3.2181 \(\int \frac{(a+b \sqrt{x})^{15}}{x^8} \, dx\)

Optimal. Leaf size=198 \[ -\frac{35 a^{13} b^2}{2 x^6}-\frac{910 a^{12} b^3}{11 x^{11/2}}-\frac{273 a^{11} b^4}{x^5}-\frac{2002 a^{10} b^5}{3 x^{9/2}}-\frac{5005 a^9 b^6}{4 x^4}-\frac{12870 a^8 b^7}{7 x^{7/2}}-\frac{2145 a^7 b^8}{x^3}-\frac{2002 a^6 b^9}{x^{5/2}}-\frac{3003 a^5 b^{10}}{2 x^2}-\frac{910 a^4 b^{11}}{x^{3/2}}-\frac{455 a^3 b^{12}}{x}-\frac{210 a^2 b^{13}}{\sqrt{x}}-\frac{30 a^{14} b}{13 x^{13/2}}-\frac{a^{15}}{7 x^7}+15 a b^{14} \log (x)+2 b^{15} \sqrt{x} \]

[Out]

-a^15/(7*x^7) - (30*a^14*b)/(13*x^(13/2)) - (35*a^13*b^2)/(2*x^6) - (910*a^12*b^3)/(11*x^(11/2)) - (273*a^11*b
^4)/x^5 - (2002*a^10*b^5)/(3*x^(9/2)) - (5005*a^9*b^6)/(4*x^4) - (12870*a^8*b^7)/(7*x^(7/2)) - (2145*a^7*b^8)/
x^3 - (2002*a^6*b^9)/x^(5/2) - (3003*a^5*b^10)/(2*x^2) - (910*a^4*b^11)/x^(3/2) - (455*a^3*b^12)/x - (210*a^2*
b^13)/Sqrt[x] + 2*b^15*Sqrt[x] + 15*a*b^14*Log[x]

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Rubi [A]  time = 0.110218, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{35 a^{13} b^2}{2 x^6}-\frac{910 a^{12} b^3}{11 x^{11/2}}-\frac{273 a^{11} b^4}{x^5}-\frac{2002 a^{10} b^5}{3 x^{9/2}}-\frac{5005 a^9 b^6}{4 x^4}-\frac{12870 a^8 b^7}{7 x^{7/2}}-\frac{2145 a^7 b^8}{x^3}-\frac{2002 a^6 b^9}{x^{5/2}}-\frac{3003 a^5 b^{10}}{2 x^2}-\frac{910 a^4 b^{11}}{x^{3/2}}-\frac{455 a^3 b^{12}}{x}-\frac{210 a^2 b^{13}}{\sqrt{x}}-\frac{30 a^{14} b}{13 x^{13/2}}-\frac{a^{15}}{7 x^7}+15 a b^{14} \log (x)+2 b^{15} \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^8,x]

[Out]

-a^15/(7*x^7) - (30*a^14*b)/(13*x^(13/2)) - (35*a^13*b^2)/(2*x^6) - (910*a^12*b^3)/(11*x^(11/2)) - (273*a^11*b
^4)/x^5 - (2002*a^10*b^5)/(3*x^(9/2)) - (5005*a^9*b^6)/(4*x^4) - (12870*a^8*b^7)/(7*x^(7/2)) - (2145*a^7*b^8)/
x^3 - (2002*a^6*b^9)/x^(5/2) - (3003*a^5*b^10)/(2*x^2) - (910*a^4*b^11)/x^(3/2) - (455*a^3*b^12)/x - (210*a^2*
b^13)/Sqrt[x] + 2*b^15*Sqrt[x] + 15*a*b^14*Log[x]

