3.2184 \(\int \frac{(a+b \sqrt{x})^{15}}{x^{11}} \, dx\)

Optimal. Leaf size=120 \[ -\frac{b^4 \left (a+b \sqrt{x}\right )^{16}}{38760 a^5 x^8}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{4845 a^4 x^{17/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{285 a^3 x^9}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{95 a^2 x^{19/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{10 a x^{10}} \]

[Out]

-(a + b*Sqrt[x])^16/(10*a*x^10) + (2*b*(a + b*Sqrt[x])^16)/(95*a^2*x^(19/2)) - (b^2*(a + b*Sqrt[x])^16)/(285*a
^3*x^9) + (2*b^3*(a + b*Sqrt[x])^16)/(4845*a^4*x^(17/2)) - (b^4*(a + b*Sqrt[x])^16)/(38760*a^5*x^8)

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Rubi [A]  time = 0.0452624, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ -\frac{b^4 \left (a+b \sqrt{x}\right )^{16}}{38760 a^5 x^8}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{4845 a^4 x^{17/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{285 a^3 x^9}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{95 a^2 x^{19/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^11,x]

[Out]

-(a + b*Sqrt[x])^16/(10*a*x^10) + (2*b*(a + b*Sqrt[x])^16)/(95*a^2*x^(19/2)) - (b^2*(a + b*Sqrt[x])^16)/(285*a
^3*x^9) + (2*b^3*(a + b*Sqrt[x])^16)/(4845*a^4*x^(17/2)) - (b^4*(a + b*Sqrt[x])^16)/(38760*a^5*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^{11}} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{21}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{10 a x^{10}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{20}} \, dx,x,\sqrt{x}\right )}{5 a}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{10 a x^{10}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{95 a^2 x^{19/2}}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt{x}\right )}{95 a^2}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{10 a x^{10}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{95 a^2 x^{19/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{285 a^3 x^9}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt{x}\right )}{285 a^3}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{10 a x^{10}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{95 a^2 x^{19/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{285 a^3 x^9}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{4845 a^4 x^{17/2}}+\frac{\left (2 b^4\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt{x}\right )}{4845 a^4}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{10 a x^{10}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{95 a^2 x^{19/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{285 a^3 x^9}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{4845 a^4 x^{17/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{16}}{38760 a^5 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0143544, size = 65, normalized size = 0.54 \[ -\frac{\left (a+b \sqrt{x}\right )^{16} \left (136 a^2 b^2 x-816 a^3 b \sqrt{x}+3876 a^4-16 a b^3 x^{3/2}+b^4 x^2\right )}{38760 a^5 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^11,x]

[Out]

-((a + b*Sqrt[x])^16*(3876*a^4 - 816*a^3*b*Sqrt[x] + 136*a^2*b^2*x - 16*a*b^3*x^(3/2) + b^4*x^2))/(38760*a^5*x
^10)

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Maple [A]  time = 0.004, size = 168, normalized size = 1.4 \begin{align*} -{\frac{2\,{b}^{15}}{5}{x}^{-{\frac{5}{2}}}}-5\,{\frac{a{b}^{14}}{{x}^{3}}}-30\,{\frac{{a}^{2}{b}^{13}}{{x}^{7/2}}}-{\frac{455\,{a}^{3}{b}^{12}}{4\,{x}^{4}}}-{\frac{910\,{a}^{4}{b}^{11}}{3}{x}^{-{\frac{9}{2}}}}-{\frac{3003\,{a}^{5}{b}^{10}}{5\,{x}^{5}}}-910\,{\frac{{a}^{6}{b}^{9}}{{x}^{11/2}}}-{\frac{2145\,{a}^{7}{b}^{8}}{2\,{x}^{6}}}-990\,{\frac{{a}^{8}{b}^{7}}{{x}^{13/2}}}-715\,{\frac{{a}^{9}{b}^{6}}{{x}^{7}}}-{\frac{2002\,{a}^{10}{b}^{5}}{5}{x}^{-{\frac{15}{2}}}}-{\frac{1365\,{a}^{11}{b}^{4}}{8\,{x}^{8}}}-{\frac{910\,{a}^{12}{b}^{3}}{17}{x}^{-{\frac{17}{2}}}}-{\frac{35\,{a}^{13}{b}^{2}}{3\,{x}^{9}}}-{\frac{30\,{a}^{14}b}{19}{x}^{-{\frac{19}{2}}}}-{\frac{{a}^{15}}{10\,{x}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^11,x)

