3.2166 \(\int \frac{(a+b \sqrt{x})^{10}}{x^9} \, dx\)

Optimal. Leaf size=146 \[ \frac{b^5 \left (a+b \sqrt{x}\right )^{11}}{24024 a^6 x^{11/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{11}}{2184 a^5 x^6}+\frac{b^3 \left (a+b \sqrt{x}\right )^{11}}{364 a^4 x^{13/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{84 a^3 x^7}+\frac{b \left (a+b \sqrt{x}\right )^{11}}{24 a^2 x^{15/2}}-\frac{\left (a+b \sqrt{x}\right )^{11}}{8 a x^8} \]

[Out]

-(a + b*Sqrt[x])^11/(8*a*x^8) + (b*(a + b*Sqrt[x])^11)/(24*a^2*x^(15/2)) - (b^2*(a + b*Sqrt[x])^11)/(84*a^3*x^
7) + (b^3*(a + b*Sqrt[x])^11)/(364*a^4*x^(13/2)) - (b^4*(a + b*Sqrt[x])^11)/(2184*a^5*x^6) + (b^5*(a + b*Sqrt[
x])^11)/(24024*a^6*x^(11/2))

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Rubi [A]  time = 0.0572555, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ \frac{b^5 \left (a+b \sqrt{x}\right )^{11}}{24024 a^6 x^{11/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{11}}{2184 a^5 x^6}+\frac{b^3 \left (a+b \sqrt{x}\right )^{11}}{364 a^4 x^{13/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{84 a^3 x^7}+\frac{b \left (a+b \sqrt{x}\right )^{11}}{24 a^2 x^{15/2}}-\frac{\left (a+b \sqrt{x}\right )^{11}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10/x^9,x]

[Out]

-(a + b*Sqrt[x])^11/(8*a*x^8) + (b*(a + b*Sqrt[x])^11)/(24*a^2*x^(15/2)) - (b^2*(a + b*Sqrt[x])^11)/(84*a^3*x^
7) + (b^3*(a + b*Sqrt[x])^11)/(364*a^4*x^(13/2)) - (b^4*(a + b*Sqrt[x])^11)/(2184*a^5*x^6) + (b^5*(a + b*Sqrt[
x])^11)/(24024*a^6*x^(11/2))

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 )^{10}}{x^9} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{17}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{8 a x^8}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{16}} \, dx,x,\sqrt{x}\right )}{8 a}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{8 a x^8}+\frac{b \left (a+b \sqrt{x}\right )^{11}}{24 a^2 x^{15/2}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{15}} \, dx,x,\sqrt{x}\right )}{6 a^2}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{8 a x^8}+\frac{b \left (a+b \sqrt{x}\right )^{11}}{24 a^2 x^{15/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{84 a^3 x^7}-\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{14}} \, dx,x,\sqrt{x}\right )}{28 a^3}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{8 a x^8}+\frac{b \left (a+b \sqrt{x}\right )^{11}}{24 a^2 x^{15/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{84 a^3 x^7}+\frac{b^3 \left (a+b \sqrt{x}\right )^{11}}{364 a^4 x^{13/2}}+\frac{b^4 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{13}} \, dx,x,\sqrt{x}\right )}{182 a^4}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{8 a x^8}+\frac{b \left (a+b \sqrt{x}\right )^{11}}{24 a^2 x^{15/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{84 a^3 x^7}+\frac{b^3 \left (a+b \sqrt{x}\right )^{11}}{364 a^4 x^{13/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{11}}{2184 a^5 x^6}-\frac{b^5 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{12}} \, dx,x,\sqrt{x}\right )}{2184 a^5}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{8 a x^8}+\frac{b \left (a+b \sqrt{x}\right )^{11}}{24 a^2 x^{15/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{84 a^3 x^7}+\frac{b^3 \left (a+b \sqrt{x}\right )^{11}}{364 a^4 x^{13/2}}-\frac{b^4 \left (a+b \sqrt{x}\right )^{11}}{2184 a^5 x^6}+\frac{b^5 \left (a+b \sqrt{x}\right )^{11}}{24024 a^6 x^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.0265123, size = 140, normalized size = 0.96 \[ -\frac{45 a^8 b^2}{7 x^7}-\frac{240 a^7 b^3}{13 x^{13/2}}-\frac{35 a^6 b^4}{x^6}-\frac{504 a^5 b^5}{11 x^{11/2}}-\frac{42 a^4 b^6}{x^5}-\frac{80 a^3 b^7}{3 x^{9/2}}-\frac{45 a^2 b^8}{4 x^4}-\frac{4 a^9 b}{3 x^{15/2}}-\frac{a^{10}}{8 x^8}-\frac{20 a b^9}{7 x^{7/2}}-\frac{b^{10}}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10/x^9,x]

