3.2165 \(\int \frac{(a+b \sqrt{x})^{10}}{x^8} \, dx\)

Optimal. Leaf size=96 \[ \frac{b^3 \left (a+b \sqrt{x}\right )^{11}}{2002 a^4 x^{11/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{182 a^3 x^6}+\frac{3 b \left (a+b \sqrt{x}\right )^{11}}{91 a^2 x^{13/2}}-\frac{\left (a+b \sqrt{x}\right )^{11}}{7 a x^7} \]

[Out]

-(a + b*Sqrt[x])^11/(7*a*x^7) + (3*b*(a + b*Sqrt[x])^11)/(91*a^2*x^(13/2)) - (b^2*(a + b*Sqrt[x])^11)/(182*a^3
*x^6) + (b^3*(a + b*Sqrt[x])^11)/(2002*a^4*x^(11/2))

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Rubi [A]  time = 0.0341753, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ \frac{b^3 \left (a+b \sqrt{x}\right )^{11}}{2002 a^4 x^{11/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{182 a^3 x^6}+\frac{3 b \left (a+b \sqrt{x}\right )^{11}}{91 a^2 x^{13/2}}-\frac{\left (a+b \sqrt{x}\right )^{11}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10/x^8,x]

[Out]

-(a + b*Sqrt[x])^11/(7*a*x^7) + (3*b*(a + b*Sqrt[x])^11)/(91*a^2*x^(13/2)) - (b^2*(a + b*Sqrt[x])^11)/(182*a^3
*x^6) + (b^3*(a + b*Sqrt[x])^11)/(2002*a^4*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^8} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{15}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{7 a x^7}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{14}} \, dx,x,\sqrt{x}\right )}{7 a}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{7 a x^7}+\frac{3 b \left (a+b \sqrt{x}\right )^{11}}{91 a^2 x^{13/2}}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{13}} \, dx,x,\sqrt{x}\right )}{91 a^2}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{7 a x^7}+\frac{3 b \left (a+b \sqrt{x}\right )^{11}}{91 a^2 x^{13/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{182 a^3 x^6}-\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{12}} \, dx,x,\sqrt{x}\right )}{182 a^3}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{7 a x^7}+\frac{3 b \left (a+b \sqrt{x}\right )^{11}}{91 a^2 x^{13/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^{11}}{182 a^3 x^6}+\frac{b^3 \left (a+b \sqrt{x}\right )^{11}}{2002 a^4 x^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.0109926, size = 54, normalized size = 0.56 \[ \frac{\left (a+b \sqrt{x}\right )^{11} \left (66 a^2 b \sqrt{x}-286 a^3-11 a b^2 x+b^3 x^{3/2}\right )}{2002 a^4 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10/x^8,x]

[Out]

((a + b*Sqrt[x])^11*(-286*a^3 + 66*a^2*b*Sqrt[x] - 11*a*b^2*x + b^3*x^(3/2)))/(2002*a^4*x^7)

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Maple [A]  time = 0.003, size = 113, normalized size = 1.2 \begin{align*} -{\frac{{b}^{10}}{2\,{x}^{2}}}-4\,{\frac{a{b}^{9}}{{x}^{5/2}}}-15\,{\frac{{a}^{2}{b}^{8}}{{x}^{3}}}-{\frac{240\,{a}^{3}{b}^{7}}{7}{x}^{-{\frac{7}{2}}}}-{\frac{105\,{a}^{4}{b}^{6}}{2\,{x}^{4}}}-56\,{\frac{{a}^{5}{b}^{5}}{{x}^{9/2}}}-42\,{\frac{{a}^{6}{b}^{4}}{{x}^{5}}}-{\frac{240\,{a}^{7}{b}^{3}}{11}{x}^{-{\frac{11}{2}}}}-{\frac{15\,{a}^{8}{b}^{2}}{2\,{x}^{6}}}-{\frac{20\,{a}^{9}b}{13}{x}^{-{\frac{13}{2}}}}-{\frac{{a}^{10}}{7\,{x}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^10/x^8,x)

[Out]

-1/2*b^10/x^2-4*a*b^9/x^(5/2)-15*a^2*b^8/x^3-240/7*a^3*b^7/x^(7/2)-105/2*a^4*b^6/x^4-56*a^5*b^5/x^(9/2)-42*a^6
*b^4/x^5-240/11*a^7*b^3/x^(11/2)-15/2*a^8*b^2/x^6-20/13*a^9*b/x^(13/2)-1/7*a^10/x^7

