3.2187 \(\int \frac{(a+b \sqrt{x})^{15}}{x^{14}} \, dx\)

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

[Out]

-(a + b*Sqrt[x])^16/(13*a*x^13) + (2*b*(a + b*Sqrt[x])^16)/(65*a^2*x^(25/2)) - (3*b^2*(a + b*Sqrt[x])^16)/(260
*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*S
qrt[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)/(53
11735*a^10*x^(17/2)) - (b^10*(a + b*Sqrt[x])^16)/(42493880*a^11*x^8)

________________________________________________________________________________________

Rubi [A]  time = 0.150993, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ -\frac{b^{10} \left (a+b \sqrt{x}\right )^{16}}{42493880 a^{11} x^8}+\frac{2 b^9 \left (a+b \sqrt{x}\right )^{16}}{5311735 a^{10} x^{17/2}}-\frac{b^8 \left (a+b \sqrt{x}\right )^{16}}{312455 a^9 x^9}+\frac{6 b^7 \left (a+b \sqrt{x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac{3 b^6 \left (a+b \sqrt{x}\right )^{16}}{32890 a^7 x^{10}}+\frac{6 b^5 \left (a+b \sqrt{x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac{21 b^4 \left (a+b \sqrt{x}\right )^{16}}{16445 a^5 x^{11}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^14,x]

[Out]

-(a + b*Sqrt[x])^16/(13*a*x^13) + (2*b*(a + b*Sqrt[x])^16)/(65*a^2*x^(25/2)) - (3*b^2*(a + b*Sqrt[x])^16)/(260
*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*S
qrt[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)/(53
11735*a^10*x^(17/2)) - (b^10*(a + b*Sqrt[x])^16)/(42493880*a^11*x^8)

Rule 266

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]]

Rule 45

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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^{14}} \, dx &=2 \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.107446, size = 207, normalized size = 0.77 \[ -\frac{35 a^{13} b^2}{4 x^{12}}-\frac{910 a^{12} b^3}{23 x^{23/2}}-\frac{1365 a^{11} b^4}{11 x^{11}}-\frac{286 a^{10} b^5}{x^{21/2}}-\frac{1001 a^9 b^6}{2 x^{10}}-\frac{12870 a^8 b^7}{19 x^{19/2}}-\frac{715 a^7 b^8}{x^9}-\frac{10010 a^6 b^9}{17 x^{17/2}}-\frac{3003 a^5 b^{10}}{8 x^8}-\frac{182 a^4 b^{11}}{x^{15/2}}-\frac{65 a^3 b^{12}}{x^7}-\frac{210 a^2 b^{13}}{13 x^{13/2}}-\frac{6 a^{14} b}{5 x^{25/2}}-\frac{a^{15}}{13 x^{13}}-\frac{5 a b^{14}}{2 x^6}-\frac{2 b^{15}}{11 x^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^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))

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Maple [A]  time = 0.004, size = 168, normalized size = 0.6 \begin{align*} -{\frac{2\,{b}^{15}}{11}{x}^{-{\frac{11}{2}}}}-{\frac{5\,a{b}^{14}}{2\,{x}^{6}}}-{\frac{210\,{a}^{2}{b}^{13}}{13}{x}^{-{\frac{13}{2}}}}-65\,{\frac{{a}^{3}{b}^{12}}{{x}^{7}}}-182\,{\frac{{a}^{4}{b}^{11}}{{x}^{15/2}}}-{\frac{3003\,{a}^{5}{b}^{10}}{8\,{x}^{8}}}-{\frac{10010\,{a}^{6}{b}^{9}}{17}{x}^{-{\frac{17}{2}}}}-715\,{\frac{{a}^{7}{b}^{8}}{{x}^{9}}}-{\frac{12870\,{a}^{8}{b}^{7}}{19}{x}^{-{\frac{19}{2}}}}-{\frac{1001\,{a}^{9}{b}^{6}}{2\,{x}^{10}}}-286\,{\frac{{a}^{10}{b}^{5}}{{x}^{21/2}}}-{\frac{1365\,{a}^{11}{b}^{4}}{11\,{x}^{11}}}-{\frac{910\,{a}^{12}{b}^{3}}{23}{x}^{-{\frac{23}{2}}}}-{\frac{35\,{a}^{13}{b}^{2}}{4\,{x}^{12}}}-{\frac{6\,{a}^{14}b}{5}{x}^{-{\frac{25}{2}}}}-{\frac{{a}^{15}}{13\,{x}^{13}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*b^15/x^(11/2)-5/2*a*b^14/x^6-210/13*a^2*b^13/x^(13/2)-65*a^3*b^12/x^7-182*a^4*b^11/x^(15/2)-3003/8*a^5*b
^10/x^8-10010/17*a^6*b^9/x^(17/2)-715*a^7*b^8/x^9-12870/19*a^8*b^7/x^(19/2)-1001/2*a^9*b^6/x^10-286*a^10*b^5/x
^(21/2)-1365/11*a^11*b^4/x^11-910/23*a^12*b^3/x^(23/2)-35/4*a^13*b^2/x^12-6/5*a^14*b/x^(25/2)-1/13*a^15/x^13

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Maxima [A]  time = 1.04356, size = 225, normalized size = 0.83 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.32377, size = 547, normalized size = 2.03 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 24.4415, size = 212, normalized size = 0.79 \begin{align*} - \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.12144, size = 225, normalized size = 0.83 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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