3.2164 \(\int \frac{(a+b \sqrt{x})^{10}}{x^7} \, dx\)

Optimal. Leaf size=46 \[ \frac{b \left (a+b \sqrt{x}\right )^{11}}{66 a^2 x^{11/2}}-\frac{\left (a+b \sqrt{x}\right )^{11}}{6 a x^6} \]

[Out]

-(a + b*Sqrt[x])^11/(6*a*x^6) + (b*(a + b*Sqrt[x])^11)/(66*a^2*x^(11/2))

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Rubi [A]  time = 0.013311, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ \frac{b \left (a+b \sqrt{x}\right )^{11}}{66 a^2 x^{11/2}}-\frac{\left (a+b \sqrt{x}\right )^{11}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10/x^7,x]

[Out]

-(a + b*Sqrt[x])^11/(6*a*x^6) + (b*(a + b*Sqrt[x])^11)/(66*a^2*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^7} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{13}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{6 a x^6}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{12}} \, dx,x,\sqrt{x}\right )}{6 a}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{11}}{6 a x^6}+\frac{b \left (a+b \sqrt{x}\right )^{11}}{66 a^2 x^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.0078622, size = 32, normalized size = 0.7 \[ \frac{\left (b \sqrt{x}-11 a\right ) \left (a+b \sqrt{x}\right )^{11}}{66 a^2 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10/x^7,x]

[Out]

((-11*a + b*Sqrt[x])*(a + b*Sqrt[x])^11)/(66*a^2*x^6)

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Maple [B]  time = 0.002, size = 113, normalized size = 2.5 \begin{align*} -{\frac{{b}^{10}}{x}}-{\frac{20\,a{b}^{9}}{3}{x}^{-{\frac{3}{2}}}}-{\frac{45\,{a}^{2}{b}^{8}}{2\,{x}^{2}}}-48\,{\frac{{a}^{3}{b}^{7}}{{x}^{5/2}}}-70\,{\frac{{a}^{4}{b}^{6}}{{x}^{3}}}-72\,{\frac{{a}^{5}{b}^{5}}{{x}^{7/2}}}-{\frac{105\,{a}^{6}{b}^{4}}{2\,{x}^{4}}}-{\frac{80\,{a}^{7}{b}^{3}}{3}{x}^{-{\frac{9}{2}}}}-9\,{\frac{{a}^{8}{b}^{2}}{{x}^{5}}}-{\frac{20\,{a}^{9}b}{11}{x}^{-{\frac{11}{2}}}}-{\frac{{a}^{10}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^10/x^7,x)

[Out]

-b^10/x-20/3*a*b^9/x^(3/2)-45/2*a^2*b^8/x^2-48*a^3*b^7/x^(5/2)-70*a^4*b^6/x^3-72*a^5*b^5/x^(7/2)-105/2*a^6*b^4
/x^4-80/3*a^7*b^3/x^(9/2)-9*a^8*b^2/x^5-20/11*a^9*b/x^(11/2)-1/6*a^10/x^6

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Maxima [B]  time = 0.984086, size = 151, normalized size = 3.28 \begin{align*} -\frac{66 \, b^{10} x^{5} + 440 \, a b^{9} x^{\frac{9}{2}} + 1485 \, a^{2} b^{8} x^{4} + 3168 \, a^{3} b^{7} x^{\frac{7}{2}} + 4620 \, a^{4} b^{6} x^{3} + 4752 \, a^{5} b^{5} x^{\frac{5}{2}} + 3465 \, a^{6} b^{4} x^{2} + 1760 \, a^{7} b^{3} x^{\frac{3}{2}} + 594 \, a^{8} b^{2} x + 120 \, a^{9} b \sqrt{x} + 11 \, a^{10}}{66 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^7,x, algorithm="maxima")

[Out]

-1/66*(66*b^10*x^5 + 440*a*b^9*x^(9/2) + 1485*a^2*b^8*x^4 + 3168*a^3*b^7*x^(7/2) + 4620*a^4*b^6*x^3 + 4752*a^5
*b^5*x^(5/2) + 3465*a^6*b^4*x^2 + 1760*a^7*b^3*x^(3/2) + 594*a^8*b^2*x + 120*a^9*b*sqrt(x) + 11*a^10)/x^6

