3.2185 \(\int \frac{(a+b \sqrt{x})^{15}}{x^{12}} \, dx\)

Optimal. Leaf size=170 \[ -\frac{b^6 \left (a+b \sqrt{x}\right )^{16}}{596904 a^7 x^8}+\frac{2 b^5 \left (a+b \sqrt{x}\right )^{16}}{74613 a^6 x^{17/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{16}}{4389 a^5 x^9}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{1463 a^4 x^{19/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{154 a^3 x^{10}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{77 a^2 x^{21/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{11 a x^{11}} \]

[Out]

-(a + b*Sqrt[x])^16/(11*a*x^11) + (2*b*(a + b*Sqrt[x])^16)/(77*a^2*x^(21/2)) - (b^2*(a + b*Sqrt[x])^16)/(154*a
^3*x^10) + (2*b^3*(a + b*Sqrt[x])^16)/(1463*a^4*x^(19/2)) - (b^4*(a + b*Sqrt[x])^16)/(4389*a^5*x^9) + (2*b^5*(
a + b*Sqrt[x])^16)/(74613*a^6*x^(17/2)) - (b^6*(a + b*Sqrt[x])^16)/(596904*a^7*x^8)

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Rubi [A]  time = 0.0721561, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ -\frac{b^6 \left (a+b \sqrt{x}\right )^{16}}{596904 a^7 x^8}+\frac{2 b^5 \left (a+b \sqrt{x}\right )^{16}}{74613 a^6 x^{17/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{16}}{4389 a^5 x^9}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{1463 a^4 x^{19/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{154 a^3 x^{10}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{77 a^2 x^{21/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^12,x]

[Out]

-(a + b*Sqrt[x])^16/(11*a*x^11) + (2*b*(a + b*Sqrt[x])^16)/(77*a^2*x^(21/2)) - (b^2*(a + b*Sqrt[x])^16)/(154*a
^3*x^10) + (2*b^3*(a + b*Sqrt[x])^16)/(1463*a^4*x^(19/2)) - (b^4*(a + b*Sqrt[x])^16)/(4389*a^5*x^9) + (2*b^5*(
a + b*Sqrt[x])^16)/(74613*a^6*x^(17/2)) - (b^6*(a + b*Sqrt[x])^16)/(596904*a^7*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^{12}} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{23}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{11 a x^{11}}-\frac{(6 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{22}} \, dx,x,\sqrt{x}\right )}{11 a}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{11 a x^{11}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{77 a^2 x^{21/2}}+\frac{\left (10 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{21}} \, dx,x,\sqrt{x}\right )}{77 a^2}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{11 a x^{11}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{77 a^2 x^{21/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{154 a^3 x^{10}}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{20}} \, dx,x,\sqrt{x}\right )}{77 a^3}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{11 a x^{11}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{77 a^2 x^{21/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{154 a^3 x^{10}}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{1463 a^4 x^{19/2}}+\frac{\left (6 b^4\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt{x}\right )}{1463 a^4}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{11 a x^{11}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{77 a^2 x^{21/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{154 a^3 x^{10}}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{1463 a^4 x^{19/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{16}}{4389 a^5 x^9}-\frac{\left (2 b^5\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt{x}\right )}{4389 a^5}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{11 a x^{11}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{77 a^2 x^{21/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{154 a^3 x^{10}}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{1463 a^4 x^{19/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{16}}{4389 a^5 x^9}+\frac{2 b^5 \left (a+b \sqrt{x}\right )^{16}}{74613 a^6 x^{17/2}}+\frac{\left (2 b^6\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt{x}\right )}{74613 a^6}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{11 a x^{11}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{77 a^2 x^{21/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{16}}{154 a^3 x^{10}}+\frac{2 b^3 \left (a+b \sqrt{x}\right )^{16}}{1463 a^4 x^{19/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{16}}{4389 a^5 x^9}+\frac{2 b^5 \left (a+b \sqrt{x}\right )^{16}}{74613 a^6 x^{17/2}}-\frac{b^6 \left (a+b \sqrt{x}\right )^{16}}{596904 a^7 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0179323, size = 89, normalized size = 0.52 \[ -\frac{\left (a+b \sqrt{x}\right )^{16} \left (-816 a^3 b^3 x^{3/2}+136 a^2 b^4 x^2+3876 a^4 b^2 x-15504 a^5 b \sqrt{x}+54264 a^6-16 a b^5 x^{5/2}+b^6 x^3\right )}{596904 a^7 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^12,x]

[Out]

-((a + b*Sqrt[x])^16*(54264*a^6 - 15504*a^5*b*Sqrt[x] + 3876*a^4*b^2*x - 816*a^3*b^3*x^(3/2) + 136*a^2*b^4*x^2
- 16*a*b^5*x^(5/2) + b^6*x^3))/(596904*a^7*x^11)