[Out]

-2/5*b^15/x^(5/2)-5*a*b^14/x^3-30*a^2*b^13/x^(7/2)-455/4*a^3*b^12/x^4-910/3*a^4*b^11/x^(9/2)-3003/5*a^5*b^10/x
^5-910*a^6*b^9/x^(11/2)-2145/2*a^7*b^8/x^6-990*a^8*b^7/x^(13/2)-715*a^9*b^6/x^7-2002/5*a^10*b^5/x^(15/2)-1365/
8*a^11*b^4/x^8-910/17*a^12*b^3/x^(17/2)-35/3*a^13*b^2/x^9-30/19*a^14*b/x^(19/2)-1/10*a^15/x^10

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Maxima [A]  time = 0.972983, size = 225, normalized size = 1.88 \begin{align*} -\frac{15504 \, b^{15} x^{\frac{15}{2}} + 193800 \, a b^{14} x^{7} + 1162800 \, a^{2} b^{13} x^{\frac{13}{2}} + 4408950 \, a^{3} b^{12} x^{6} + 11757200 \, a^{4} b^{11} x^{\frac{11}{2}} + 23279256 \, a^{5} b^{10} x^{5} + 35271600 \, a^{6} b^{9} x^{\frac{9}{2}} + 41570100 \, a^{7} b^{8} x^{4} + 38372400 \, a^{8} b^{7} x^{\frac{7}{2}} + 27713400 \, a^{9} b^{6} x^{3} + 15519504 \, a^{10} b^{5} x^{\frac{5}{2}} + 6613425 \, a^{11} b^{4} x^{2} + 2074800 \, a^{12} b^{3} x^{\frac{3}{2}} + 452200 \, a^{13} b^{2} x + 61200 \, a^{14} b \sqrt{x} + 3876 \, a^{15}}{38760 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^11,x, algorithm="maxima")

[Out]

-1/38760*(15504*b^15*x^(15/2) + 193800*a*b^14*x^7 + 1162800*a^2*b^13*x^(13/2) + 4408950*a^3*b^12*x^6 + 1175720
0*a^4*b^11*x^(11/2) + 23279256*a^5*b^10*x^5 + 35271600*a^6*b^9*x^(9/2) + 41570100*a^7*b^8*x^4 + 38372400*a^8*b
^7*x^(7/2) + 27713400*a^9*b^6*x^3 + 15519504*a^10*b^5*x^(5/2) + 6613425*a^11*b^4*x^2 + 2074800*a^12*b^3*x^(3/2
) + 452200*a^13*b^2*x + 61200*a^14*b*sqrt(x) + 3876*a^15)/x^10

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Fricas [A]  time = 1.13332, size = 478, normalized size = 3.98 \begin{align*} -\frac{193800 \, a b^{14} x^{7} + 4408950 \, a^{3} b^{12} x^{6} + 23279256 \, a^{5} b^{10} x^{5} + 41570100 \, a^{7} b^{8} x^{4} + 27713400 \, a^{9} b^{6} x^{3} + 6613425 \, a^{11} b^{4} x^{2} + 452200 \, a^{13} b^{2} x + 3876 \, a^{15} + 16 \,{\left (969 \, b^{15} x^{7} + 72675 \, a^{2} b^{13} x^{6} + 734825 \, a^{4} b^{11} x^{5} + 2204475 \, a^{6} b^{9} x^{4} + 2398275 \, a^{8} b^{7} x^{3} + 969969 \, a^{10} b^{5} x^{2} + 129675 \, a^{12} b^{3} x + 3825 \, a^{14} b\right )} \sqrt{x}}{38760 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^11,x, algorithm="fricas")

[Out]

-1/38760*(193800*a*b^14*x^7 + 4408950*a^3*b^12*x^6 + 23279256*a^5*b^10*x^5 + 41570100*a^7*b^8*x^4 + 27713400*a
^9*b^6*x^3 + 6613425*a^11*b^4*x^2 + 452200*a^13*b^2*x + 3876*a^15 + 16*(969*b^15*x^7 + 72675*a^2*b^13*x^6 + 73
4825*a^4*b^11*x^5 + 2204475*a^6*b^9*x^4 + 2398275*a^8*b^7*x^3 + 969969*a^10*b^5*x^2 + 129675*a^12*b^3*x + 3825
*a^14*b)*sqrt(x))/x^10