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^8} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{15}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (b^{15}+\frac{a^{15}}{x^{15}}+\frac{15 a^{14} b}{x^{14}}+\frac{105 a^{13} b^2}{x^{13}}+\frac{455 a^{12} b^3}{x^{12}}+\frac{1365 a^{11} b^4}{x^{11}}+\frac{3003 a^{10} b^5}{x^{10}}+\frac{5005 a^9 b^6}{x^9}+\frac{6435 a^8 b^7}{x^8}+\frac{6435 a^7 b^8}{x^7}+\frac{5005 a^6 b^9}{x^6}+\frac{3003 a^5 b^{10}}{x^5}+\frac{1365 a^4 b^{11}}{x^4}+\frac{455 a^3 b^{12}}{x^3}+\frac{105 a^2 b^{13}}{x^2}+\frac{15 a b^{14}}{x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{15}}{7 x^7}-\frac{30 a^{14} b}{13 x^{13/2}}-\frac{35 a^{13} b^2}{2 x^6}-\frac{910 a^{12} b^3}{11 x^{11/2}}-\frac{273 a^{11} b^4}{x^5}-\frac{2002 a^{10} b^5}{3 x^{9/2}}-\frac{5005 a^9 b^6}{4 x^4}-\frac{12870 a^8 b^7}{7 x^{7/2}}-\frac{2145 a^7 b^8}{x^3}-\frac{2002 a^6 b^9}{x^{5/2}}-\frac{3003 a^5 b^{10}}{2 x^2}-\frac{910 a^4 b^{11}}{x^{3/2}}-\frac{455 a^3 b^{12}}{x}-\frac{210 a^2 b^{13}}{\sqrt{x}}+2 b^{15} \sqrt{x}+15 a b^{14} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.119079, size = 198, normalized size = 1. \[ -\frac{35 a^{13} b^2}{2 x^6}-\frac{910 a^{12} b^3}{11 x^{11/2}}-\frac{273 a^{11} b^4}{x^5}-\frac{2002 a^{10} b^5}{3 x^{9/2}}-\frac{5005 a^9 b^6}{4 x^4}-\frac{12870 a^8 b^7}{7 x^{7/2}}-\frac{2145 a^7 b^8}{x^3}-\frac{2002 a^6 b^9}{x^{5/2}}-\frac{3003 a^5 b^{10}}{2 x^2}-\frac{910 a^4 b^{11}}{x^{3/2}}-\frac{455 a^3 b^{12}}{x}-\frac{210 a^2 b^{13}}{\sqrt{x}}-\frac{30 a^{14} b}{13 x^{13/2}}-\frac{a^{15}}{7 x^7}+15 a b^{14} \log (x)+2 b^{15} \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^8,x]

[Out]

-a^15/(7*x^7) - (30*a^14*b)/(13*x^(13/2)) - (35*a^13*b^2)/(2*x^6) - (910*a^12*b^3)/(11*x^(11/2)) - (273*a^11*b
^4)/x^5 - (2002*a^10*b^5)/(3*x^(9/2)) - (5005*a^9*b^6)/(4*x^4) - (12870*a^8*b^7)/(7*x^(7/2)) - (2145*a^7*b^8)/
x^3 - (2002*a^6*b^9)/x^(5/2) - (3003*a^5*b^10)/(2*x^2) - (910*a^4*b^11)/x^(3/2) - (455*a^3*b^12)/x - (210*a^2*
b^13)/Sqrt[x] + 2*b^15*Sqrt[x] + 15*a*b^14*Log[x]

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Maple [A]  time = 0.005, size = 167, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{7\,{x}^{7}}}-{\frac{30\,{a}^{14}b}{13}{x}^{-{\frac{13}{2}}}}-{\frac{35\,{a}^{13}{b}^{2}}{2\,{x}^{6}}}-{\frac{910\,{a}^{12}{b}^{3}}{11}{x}^{-{\frac{11}{2}}}}-273\,{\frac{{a}^{11}{b}^{4}}{{x}^{5}}}-{\frac{2002\,{a}^{10}{b}^{5}}{3}{x}^{-{\frac{9}{2}}}}-{\frac{5005\,{a}^{9}{b}^{6}}{4\,{x}^{4}}}-{\frac{12870\,{a}^{8}{b}^{7}}{7}{x}^{-{\frac{7}{2}}}}-2145\,{\frac{{a}^{7}{b}^{8}}{{x}^{3}}}-2002\,{\frac{{a}^{6}{b}^{9}}{{x}^{5/2}}}-{\frac{3003\,{a}^{5}{b}^{10}}{2\,{x}^{2}}}-910\,{\frac{{a}^{4}{b}^{11}}{{x}^{3/2}}}-455\,{\frac{{a}^{3}{b}^{12}}{x}}+15\,a{b}^{14}\ln \left ( x \right ) -210\,{\frac{{a}^{2}{b}^{13}}{\sqrt{x}}}+2\,{b}^{15}\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^8,x)