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Maple [A]  time = 0.005, size = 168, normalized size = 1. \begin{align*} -{\frac{2\,{b}^{15}}{7}{x}^{-{\frac{7}{2}}}}-{\frac{15\,a{b}^{14}}{4\,{x}^{4}}}-{\frac{70\,{a}^{2}{b}^{13}}{3}{x}^{-{\frac{9}{2}}}}-91\,{\frac{{a}^{3}{b}^{12}}{{x}^{5}}}-{\frac{2730\,{a}^{4}{b}^{11}}{11}{x}^{-{\frac{11}{2}}}}-{\frac{1001\,{a}^{5}{b}^{10}}{2\,{x}^{6}}}-770\,{\frac{{a}^{6}{b}^{9}}{{x}^{13/2}}}-{\frac{6435\,{a}^{7}{b}^{8}}{7\,{x}^{7}}}-858\,{\frac{{a}^{8}{b}^{7}}{{x}^{15/2}}}-{\frac{5005\,{a}^{9}{b}^{6}}{8\,{x}^{8}}}-{\frac{6006\,{a}^{10}{b}^{5}}{17}{x}^{-{\frac{17}{2}}}}-{\frac{455\,{a}^{11}{b}^{4}}{3\,{x}^{9}}}-{\frac{910\,{a}^{12}{b}^{3}}{19}{x}^{-{\frac{19}{2}}}}-{\frac{21\,{a}^{13}{b}^{2}}{2\,{x}^{10}}}-{\frac{10\,{a}^{14}b}{7}{x}^{-{\frac{21}{2}}}}-{\frac{{a}^{15}}{11\,{x}^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^12,x)

[Out]

-2/7*b^15/x^(7/2)-15/4*a*b^14/x^4-70/3*a^2*b^13/x^(9/2)-91*a^3*b^12/x^5-2730/11*a^4*b^11/x^(11/2)-1001/2*a^5*b
^10/x^6-770*a^6*b^9/x^(13/2)-6435/7*a^7*b^8/x^7-858*a^8*b^7/x^(15/2)-5005/8*a^9*b^6/x^8-6006/17*a^10*b^5/x^(17
/2)-455/3*a^11*b^4/x^9-910/19*a^12*b^3/x^(19/2)-21/2*a^13*b^2/x^10-10/7*a^14*b/x^(21/2)-1/11*a^15/x^11

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Maxima [A]  time = 0.98954, size = 225, normalized size = 1.32 \begin{align*} -\frac{170544 \, b^{15} x^{\frac{15}{2}} + 2238390 \, a b^{14} x^{7} + 13927760 \, a^{2} b^{13} x^{\frac{13}{2}} + 54318264 \, a^{3} b^{12} x^{6} + 148140720 \, a^{4} b^{11} x^{\frac{11}{2}} + 298750452 \, a^{5} b^{10} x^{5} + 459616080 \, a^{6} b^{9} x^{\frac{9}{2}} + 548725320 \, a^{7} b^{8} x^{4} + 512143632 \, a^{8} b^{7} x^{\frac{7}{2}} + 373438065 \, a^{9} b^{6} x^{3} + 210882672 \, a^{10} b^{5} x^{\frac{5}{2}} + 90530440 \, a^{11} b^{4} x^{2} + 28588560 \, a^{12} b^{3} x^{\frac{3}{2}} + 6267492 \, a^{13} b^{2} x + 852720 \, a^{14} b \sqrt{x} + 54264 \, a^{15}}{596904 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^12,x, algorithm="maxima")

[Out]

-1/596904*(170544*b^15*x^(15/2) + 2238390*a*b^14*x^7 + 13927760*a^2*b^13*x^(13/2) + 54318264*a^3*b^12*x^6 + 14
8140720*a^4*b^11*x^(11/2) + 298750452*a^5*b^10*x^5 + 459616080*a^6*b^9*x^(9/2) + 548725320*a^7*b^8*x^4 + 51214
3632*a^8*b^7*x^(7/2) + 373438065*a^9*b^6*x^3 + 210882672*a^10*b^5*x^(5/2) + 90530440*a^11*b^4*x^2 + 28588560*a
^12*b^3*x^(3/2) + 6267492*a^13*b^2*x + 852720*a^14*b*sqrt(x) + 54264*a^15)/x^11

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Fricas [A]  time = 1.29809, size = 504, normalized size = 2.96 \begin{align*} -\frac{2238390 \, a b^{14} x^{7} + 54318264 \, a^{3} b^{12} x^{6} + 298750452 \, a^{5} b^{10} x^{5} + 548725320 \, a^{7} b^{8} x^{4} + 373438065 \, a^{9} b^{6} x^{3} + 90530440 \, a^{11} b^{4} x^{2} + 6267492 \, a^{13} b^{2} x + 54264 \, a^{15} + 16 \,{\left (10659 \, b^{15} x^{7} + 870485 \, a^{2} b^{13} x^{6} + 9258795 \, a^{4} b^{11} x^{5} + 28726005 \, a^{6} b^{9} x^{4} + 32008977 \, a^{8} b^{7} x^{3} + 13180167 \, a^{10} b^{5} x^{2} + 1786785 \, a^{12} b^{3} x + 53295 \, a^{14} b\right )} \sqrt{x}}{596904 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^12,x, algorithm="fricas")