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Sympy [A]  time = 13.2374, size = 211, normalized size = 1.76 \begin{align*} - \frac{a^{15}}{10 x^{10}} - \frac{30 a^{14} b}{19 x^{\frac{19}{2}}} - \frac{35 a^{13} b^{2}}{3 x^{9}} - \frac{910 a^{12} b^{3}}{17 x^{\frac{17}{2}}} - \frac{1365 a^{11} b^{4}}{8 x^{8}} - \frac{2002 a^{10} b^{5}}{5 x^{\frac{15}{2}}} - \frac{715 a^{9} b^{6}}{x^{7}} - \frac{990 a^{8} b^{7}}{x^{\frac{13}{2}}} - \frac{2145 a^{7} b^{8}}{2 x^{6}} - \frac{910 a^{6} b^{9}}{x^{\frac{11}{2}}} - \frac{3003 a^{5} b^{10}}{5 x^{5}} - \frac{910 a^{4} b^{11}}{3 x^{\frac{9}{2}}} - \frac{455 a^{3} b^{12}}{4 x^{4}} - \frac{30 a^{2} b^{13}}{x^{\frac{7}{2}}} - \frac{5 a b^{14}}{x^{3}} - \frac{2 b^{15}}{5 x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**15/x**11,x)

[Out]

-a**15/(10*x**10) - 30*a**14*b/(19*x**(19/2)) - 35*a**13*b**2/(3*x**9) - 910*a**12*b**3/(17*x**(17/2)) - 1365*
a**11*b**4/(8*x**8) - 2002*a**10*b**5/(5*x**(15/2)) - 715*a**9*b**6/x**7 - 990*a**8*b**7/x**(13/2) - 2145*a**7
*b**8/(2*x**6) - 910*a**6*b**9/x**(11/2) - 3003*a**5*b**10/(5*x**5) - 910*a**4*b**11/(3*x**(9/2)) - 455*a**3*b
**12/(4*x**4) - 30*a**2*b**13/x**(7/2) - 5*a*b**14/x**3 - 2*b**15/(5*x**(5/2))

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Giac [A]  time = 1.1071, size = 225, normalized size = 1.88 \begin{align*} -\frac{15504 \, b^{15} x^{\frac{15}{2}} + 193800 \, a b^{14} x^{7} + 1162800 \, a^{2} b^{13} x^{\frac{13}{2}} + 4408950 \, a^{3} b^{12} x^{6} + 11757200 \, a^{4} b^{11} x^{\frac{11}{2}} + 23279256 \, a^{5} b^{10} x^{5} + 35271600 \, a^{6} b^{9} x^{\frac{9}{2}} + 41570100 \, a^{7} b^{8} x^{4} + 38372400 \, a^{8} b^{7} x^{\frac{7}{2}} + 27713400 \, a^{9} b^{6} x^{3} + 15519504 \, a^{10} b^{5} x^{\frac{5}{2}} + 6613425 \, a^{11} b^{4} x^{2} + 2074800 \, a^{12} b^{3} x^{\frac{3}{2}} + 452200 \, a^{13} b^{2} x + 61200 \, a^{14} b \sqrt{x} + 3876 \, a^{15}}{38760 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^11,x, algorithm="giac")

[Out]

-1/38760*(15504*b^15*x^(15/2) + 193800*a*b^14*x^7 + 1162800*a^2*b^13*x^(13/2) + 4408950*a^3*b^12*x^6 + 1175720
0*a^4*b^11*x^(11/2) + 23279256*a^5*b^10*x^5 + 35271600*a^6*b^9*x^(9/2) + 41570100*a^7*b^8*x^4 + 38372400*a^8*b
^7*x^(7/2) + 27713400*a^9*b^6*x^3 + 15519504*a^10*b^5*x^(5/2) + 6613425*a^11*b^4*x^2 + 2074800*a^12*b^3*x^(3/2
) + 452200*a^13*b^2*x + 61200*a^14*b*sqrt(x) + 3876*a^15)/x^10