[Out]

-a^10/(8*x^8) - (4*a^9*b)/(3*x^(15/2)) - (45*a^8*b^2)/(7*x^7) - (240*a^7*b^3)/(13*x^(13/2)) - (35*a^6*b^4)/x^6
- (504*a^5*b^5)/(11*x^(11/2)) - (42*a^4*b^6)/x^5 - (80*a^3*b^7)/(3*x^(9/2)) - (45*a^2*b^8)/(4*x^4) - (20*a*b^
9)/(7*x^(7/2)) - b^10/(3*x^3)

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Maple [A]  time = 0.003, size = 113, normalized size = 0.8 \begin{align*} -{\frac{{b}^{10}}{3\,{x}^{3}}}-{\frac{20\,a{b}^{9}}{7}{x}^{-{\frac{7}{2}}}}-{\frac{45\,{a}^{2}{b}^{8}}{4\,{x}^{4}}}-{\frac{80\,{a}^{3}{b}^{7}}{3}{x}^{-{\frac{9}{2}}}}-42\,{\frac{{a}^{4}{b}^{6}}{{x}^{5}}}-{\frac{504\,{a}^{5}{b}^{5}}{11}{x}^{-{\frac{11}{2}}}}-35\,{\frac{{a}^{6}{b}^{4}}{{x}^{6}}}-{\frac{240\,{a}^{7}{b}^{3}}{13}{x}^{-{\frac{13}{2}}}}-{\frac{45\,{a}^{8}{b}^{2}}{7\,{x}^{7}}}-{\frac{4\,{a}^{9}b}{3}{x}^{-{\frac{15}{2}}}}-{\frac{{a}^{10}}{8\,{x}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^10/x^9,x)

[Out]

-1/3*b^10/x^3-20/7*a*b^9/x^(7/2)-45/4*a^2*b^8/x^4-80/3*a^3*b^7/x^(9/2)-42*a^4*b^6/x^5-504/11*a^5*b^5/x^(11/2)-
35*a^6*b^4/x^6-240/13*a^7*b^3/x^(13/2)-45/7*a^8*b^2/x^7-4/3*a^9*b/x^(15/2)-1/8*a^10/x^8

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Maxima [A]  time = 0.994168, size = 151, normalized size = 1.03 \begin{align*} -\frac{8008 \, b^{10} x^{5} + 68640 \, a b^{9} x^{\frac{9}{2}} + 270270 \, a^{2} b^{8} x^{4} + 640640 \, a^{3} b^{7} x^{\frac{7}{2}} + 1009008 \, a^{4} b^{6} x^{3} + 1100736 \, a^{5} b^{5} x^{\frac{5}{2}} + 840840 \, a^{6} b^{4} x^{2} + 443520 \, a^{7} b^{3} x^{\frac{3}{2}} + 154440 \, a^{8} b^{2} x + 32032 \, a^{9} b \sqrt{x} + 3003 \, a^{10}}{24024 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^9,x, algorithm="maxima")

[Out]

-1/24024*(8008*b^10*x^5 + 68640*a*b^9*x^(9/2) + 270270*a^2*b^8*x^4 + 640640*a^3*b^7*x^(7/2) + 1009008*a^4*b^6*
x^3 + 1100736*a^5*b^5*x^(5/2) + 840840*a^6*b^4*x^2 + 443520*a^7*b^3*x^(3/2) + 154440*a^8*b^2*x + 32032*a^9*b*s
qrt(x) + 3003*a^10)/x^8

