3.2183 \(\int \frac{(a+b \sqrt{x})^{15}}{x^{10}} \, dx\)

Optimal. Leaf size=70 \[ -\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{1224 a^3 x^8}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{153 a^2 x^{17/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{9 a x^9} \]

[Out]

-(a + b*Sqrt[x])^16/(9*a*x^9) + (2*b*(a + b*Sqrt[x])^16)/(153*a^2*x^(17/2)) - (b^2*(a + b*Sqrt[x])^16)/(1224*a
^3*x^8)

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Rubi [A]  time = 0.0236033, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ -\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{1224 a^3 x^8}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{153 a^2 x^{17/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^10,x]

[Out]

-(a + b*Sqrt[x])^16/(9*a*x^9) + (2*b*(a + b*Sqrt[x])^16)/(153*a^2*x^(17/2)) - (b^2*(a + b*Sqrt[x])^16)/(1224*a
^3*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^{10}} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{9 a x^9}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt{x}\right )}{9 a}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{9 a x^9}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{153 a^2 x^{17/2}}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt{x}\right )}{153 a^2}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{9 a x^9}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{153 a^2 x^{17/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{1224 a^3 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0099028, size = 41, normalized size = 0.59 \[ -\frac{\left (a+b \sqrt{x}\right )^{16} \left (136 a^2-16 a b \sqrt{x}+b^2 x\right )}{1224 a^3 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^10,x]

[Out]

-((a + b*Sqrt[x])^16*(136*a^2 - 16*a*b*Sqrt[x] + b^2*x))/(1224*a^3*x^9)

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Maple [B]  time = 0.003, size = 168, normalized size = 2.4 \begin{align*} -{\frac{2\,{b}^{15}}{3}{x}^{-{\frac{3}{2}}}}-{\frac{15\,a{b}^{14}}{2\,{x}^{2}}}-42\,{\frac{{a}^{2}{b}^{13}}{{x}^{5/2}}}-{\frac{455\,{a}^{3}{b}^{12}}{3\,{x}^{3}}}-390\,{\frac{{a}^{4}{b}^{11}}{{x}^{7/2}}}-{\frac{3003\,{a}^{5}{b}^{10}}{4\,{x}^{4}}}-{\frac{10010\,{a}^{6}{b}^{9}}{9}{x}^{-{\frac{9}{2}}}}-1287\,{\frac{{a}^{7}{b}^{8}}{{x}^{5}}}-1170\,{\frac{{a}^{8}{b}^{7}}{{x}^{11/2}}}-{\frac{5005\,{a}^{9}{b}^{6}}{6\,{x}^{6}}}-462\,{\frac{{a}^{10}{b}^{5}}{{x}^{13/2}}}-195\,{\frac{{a}^{11}{b}^{4}}{{x}^{7}}}-{\frac{182\,{a}^{12}{b}^{3}}{3}{x}^{-{\frac{15}{2}}}}-{\frac{105\,{a}^{13}{b}^{2}}{8\,{x}^{8}}}-{\frac{30\,{a}^{14}b}{17}{x}^{-{\frac{17}{2}}}}-{\frac{{a}^{15}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^10,x)

[Out]

-2/3*b^15/x^(3/2)-15/2*a*b^14/x^2-42*a^2*b^13/x^(5/2)-455/3*a^3*b^12/x^3-390*a^4*b^11/x^(7/2)-3003/4*a^5*b^10/
x^4-10010/9*a^6*b^9/x^(9/2)-1287*a^7*b^8/x^5-1170*a^8*b^7/x^(11/2)-5005/6*a^9*b^6/x^6-462*a^10*b^5/x^(13/2)-19
5*a^11*b^4/x^7-182/3*a^12*b^3/x^(15/2)-105/8*a^13*b^2/x^8-30/17*a^14*b/x^(17/2)-1/9*a^15/x^9

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Maxima [B]  time = 0.968305, size = 225, normalized size = 3.21 \begin{align*} -\frac{816 \, b^{15} x^{\frac{15}{2}} + 9180 \, a b^{14} x^{7} + 51408 \, a^{2} b^{13} x^{\frac{13}{2}} + 185640 \, a^{3} b^{12} x^{6} + 477360 \, a^{4} b^{11} x^{\frac{11}{2}} + 918918 \, a^{5} b^{10} x^{5} + 1361360 \, a^{6} b^{9} x^{\frac{9}{2}} + 1575288 \, a^{7} b^{8} x^{4} + 1432080 \, a^{8} b^{7} x^{\frac{7}{2}} + 1021020 \, a^{9} b^{6} x^{3} + 565488 \, a^{10} b^{5} x^{\frac{5}{2}} + 238680 \, a^{11} b^{4} x^{2} + 74256 \, a^{12} b^{3} x^{\frac{3}{2}} + 16065 \, a^{13} b^{2} x + 2160 \, a^{14} b \sqrt{x} + 136 \, a^{15}}{1224 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^10,x, algorithm="maxima")

[Out]

-1/1224*(816*b^15*x^(15/2) + 9180*a*b^14*x^7 + 51408*a^2*b^13*x^(13/2) + 185640*a^3*b^12*x^6 + 477360*a^4*b^11
*x^(11/2) + 918918*a^5*b^10*x^5 + 1361360*a^6*b^9*x^(9/2) + 1575288*a^7*b^8*x^4 + 1432080*a^8*b^7*x^(7/2) + 10
21020*a^9*b^6*x^3 + 565488*a^10*b^5*x^(5/2) + 238680*a^11*b^4*x^2 + 74256*a^12*b^3*x^(3/2) + 16065*a^13*b^2*x
+ 2160*a^14*b*sqrt(x) + 136*a^15)/x^9