[Out]

-1/596904*(2238390*a*b^14*x^7 + 54318264*a^3*b^12*x^6 + 298750452*a^5*b^10*x^5 + 548725320*a^7*b^8*x^4 + 37343
8065*a^9*b^6*x^3 + 90530440*a^11*b^4*x^2 + 6267492*a^13*b^2*x + 54264*a^15 + 16*(10659*b^15*x^7 + 870485*a^2*b
^13*x^6 + 9258795*a^4*b^11*x^5 + 28726005*a^6*b^9*x^4 + 32008977*a^8*b^7*x^3 + 13180167*a^10*b^5*x^2 + 1786785
*a^12*b^3*x + 53295*a^14*b)*sqrt(x))/x^11

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Sympy [A]  time = 21.2609, size = 214, normalized size = 1.26 \begin{align*} - \frac{a^{15}}{11 x^{11}} - \frac{10 a^{14} b}{7 x^{\frac{21}{2}}} - \frac{21 a^{13} b^{2}}{2 x^{10}} - \frac{910 a^{12} b^{3}}{19 x^{\frac{19}{2}}} - \frac{455 a^{11} b^{4}}{3 x^{9}} - \frac{6006 a^{10} b^{5}}{17 x^{\frac{17}{2}}} - \frac{5005 a^{9} b^{6}}{8 x^{8}} - \frac{858 a^{8} b^{7}}{x^{\frac{15}{2}}} - \frac{6435 a^{7} b^{8}}{7 x^{7}} - \frac{770 a^{6} b^{9}}{x^{\frac{13}{2}}} - \frac{1001 a^{5} b^{10}}{2 x^{6}} - \frac{2730 a^{4} b^{11}}{11 x^{\frac{11}{2}}} - \frac{91 a^{3} b^{12}}{x^{5}} - \frac{70 a^{2} b^{13}}{3 x^{\frac{9}{2}}} - \frac{15 a b^{14}}{4 x^{4}} - \frac{2 b^{15}}{7 x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**15/x**12,x)

[Out]

-a**15/(11*x**11) - 10*a**14*b/(7*x**(21/2)) - 21*a**13*b**2/(2*x**10) - 910*a**12*b**3/(19*x**(19/2)) - 455*a
**11*b**4/(3*x**9) - 6006*a**10*b**5/(17*x**(17/2)) - 5005*a**9*b**6/(8*x**8) - 858*a**8*b**7/x**(15/2) - 6435
*a**7*b**8/(7*x**7) - 770*a**6*b**9/x**(13/2) - 1001*a**5*b**10/(2*x**6) - 2730*a**4*b**11/(11*x**(11/2)) - 91
*a**3*b**12/x**5 - 70*a**2*b**13/(3*x**(9/2)) - 15*a*b**14/(4*x**4) - 2*b**15/(7*x**(7/2))

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Giac [A]  time = 1.13195, size = 225, normalized size = 1.32 \begin{align*} -\frac{170544 \, b^{15} x^{\frac{15}{2}} + 2238390 \, a b^{14} x^{7} + 13927760 \, a^{2} b^{13} x^{\frac{13}{2}} + 54318264 \, a^{3} b^{12} x^{6} + 148140720 \, a^{4} b^{11} x^{\frac{11}{2}} + 298750452 \, a^{5} b^{10} x^{5} + 459616080 \, a^{6} b^{9} x^{\frac{9}{2}} + 548725320 \, a^{7} b^{8} x^{4} + 512143632 \, a^{8} b^{7} x^{\frac{7}{2}} + 373438065 \, a^{9} b^{6} x^{3} + 210882672 \, a^{10} b^{5} x^{\frac{5}{2}} + 90530440 \, a^{11} b^{4} x^{2} + 28588560 \, a^{12} b^{3} x^{\frac{3}{2}} + 6267492 \, a^{13} b^{2} x + 852720 \, a^{14} b \sqrt{x} + 54264 \, a^{15}}{596904 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^12,x, algorithm="giac")

[Out]

-1/596904*(170544*b^15*x^(15/2) + 2238390*a*b^14*x^7 + 13927760*a^2*b^13*x^(13/2) + 54318264*a^3*b^12*x^6 + 14
8140720*a^4*b^11*x^(11/2) + 298750452*a^5*b^10*x^5 + 459616080*a^6*b^9*x^(9/2) + 548725320*a^7*b^8*x^4 + 51214
3632*a^8*b^7*x^(7/2) + 373438065*a^9*b^6*x^3 + 210882672*a^10*b^5*x^(5/2) + 90530440*a^11*b^4*x^2 + 28588560*a
^12*b^3*x^(3/2) + 6267492*a^13*b^2*x + 852720*a^14*b*sqrt(x) + 54264*a^15)/x^11