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Fricas [B]  time = 1.32821, size = 267, normalized size = 5.8 \begin{align*} -\frac{66 \, b^{10} x^{5} + 1485 \, a^{2} b^{8} x^{4} + 4620 \, a^{4} b^{6} x^{3} + 3465 \, a^{6} b^{4} x^{2} + 594 \, a^{8} b^{2} x + 11 \, a^{10} + 8 \,{\left (55 \, a b^{9} x^{4} + 396 \, a^{3} b^{7} x^{3} + 594 \, a^{5} b^{5} x^{2} + 220 \, a^{7} b^{3} x + 15 \, a^{9} b\right )} \sqrt{x}}{66 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^7,x, algorithm="fricas")

[Out]

-1/66*(66*b^10*x^5 + 1485*a^2*b^8*x^4 + 4620*a^4*b^6*x^3 + 3465*a^6*b^4*x^2 + 594*a^8*b^2*x + 11*a^10 + 8*(55*
a*b^9*x^4 + 396*a^3*b^7*x^3 + 594*a^5*b^5*x^2 + 220*a^7*b^3*x + 15*a^9*b)*sqrt(x))/x^6

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Sympy [B]  time = 4.70582, size = 134, normalized size = 2.91 \begin{align*} - \frac{a^{10}}{6 x^{6}} - \frac{20 a^{9} b}{11 x^{\frac{11}{2}}} - \frac{9 a^{8} b^{2}}{x^{5}} - \frac{80 a^{7} b^{3}}{3 x^{\frac{9}{2}}} - \frac{105 a^{6} b^{4}}{2 x^{4}} - \frac{72 a^{5} b^{5}}{x^{\frac{7}{2}}} - \frac{70 a^{4} b^{6}}{x^{3}} - \frac{48 a^{3} b^{7}}{x^{\frac{5}{2}}} - \frac{45 a^{2} b^{8}}{2 x^{2}} - \frac{20 a b^{9}}{3 x^{\frac{3}{2}}} - \frac{b^{10}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**10/x**7,x)

[Out]

-a**10/(6*x**6) - 20*a**9*b/(11*x**(11/2)) - 9*a**8*b**2/x**5 - 80*a**7*b**3/(3*x**(9/2)) - 105*a**6*b**4/(2*x
**4) - 72*a**5*b**5/x**(7/2) - 70*a**4*b**6/x**3 - 48*a**3*b**7/x**(5/2) - 45*a**2*b**8/(2*x**2) - 20*a*b**9/(
3*x**(3/2)) - b**10/x

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Giac [B]  time = 1.10787, size = 151, normalized size = 3.28 \begin{align*} -\frac{66 \, b^{10} x^{5} + 440 \, a b^{9} x^{\frac{9}{2}} + 1485 \, a^{2} b^{8} x^{4} + 3168 \, a^{3} b^{7} x^{\frac{7}{2}} + 4620 \, a^{4} b^{6} x^{3} + 4752 \, a^{5} b^{5} x^{\frac{5}{2}} + 3465 \, a^{6} b^{4} x^{2} + 1760 \, a^{7} b^{3} x^{\frac{3}{2}} + 594 \, a^{8} b^{2} x + 120 \, a^{9} b \sqrt{x} + 11 \, a^{10}}{66 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^10/x^7,x, algorithm="giac")

[Out]

-1/66*(66*b^10*x^5 + 440*a*b^9*x^(9/2) + 1485*a^2*b^8*x^4 + 3168*a^3*b^7*x^(7/2) + 4620*a^4*b^6*x^3 + 4752*a^5
*b^5*x^(5/2) + 3465*a^6*b^4*x^2 + 1760*a^7*b^3*x^(3/2) + 594*a^8*b^2*x + 120*a^9*b*sqrt(x) + 11*a^10)/x^6