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Fricas [B]  time = 1.32313, size = 447, normalized size = 6.39 \begin{align*} -\frac{9180 \, a b^{14} x^{7} + 185640 \, a^{3} b^{12} x^{6} + 918918 \, a^{5} b^{10} x^{5} + 1575288 \, a^{7} b^{8} x^{4} + 1021020 \, a^{9} b^{6} x^{3} + 238680 \, a^{11} b^{4} x^{2} + 16065 \, a^{13} b^{2} x + 136 \, a^{15} + 16 \,{\left (51 \, b^{15} x^{7} + 3213 \, a^{2} b^{13} x^{6} + 29835 \, a^{4} b^{11} x^{5} + 85085 \, a^{6} b^{9} x^{4} + 89505 \, a^{8} b^{7} x^{3} + 35343 \, a^{10} b^{5} x^{2} + 4641 \, a^{12} b^{3} x + 135 \, a^{14} b\right )} \sqrt{x}}{1224 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^10,x, algorithm="fricas")

[Out]

-1/1224*(9180*a*b^14*x^7 + 185640*a^3*b^12*x^6 + 918918*a^5*b^10*x^5 + 1575288*a^7*b^8*x^4 + 1021020*a^9*b^6*x
^3 + 238680*a^11*b^4*x^2 + 16065*a^13*b^2*x + 136*a^15 + 16*(51*b^15*x^7 + 3213*a^2*b^13*x^6 + 29835*a^4*b^11*
x^5 + 85085*a^6*b^9*x^4 + 89505*a^8*b^7*x^3 + 35343*a^10*b^5*x^2 + 4641*a^12*b^3*x + 135*a^14*b)*sqrt(x))/x^9

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Sympy [B]  time = 8.99436, size = 209, normalized size = 2.99 \begin{align*} - \frac{a^{15}}{9 x^{9}} - \frac{30 a^{14} b}{17 x^{\frac{17}{2}}} - \frac{105 a^{13} b^{2}}{8 x^{8}} - \frac{182 a^{12} b^{3}}{3 x^{\frac{15}{2}}} - \frac{195 a^{11} b^{4}}{x^{7}} - \frac{462 a^{10} b^{5}}{x^{\frac{13}{2}}} - \frac{5005 a^{9} b^{6}}{6 x^{6}} - \frac{1170 a^{8} b^{7}}{x^{\frac{11}{2}}} - \frac{1287 a^{7} b^{8}}{x^{5}} - \frac{10010 a^{6} b^{9}}{9 x^{\frac{9}{2}}} - \frac{3003 a^{5} b^{10}}{4 x^{4}} - \frac{390 a^{4} b^{11}}{x^{\frac{7}{2}}} - \frac{455 a^{3} b^{12}}{3 x^{3}} - \frac{42 a^{2} b^{13}}{x^{\frac{5}{2}}} - \frac{15 a b^{14}}{2 x^{2}} - \frac{2 b^{15}}{3 x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**15/x**10,x)

[Out]

-a**15/(9*x**9) - 30*a**14*b/(17*x**(17/2)) - 105*a**13*b**2/(8*x**8) - 182*a**12*b**3/(3*x**(15/2)) - 195*a**
11*b**4/x**7 - 462*a**10*b**5/x**(13/2) - 5005*a**9*b**6/(6*x**6) - 1170*a**8*b**7/x**(11/2) - 1287*a**7*b**8/
x**5 - 10010*a**6*b**9/(9*x**(9/2)) - 3003*a**5*b**10/(4*x**4) - 390*a**4*b**11/x**(7/2) - 455*a**3*b**12/(3*x
**3) - 42*a**2*b**13/x**(5/2) - 15*a*b**14/(2*x**2) - 2*b**15/(3*x**(3/2))

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Giac [B]  time = 1.11694, size = 225, normalized size = 3.21 \begin{align*} -\frac{816 \, b^{15} x^{\frac{15}{2}} + 9180 \, a b^{14} x^{7} + 51408 \, a^{2} b^{13} x^{\frac{13}{2}} + 185640 \, a^{3} b^{12} x^{6} + 477360 \, a^{4} b^{11} x^{\frac{11}{2}} + 918918 \, a^{5} b^{10} x^{5} + 1361360 \, a^{6} b^{9} x^{\frac{9}{2}} + 1575288 \, a^{7} b^{8} x^{4} + 1432080 \, a^{8} b^{7} x^{\frac{7}{2}} + 1021020 \, a^{9} b^{6} x^{3} + 565488 \, a^{10} b^{5} x^{\frac{5}{2}} + 238680 \, a^{11} b^{4} x^{2} + 74256 \, a^{12} b^{3} x^{\frac{3}{2}} + 16065 \, a^{13} b^{2} x + 2160 \, a^{14} b \sqrt{x} + 136 \, a^{15}}{1224 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^10,x, algorithm="giac")

[Out]

-1/1224*(816*b^15*x^(15/2) + 9180*a*b^14*x^7 + 51408*a^2*b^13*x^(13/2) + 185640*a^3*b^12*x^6 + 477360*a^4*b^11
*x^(11/2) + 918918*a^5*b^10*x^5 + 1361360*a^6*b^9*x^(9/2) + 1575288*a^7*b^8*x^4 + 1432080*a^8*b^7*x^(7/2) + 10
21020*a^9*b^6*x^3 + 565488*a^10*b^5*x^(5/2) + 238680*a^11*b^4*x^2 + 74256*a^12*b^3*x^(3/2) + 16065*a^13*b^2*x
+ 2160*a^14*b*sqrt(x) + 136*a^15)/x^9