[Out]

-1/7*a^15/x^7-30/13*a^14*b/x^(13/2)-35/2*a^13*b^2/x^6-910/11*a^12*b^3/x^(11/2)-273*a^11*b^4/x^5-2002/3*a^10*b^
5/x^(9/2)-5005/4*a^9*b^6/x^4-12870/7*a^8*b^7/x^(7/2)-2145*a^7*b^8/x^3-2002*a^6*b^9/x^(5/2)-3003/2*a^5*b^10/x^2
-910*a^4*b^11/x^(3/2)-455*a^3*b^12/x+15*a*b^14*ln(x)-210*a^2*b^13/x^(1/2)+2*b^15*x^(1/2)

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Maxima [A]  time = 0.986467, size = 225, normalized size = 1.14 \begin{align*} 15 \, a b^{14} \log \left (x\right ) + 2 \, b^{15} \sqrt{x} - \frac{2522520 \, a^{2} b^{13} x^{\frac{13}{2}} + 5465460 \, a^{3} b^{12} x^{6} + 10930920 \, a^{4} b^{11} x^{\frac{11}{2}} + 18036018 \, a^{5} b^{10} x^{5} + 24048024 \, a^{6} b^{9} x^{\frac{9}{2}} + 25765740 \, a^{7} b^{8} x^{4} + 22084920 \, a^{8} b^{7} x^{\frac{7}{2}} + 15030015 \, a^{9} b^{6} x^{3} + 8016008 \, a^{10} b^{5} x^{\frac{5}{2}} + 3279276 \, a^{11} b^{4} x^{2} + 993720 \, a^{12} b^{3} x^{\frac{3}{2}} + 210210 \, a^{13} b^{2} x + 27720 \, a^{14} b \sqrt{x} + 1716 \, a^{15}}{12012 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^8,x, algorithm="maxima")

[Out]

15*a*b^14*log(x) + 2*b^15*sqrt(x) - 1/12012*(2522520*a^2*b^13*x^(13/2) + 5465460*a^3*b^12*x^6 + 10930920*a^4*b
^11*x^(11/2) + 18036018*a^5*b^10*x^5 + 24048024*a^6*b^9*x^(9/2) + 25765740*a^7*b^8*x^4 + 22084920*a^8*b^7*x^(7
/2) + 15030015*a^9*b^6*x^3 + 8016008*a^10*b^5*x^(5/2) + 3279276*a^11*b^4*x^2 + 993720*a^12*b^3*x^(3/2) + 21021
0*a^13*b^2*x + 27720*a^14*b*sqrt(x) + 1716*a^15)/x^7

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Fricas [A]  time = 1.33782, size = 497, normalized size = 2.51 \begin{align*} \frac{360360 \, a b^{14} x^{7} \log \left (\sqrt{x}\right ) - 5465460 \, a^{3} b^{12} x^{6} - 18036018 \, a^{5} b^{10} x^{5} - 25765740 \, a^{7} b^{8} x^{4} - 15030015 \, a^{9} b^{6} x^{3} - 3279276 \, a^{11} b^{4} x^{2} - 210210 \, a^{13} b^{2} x - 1716 \, a^{15} + 8 \,{\left (3003 \, b^{15} x^{7} - 315315 \, a^{2} b^{13} x^{6} - 1366365 \, a^{4} b^{11} x^{5} - 3006003 \, a^{6} b^{9} x^{4} - 2760615 \, a^{8} b^{7} x^{3} - 1002001 \, a^{10} b^{5} x^{2} - 124215 \, a^{12} b^{3} x - 3465 \, a^{14} b\right )} \sqrt{x}}{12012 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^8,x, algorithm="fricas")

[Out]

1/12012*(360360*a*b^14*x^7*log(sqrt(x)) - 5465460*a^3*b^12*x^6 - 18036018*a^5*b^10*x^5 - 25765740*a^7*b^8*x^4
- 15030015*a^9*b^6*x^3 - 3279276*a^11*b^4*x^2 - 210210*a^13*b^2*x - 1716*a^15 + 8*(3003*b^15*x^7 - 315315*a^2*
b^13*x^6 - 1366365*a^4*b^11*x^5 - 3006003*a^6*b^9*x^4 - 2760615*a^8*b^7*x^3 - 1002001*a^10*b^5*x^2 - 124215*a^
12*b^3*x - 3465*a^14*b)*sqrt(x))/x^7

