^{4} b^{11} x^{\frac {11}{2}} + 2524136472 \, a^{5} b^{10} x^{5} + 3926434512 \, a^{6} b^{9} x^{\frac {9}{2}} + 4732755885 \, a^{7} b^{8} x^{4} + 4454358480 \, a^{8} b^{7} x^{\frac {7}{2}} + 3272028760 \, a^{9} b^{6} x^{3} + 1859890032 \, a^{10} b^{5} x^{\frac {5}{2}} + 803134332 \, a^{11} b^{4} x^{2} + 254963280 \, a^{12} b^{3} x^{\frac {3}{2}} + 56163240 \, a^{13} b^{2} x + 7674480 \, a^{14} b \sqrt {x} + 490314 \, a^{15}}{5883768 \, x^{12}} \]

[In]

integrate((a+b*x^(1/2))^15/x^13,x, algorithm="maxima")

[Out]

-1/5883768*(1307504*b^15*x^(15/2) + 17651304*a*b^14*x^7 + 112326480*a^2*b^13*x^(13/2) + 446185740*a^3*b^12*x^6
+ 1235591280*a^4*b^11*x^(11/2) + 2524136472*a^5*b^10*x^5 + 3926434512*a^6*b^9*x^(9/2) + 4732755885*a^7*b^8*x^
4 + 4454358480*a^8*b^7*x^(7/2) + 3272028760*a^9*b^6*x^3 + 1859890032*a^10*b^5*x^(5/2) + 803134332*a^11*b^4*x^2
+ 254963280*a^12*b^3*x^(3/2) + 56163240*a^13*b^2*x + 7674480*a^14*b*sqrt(x) + 490314*a^15)/x^12

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {1307504 \, b^{15} x^{\frac {15}{2}} + 17651304 \, a b^{14} x^{7} + 112326480 \, a^{2} b^{13} x^{\frac {13}{2}} + 446185740 \, a^{3} b^{12} x^{6} + 1235591280 \, a^{4} b^{11} x^{\frac {11}{2}} + 2524136472 \, a^{5} b^{10} x^{5} + 3926434512 \, a^{6} b^{9} x^{\frac {9}{2}} + 4732755885 \, a^{7} b^{8} x^{4} + 4454358480 \, a^{8} b^{7} x^{\frac {7}{2}} + 3272028760 \, a^{9} b^{6} x^{3} + 1859890032 \, a^{10} b^{5} x^{\frac {5}{2}} + 803134332 \, a^{11} b^{4} x^{2} + 254963280 \, a^{12} b^{3} x^{\frac {3}{2}} + 56163240 \, a^{13} b^{2} x + 7674480 \, a^{14} b \sqrt {x} + 490314 \, a^{15}}{5883768 \, x^{12}} \]

[In]

integrate((a+b*x^(1/2))^15/x^13,x, algorithm="giac")

[Out]

-1/5883768*(1307504*b^15*x^(15/2) + 17651304*a*b^14*x^7 + 112326480*a^2*b^13*x^(13/2) + 446185740*a^3*b^12*x^6
+ 1235591280*a^4*b^11*x^(11/2) + 2524136472*a^5*b^10*x^5 + 3926434512*a^6*b^9*x^(9/2) + 4732755885*a^7*b^8*x^
4 + 4454358480*a^8*b^7*x^(7/2) + 3272028760*a^9*b^6*x^3 + 1859890032*a^10*b^5*x^(5/2) + 803134332*a^11*b^4*x^2
+ 254963280*a^12*b^3*x^(3/2) + 56163240*a^13*b^2*x + 7674480*a^14*b*sqrt(x) + 490314*a^15)/x^12

Mupad [B] (verification not implemented)

Time = 5.82 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {\frac {a^{15}}{12}+\frac {2\,b^{15}\,x^{15/2}}{9}+\frac {105\,a^{13}\,b^2\,x}{11}+\frac {30\,a^{14}\,b\,\sqrt {x}}{23}+3\,a\,b^{14}\,x^7+\frac {273\,a^{11}\,b^4\,x^2}{2}+\frac {5005\,a^9\,b^6\,x^3}{9}+\frac {6435\,a^7\,b^8\,x^4}{8}+429\,a^5\,b^{10}\,x^5+\frac {130\,a^{12}\,b^3\,x^{3/2}}{3}+\frac {455\,a^3\,b^{12}\,x^6}{6}+\frac {6006\,a^{10}\,b^5\,x^{5/2}}{19}+\frac {12870\,a^8\,b^7\,x^{7/2}}{17}+\frac {2002\,a^6\,b^9\,x^{9/2}}{3}+210\,a^4\,b^{11}\,x^{11/2}+\frac {210\,a^2\,b^{13}\,x^{13/2}}{11}}{x^{12}} \]

[In]

int((a + b*x^(1/2))^15/x^13,x)

[Out]

-(a^15/12 + (2*b^15*x^(15/2))/9 + (105*a^13*b^2*x)/11 + (30*a^14*b*x^(1/2))/23 + 3*a*b^14*x^7 + (273*a^11*b^4*
x^2)/2 + (5005*a^9*b^6*x^3)/9 + (6435*a^7*b^8*x^4)/8 + 429*a^5*b^10*x^5 + (130*a^12*b^3*x^(3/2))/3 + (455*a^3*
b^12*x^6)/6 + (6006*a^10*b^5*x^(5/2))/19 + (12870*a^8*b^7*x^(7/2))/17 + (2002*a^6*b^9*x^(9/2))/3 + 210*a^4*b^1
1*x^(11/2) + (210*a^2*b^13*x^(13/2))/11)/x^12