3.2182 \(\int \frac{(a+b \sqrt{x})^{15}}{x^9} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b \sqrt{x}\right )^{16}}{8 a x^8} \]

[Out]

-(a + b*Sqrt[x])^16/(8*a*x^8)

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Rubi [A]  time = 0.0032669, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\left (a+b \sqrt{x}\right )^{16}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^9,x]

[Out]

-(a + b*Sqrt[x])^16/(8*a*x^8)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^9} \, dx &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{8 a x^8}\\ \end{align*}

Mathematica [A]  time = 0.0070721, size = 21, normalized size = 1. \[ -\frac{\left (a+b \sqrt{x}\right )^{16}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^9,x]

[Out]

-(a + b*Sqrt[x])^16/(8*a*x^8)

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Maple [B]  time = 0.004, size = 168, normalized size = 8. \begin{align*} -2\,{\frac{{b}^{15}}{\sqrt{x}}}-15\,{\frac{a{b}^{14}}{x}}-70\,{\frac{{a}^{2}{b}^{13}}{{x}^{3/2}}}-{\frac{455\,{a}^{3}{b}^{12}}{2\,{x}^{2}}}-546\,{\frac{{a}^{4}{b}^{11}}{{x}^{5/2}}}-1001\,{\frac{{a}^{5}{b}^{10}}{{x}^{3}}}-1430\,{\frac{{a}^{6}{b}^{9}}{{x}^{7/2}}}-{\frac{6435\,{a}^{7}{b}^{8}}{4\,{x}^{4}}}-1430\,{\frac{{a}^{8}{b}^{7}}{{x}^{9/2}}}-1001\,{\frac{{a}^{9}{b}^{6}}{{x}^{5}}}-546\,{\frac{{a}^{10}{b}^{5}}{{x}^{11/2}}}-{\frac{455\,{a}^{11}{b}^{4}}{2\,{x}^{6}}}-70\,{\frac{{a}^{12}{b}^{3}}{{x}^{13/2}}}-15\,{\frac{{a}^{13}{b}^{2}}{{x}^{7}}}-2\,{\frac{{a}^{14}b}{{x}^{15/2}}}-{\frac{{a}^{15}}{8\,{x}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^9,x)

[Out]

-2*b^15/x^(1/2)-15*a*b^14/x-70*a^2*b^13/x^(3/2)-455/2*a^3*b^12/x^2-546*a^4*b^11/x^(5/2)-1001*a^5*b^10/x^3-1430
*a^6*b^9/x^(7/2)-6435/4*a^7*b^8/x^4-1430*a^8*b^7/x^(9/2)-1001*a^9*b^6/x^5-546*a^10*b^5/x^(11/2)-455/2*a^11*b^4
/x^6-70*a^12*b^3/x^(13/2)-15*a^13*b^2/x^7-2*a^14*b/x^(15/2)-1/8*a^15/x^8

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Maxima [B]  time = 0.969159, size = 223, normalized size = 10.62 \begin{align*} -\frac{16 \, b^{15} x^{\frac{15}{2}} + 120 \, a b^{14} x^{7} + 560 \, a^{2} b^{13} x^{\frac{13}{2}} + 1820 \, a^{3} b^{12} x^{6} + 4368 \, a^{4} b^{11} x^{\frac{11}{2}} + 8008 \, a^{5} b^{10} x^{5} + 11440 \, a^{6} b^{9} x^{\frac{9}{2}} + 12870 \, a^{7} b^{8} x^{4} + 11440 \, a^{8} b^{7} x^{\frac{7}{2}} + 8008 \, a^{9} b^{6} x^{3} + 4368 \, a^{10} b^{5} x^{\frac{5}{2}} + 1820 \, a^{11} b^{4} x^{2} + 560 \, a^{12} b^{3} x^{\frac{3}{2}} + 120 \, a^{13} b^{2} x + 16 \, a^{14} b \sqrt{x} + a^{15}}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^9,x, algorithm="maxima")

[Out]

-1/8*(16*b^15*x^(15/2) + 120*a*b^14*x^7 + 560*a^2*b^13*x^(13/2) + 1820*a^3*b^12*x^6 + 4368*a^4*b^11*x^(11/2) +
8008*a^5*b^10*x^5 + 11440*a^6*b^9*x^(9/2) + 12870*a^7*b^8*x^4 + 11440*a^8*b^7*x^(7/2) + 8008*a^9*b^6*x^3 + 43
68*a^10*b^5*x^(5/2) + 1820*a^11*b^4*x^2 + 560*a^12*b^3*x^(3/2) + 120*a^13*b^2*x + 16*a^14*b*sqrt(x) + a^15)/x^
8