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Sympy [A]  time = 5.15206, size = 202, normalized size = 1.02 \begin{align*} - \frac{a^{15}}{7 x^{7}} - \frac{30 a^{14} b}{13 x^{\frac{13}{2}}} - \frac{35 a^{13} b^{2}}{2 x^{6}} - \frac{910 a^{12} b^{3}}{11 x^{\frac{11}{2}}} - \frac{273 a^{11} b^{4}}{x^{5}} - \frac{2002 a^{10} b^{5}}{3 x^{\frac{9}{2}}} - \frac{5005 a^{9} b^{6}}{4 x^{4}} - \frac{12870 a^{8} b^{7}}{7 x^{\frac{7}{2}}} - \frac{2145 a^{7} b^{8}}{x^{3}} - \frac{2002 a^{6} b^{9}}{x^{\frac{5}{2}}} - \frac{3003 a^{5} b^{10}}{2 x^{2}} - \frac{910 a^{4} b^{11}}{x^{\frac{3}{2}}} - \frac{455 a^{3} b^{12}}{x} - \frac{210 a^{2} b^{13}}{\sqrt{x}} + 15 a b^{14} \log{\left (x \right )} + 2 b^{15} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**15/x**8,x)

[Out]

-a**15/(7*x**7) - 30*a**14*b/(13*x**(13/2)) - 35*a**13*b**2/(2*x**6) - 910*a**12*b**3/(11*x**(11/2)) - 273*a**
11*b**4/x**5 - 2002*a**10*b**5/(3*x**(9/2)) - 5005*a**9*b**6/(4*x**4) - 12870*a**8*b**7/(7*x**(7/2)) - 2145*a*
*7*b**8/x**3 - 2002*a**6*b**9/x**(5/2) - 3003*a**5*b**10/(2*x**2) - 910*a**4*b**11/x**(3/2) - 455*a**3*b**12/x
- 210*a**2*b**13/sqrt(x) + 15*a*b**14*log(x) + 2*b**15*sqrt(x)

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Giac [A]  time = 1.13777, size = 227, normalized size = 1.15 \begin{align*} 15 \, a b^{14} \log \left ({\left | x \right |}\right ) + 2 \, b^{15} \sqrt{x} - \frac{2522520 \, a^{2} b^{13} x^{\frac{13}{2}} + 5465460 \, a^{3} b^{12} x^{6} + 10930920 \, a^{4} b^{11} x^{\frac{11}{2}} + 18036018 \, a^{5} b^{10} x^{5} + 24048024 \, a^{6} b^{9} x^{\frac{9}{2}} + 25765740 \, a^{7} b^{8} x^{4} + 22084920 \, a^{8} b^{7} x^{\frac{7}{2}} + 15030015 \, a^{9} b^{6} x^{3} + 8016008 \, a^{10} b^{5} x^{\frac{5}{2}} + 3279276 \, a^{11} b^{4} x^{2} + 993720 \, a^{12} b^{3} x^{\frac{3}{2}} + 210210 \, a^{13} b^{2} x + 27720 \, a^{14} b \sqrt{x} + 1716 \, a^{15}}{12012 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^8,x, algorithm="giac")

[Out]

15*a*b^14*log(abs(x)) + 2*b^15*sqrt(x) - 1/12012*(2522520*a^2*b^13*x^(13/2) + 5465460*a^3*b^12*x^6 + 10930920*
a^4*b^11*x^(11/2) + 18036018*a^5*b^10*x^5 + 24048024*a^6*b^9*x^(9/2) + 25765740*a^7*b^8*x^4 + 22084920*a^8*b^7
*x^(7/2) + 15030015*a^9*b^6*x^3 + 8016008*a^10*b^5*x^(5/2) + 3279276*a^11*b^4*x^2 + 993720*a^12*b^3*x^(3/2) +
210210*a^13*b^2*x + 27720*a^14*b*sqrt(x) + 1716*a^15)/x^7