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Maxima [A]  time = 0.961894, size = 151, normalized size = 1.57 \begin{align*} -\frac{1001 \, b^{10} x^{5} + 8008 \, a b^{9} x^{\frac{9}{2}} + 30030 \, a^{2} b^{8} x^{4} + 68640 \, a^{3} b^{7} x^{\frac{7}{2}} + 105105 \, a^{4} b^{6} x^{3} + 112112 \, a^{5} b^{5} x^{\frac{5}{2}} + 84084 \, a^{6} b^{4} x^{2} + 43680 \, a^{7} b^{3} x^{\frac{3}{2}} + 15015 \, a^{8} b^{2} x + 3080 \, a^{9} b \sqrt{x} + 286 \, a^{10}}{2002 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^8,x, algorithm="maxima")

[Out]

-1/2002*(1001*b^10*x^5 + 8008*a*b^9*x^(9/2) + 30030*a^2*b^8*x^4 + 68640*a^3*b^7*x^(7/2) + 105105*a^4*b^6*x^3 +
112112*a^5*b^5*x^(5/2) + 84084*a^6*b^4*x^2 + 43680*a^7*b^3*x^(3/2) + 15015*a^8*b^2*x + 3080*a^9*b*sqrt(x) + 2
86*a^10)/x^7

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Fricas [A]  time = 1.22923, size = 292, normalized size = 3.04 \begin{align*} -\frac{1001 \, b^{10} x^{5} + 30030 \, a^{2} b^{8} x^{4} + 105105 \, a^{4} b^{6} x^{3} + 84084 \, a^{6} b^{4} x^{2} + 15015 \, a^{8} b^{2} x + 286 \, a^{10} + 8 \,{\left (1001 \, a b^{9} x^{4} + 8580 \, a^{3} b^{7} x^{3} + 14014 \, a^{5} b^{5} x^{2} + 5460 \, a^{7} b^{3} x + 385 \, a^{9} b\right )} \sqrt{x}}{2002 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^8,x, algorithm="fricas")

[Out]

-1/2002*(1001*b^10*x^5 + 30030*a^2*b^8*x^4 + 105105*a^4*b^6*x^3 + 84084*a^6*b^4*x^2 + 15015*a^8*b^2*x + 286*a^
10 + 8*(1001*a*b^9*x^4 + 8580*a^3*b^7*x^3 + 14014*a^5*b^5*x^2 + 5460*a^7*b^3*x + 385*a^9*b)*sqrt(x))/x^7

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Sympy [A]  time = 6.68059, size = 138, normalized size = 1.44 \begin{align*} - \frac{a^{10}}{7 x^{7}} - \frac{20 a^{9} b}{13 x^{\frac{13}{2}}} - \frac{15 a^{8} b^{2}}{2 x^{6}} - \frac{240 a^{7} b^{3}}{11 x^{\frac{11}{2}}} - \frac{42 a^{6} b^{4}}{x^{5}} - \frac{56 a^{5} b^{5}}{x^{\frac{9}{2}}} - \frac{105 a^{4} b^{6}}{2 x^{4}} - \frac{240 a^{3} b^{7}}{7 x^{\frac{7}{2}}} - \frac{15 a^{2} b^{8}}{x^{3}} - \frac{4 a b^{9}}{x^{\frac{5}{2}}} - \frac{b^{10}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**10/x**8,x)

[Out]

-a**10/(7*x**7) - 20*a**9*b/(13*x**(13/2)) - 15*a**8*b**2/(2*x**6) - 240*a**7*b**3/(11*x**(11/2)) - 42*a**6*b*
*4/x**5 - 56*a**5*b**5/x**(9/2) - 105*a**4*b**6/(2*x**4) - 240*a**3*b**7/(7*x**(7/2)) - 15*a**2*b**8/x**3 - 4*
a*b**9/x**(5/2) - b**10/(2*x**2)

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Giac [A]  time = 1.11046, size = 151, normalized size = 1.57 \begin{align*} -\frac{1001 \, b^{10} x^{5} + 8008 \, a b^{9} x^{\frac{9}{2}} + 30030 \, a^{2} b^{8} x^{4} + 68640 \, a^{3} b^{7} x^{\frac{7}{2}} + 105105 \, a^{4} b^{6} x^{3} + 112112 \, a^{5} b^{5} x^{\frac{5}{2}} + 84084 \, a^{6} b^{4} x^{2} + 43680 \, a^{7} b^{3} x^{\frac{3}{2}} + 15015 \, a^{8} b^{2} x + 3080 \, a^{9} b \sqrt{x} + 286 \, a^{10}}{2002 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^8,x, algorithm="giac")

[Out]

-1/2002*(1001*b^10*x^5 + 8008*a*b^9*x^(9/2) + 30030*a^2*b^8*x^4 + 68640*a^3*b^7*x^(7/2) + 105105*a^4*b^6*x^3 +
112112*a^5*b^5*x^(5/2) + 84084*a^6*b^4*x^2 + 43680*a^7*b^3*x^(3/2) + 15015*a^8*b^2*x + 3080*a^9*b*sqrt(x) + 2
86*a^10)/x^7