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Fricas [B]  time = 1.24662, size = 393, normalized size = 18.71 \begin{align*} -\frac{120 \, a b^{14} x^{7} + 1820 \, a^{3} b^{12} x^{6} + 8008 \, a^{5} b^{10} x^{5} + 12870 \, a^{7} b^{8} x^{4} + 8008 \, a^{9} b^{6} x^{3} + 1820 \, a^{11} b^{4} x^{2} + 120 \, a^{13} b^{2} x + a^{15} + 16 \,{\left (b^{15} x^{7} + 35 \, a^{2} b^{13} x^{6} + 273 \, a^{4} b^{11} x^{5} + 715 \, a^{6} b^{9} x^{4} + 715 \, a^{8} b^{7} x^{3} + 273 \, a^{10} b^{5} x^{2} + 35 \, a^{12} b^{3} x + a^{14} b\right )} \sqrt{x}}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^9,x, algorithm="fricas")

[Out]

-1/8*(120*a*b^14*x^7 + 1820*a^3*b^12*x^6 + 8008*a^5*b^10*x^5 + 12870*a^7*b^8*x^4 + 8008*a^9*b^6*x^3 + 1820*a^1
1*b^4*x^2 + 120*a^13*b^2*x + a^15 + 16*(b^15*x^7 + 35*a^2*b^13*x^6 + 273*a^4*b^11*x^5 + 715*a^6*b^9*x^4 + 715*
a^8*b^7*x^3 + 273*a^10*b^5*x^2 + 35*a^12*b^3*x + a^14*b)*sqrt(x))/x^8

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Sympy [B]  time = 6.76106, size = 197, normalized size = 9.38 \begin{align*} - \frac{a^{15}}{8 x^{8}} - \frac{2 a^{14} b}{x^{\frac{15}{2}}} - \frac{15 a^{13} b^{2}}{x^{7}} - \frac{70 a^{12} b^{3}}{x^{\frac{13}{2}}} - \frac{455 a^{11} b^{4}}{2 x^{6}} - \frac{546 a^{10} b^{5}}{x^{\frac{11}{2}}} - \frac{1001 a^{9} b^{6}}{x^{5}} - \frac{1430 a^{8} b^{7}}{x^{\frac{9}{2}}} - \frac{6435 a^{7} b^{8}}{4 x^{4}} - \frac{1430 a^{6} b^{9}}{x^{\frac{7}{2}}} - \frac{1001 a^{5} b^{10}}{x^{3}} - \frac{546 a^{4} b^{11}}{x^{\frac{5}{2}}} - \frac{455 a^{3} b^{12}}{2 x^{2}} - \frac{70 a^{2} b^{13}}{x^{\frac{3}{2}}} - \frac{15 a b^{14}}{x} - \frac{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**9,x)

[Out]

-a**15/(8*x**8) - 2*a**14*b/x**(15/2) - 15*a**13*b**2/x**7 - 70*a**12*b**3/x**(13/2) - 455*a**11*b**4/(2*x**6)
- 546*a**10*b**5/x**(11/2) - 1001*a**9*b**6/x**5 - 1430*a**8*b**7/x**(9/2) - 6435*a**7*b**8/(4*x**4) - 1430*a
**6*b**9/x**(7/2) - 1001*a**5*b**10/x**3 - 546*a**4*b**11/x**(5/2) - 455*a**3*b**12/(2*x**2) - 70*a**2*b**13/x
**(3/2) - 15*a*b**14/x - 2*b**15/sqrt(x)

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Giac [B]  time = 1.10753, size = 223, normalized size = 10.62 \begin{align*} -\frac{16 \, b^{15} x^{\frac{15}{2}} + 120 \, a b^{14} x^{7} + 560 \, a^{2} b^{13} x^{\frac{13}{2}} + 1820 \, a^{3} b^{12} x^{6} + 4368 \, a^{4} b^{11} x^{\frac{11}{2}} + 8008 \, a^{5} b^{10} x^{5} + 11440 \, a^{6} b^{9} x^{\frac{9}{2}} + 12870 \, a^{7} b^{8} x^{4} + 11440 \, a^{8} b^{7} x^{\frac{7}{2}} + 8008 \, a^{9} b^{6} x^{3} + 4368 \, a^{10} b^{5} x^{\frac{5}{2}} + 1820 \, a^{11} b^{4} x^{2} + 560 \, a^{12} b^{3} x^{\frac{3}{2}} + 120 \, a^{13} b^{2} x + 16 \, a^{14} b \sqrt{x} + a^{15}}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^9,x, algorithm="giac")

[Out]

-1/8*(16*b^15*x^(15/2) + 120*a*b^14*x^7 + 560*a^2*b^13*x^(13/2) + 1820*a^3*b^12*x^6 + 4368*a^4*b^11*x^(11/2) +
8008*a^5*b^10*x^5 + 11440*a^6*b^9*x^(9/2) + 12870*a^7*b^8*x^4 + 11440*a^8*b^7*x^(7/2) + 8008*a^9*b^6*x^3 + 43
68*a^10*b^5*x^(5/2) + 1820*a^11*b^4*x^2 + 560*a^12*b^3*x^(3/2) + 120*a^13*b^2*x + 16*a^14*b*sqrt(x) + a^15)/x^
8