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Fricas [A]  time = 1.3138, size = 305, normalized size = 2.09 \begin{align*} -\frac{8008 \, b^{10} x^{5} + 270270 \, a^{2} b^{8} x^{4} + 1009008 \, a^{4} b^{6} x^{3} + 840840 \, a^{6} b^{4} x^{2} + 154440 \, a^{8} b^{2} x + 3003 \, a^{10} + 32 \,{\left (2145 \, a b^{9} x^{4} + 20020 \, a^{3} b^{7} x^{3} + 34398 \, a^{5} b^{5} x^{2} + 13860 \, a^{7} b^{3} x + 1001 \, a^{9} b\right )} \sqrt{x}}{24024 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^9,x, algorithm="fricas")

[Out]

-1/24024*(8008*b^10*x^5 + 270270*a^2*b^8*x^4 + 1009008*a^4*b^6*x^3 + 840840*a^6*b^4*x^2 + 154440*a^8*b^2*x + 3
003*a^10 + 32*(2145*a*b^9*x^4 + 20020*a^3*b^7*x^3 + 34398*a^5*b^5*x^2 + 13860*a^7*b^3*x + 1001*a^9*b)*sqrt(x))
/x^8

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Sympy [A]  time = 9.28918, size = 141, normalized size = 0.97 \begin{align*} - \frac{a^{10}}{8 x^{8}} - \frac{4 a^{9} b}{3 x^{\frac{15}{2}}} - \frac{45 a^{8} b^{2}}{7 x^{7}} - \frac{240 a^{7} b^{3}}{13 x^{\frac{13}{2}}} - \frac{35 a^{6} b^{4}}{x^{6}} - \frac{504 a^{5} b^{5}}{11 x^{\frac{11}{2}}} - \frac{42 a^{4} b^{6}}{x^{5}} - \frac{80 a^{3} b^{7}}{3 x^{\frac{9}{2}}} - \frac{45 a^{2} b^{8}}{4 x^{4}} - \frac{20 a b^{9}}{7 x^{\frac{7}{2}}} - \frac{b^{10}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**10/x**9,x)

[Out]

-a**10/(8*x**8) - 4*a**9*b/(3*x**(15/2)) - 45*a**8*b**2/(7*x**7) - 240*a**7*b**3/(13*x**(13/2)) - 35*a**6*b**4
/x**6 - 504*a**5*b**5/(11*x**(11/2)) - 42*a**4*b**6/x**5 - 80*a**3*b**7/(3*x**(9/2)) - 45*a**2*b**8/(4*x**4) -
20*a*b**9/(7*x**(7/2)) - b**10/(3*x**3)

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Giac [A]  time = 1.1106, size = 151, normalized size = 1.03 \begin{align*} -\frac{8008 \, b^{10} x^{5} + 68640 \, a b^{9} x^{\frac{9}{2}} + 270270 \, a^{2} b^{8} x^{4} + 640640 \, a^{3} b^{7} x^{\frac{7}{2}} + 1009008 \, a^{4} b^{6} x^{3} + 1100736 \, a^{5} b^{5} x^{\frac{5}{2}} + 840840 \, a^{6} b^{4} x^{2} + 443520 \, a^{7} b^{3} x^{\frac{3}{2}} + 154440 \, a^{8} b^{2} x + 32032 \, a^{9} b \sqrt{x} + 3003 \, a^{10}}{24024 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^9,x, algorithm="giac")

[Out]

-1/24024*(8008*b^10*x^5 + 68640*a*b^9*x^(9/2) + 270270*a^2*b^8*x^4 + 640640*a^3*b^7*x^(7/2) + 1009008*a^4*b^6*
x^3 + 1100736*a^5*b^5*x^(5/2) + 840840*a^6*b^4*x^2 + 443520*a^7*b^3*x^(3/2) + 154440*a^8*b^2*x + 32032*a^9*b*s
qrt(x) + 3003*a^10)/x^8