3.2157 \(\int (a+b \sqrt{x})^{10} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b \sqrt{x}\right )^{12}}{6 b^2}-\frac{2 a \left (a+b \sqrt{x}\right )^{11}}{11 b^2} \]

[Out]

(-2*a*(a + b*Sqrt[x])^11)/(11*b^2) + (a + b*Sqrt[x])^12/(6*b^2)

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Rubi [A]  time = 0.0146011, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{\left (a+b \sqrt{x}\right )^{12}}{6 b^2}-\frac{2 a \left (a+b \sqrt{x}\right )^{11}}{11 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10,x]

[Out]

(-2*a*(a + b*Sqrt[x])^11)/(11*b^2) + (a + b*Sqrt[x])^12/(6*b^2)

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[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 \left (a+b \sqrt{x}\right )^{10} \, dx &=2 \operatorname{Subst}\left (\int x (a+b x)^{10} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{10}}{b}+\frac{(a+b x)^{11}}{b}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 a \left (a+b \sqrt{x}\right )^{11}}{11 b^2}+\frac{\left (a+b \sqrt{x}\right )^{12}}{6 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0267577, size = 28, normalized size = 0.74 \[ -\frac{\left (a-11 b \sqrt{x}\right ) \left (a+b \sqrt{x}\right )^{11}}{66 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10,x]

[Out]

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

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Maple [B]  time = 0.002, size = 110, normalized size = 2.9 \begin{align*}{\frac{{x}^{6}{b}^{10}}{6}}+{\frac{20\,a{b}^{9}}{11}{x}^{{\frac{11}{2}}}}+9\,{x}^{5}{a}^{2}{b}^{8}+{\frac{80\,{a}^{3}{b}^{7}}{3}{x}^{{\frac{9}{2}}}}+{\frac{105\,{x}^{4}{a}^{4}{b}^{6}}{2}}+72\,{x}^{7/2}{a}^{5}{b}^{5}+70\,{x}^{3}{a}^{6}{b}^{4}+48\,{x}^{5/2}{a}^{7}{b}^{3}+{\frac{45\,{x}^{2}{a}^{8}{b}^{2}}{2}}+{\frac{20\,{a}^{9}b}{3}{x}^{{\frac{3}{2}}}}+x{a}^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.981726, size = 41, normalized size = 1.08 \begin{align*} \frac{{\left (b \sqrt{x} + a\right )}^{12}}{6 \, b^{2}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11} a}{11 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*sqrt(x) + a)^12/b^2 - 2/11*(b*sqrt(x) + a)^11*a/b^2

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Fricas [B]  time = 1.52648, size = 259, normalized size = 6.82 \begin{align*} \frac{1}{6} \, b^{10} x^{6} + 9 \, a^{2} b^{8} x^{5} + \frac{105}{2} \, a^{4} b^{6} x^{4} + 70 \, a^{6} b^{4} x^{3} + \frac{45}{2} \, a^{8} b^{2} x^{2} + a^{10} x + \frac{4}{33} \,{\left (15 \, a b^{9} x^{5} + 220 \, a^{3} b^{7} x^{4} + 594 \, a^{5} b^{5} x^{3} + 396 \, a^{7} b^{3} x^{2} + 55 \, a^{9} b x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^10*x^6 + 9*a^2*b^8*x^5 + 105/2*a^4*b^6*x^4 + 70*a^6*b^4*x^3 + 45/2*a^8*b^2*x^2 + a^10*x + 4/33*(15*a*b^9
*x^5 + 220*a^3*b^7*x^4 + 594*a^5*b^5*x^3 + 396*a^7*b^3*x^2 + 55*a^9*b*x)*sqrt(x)

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Sympy [B]  time = 1.52946, size = 133, normalized size = 3.5 \begin{align*} a^{10} x + \frac{20 a^{9} b x^{\frac{3}{2}}}{3} + \frac{45 a^{8} b^{2} x^{2}}{2} + 48 a^{7} b^{3} x^{\frac{5}{2}} + 70 a^{6} b^{4} x^{3} + 72 a^{5} b^{5} x^{\frac{7}{2}} + \frac{105 a^{4} b^{6} x^{4}}{2} + \frac{80 a^{3} b^{7} x^{\frac{9}{2}}}{3} + 9 a^{2} b^{8} x^{5} + \frac{20 a b^{9} x^{\frac{11}{2}}}{11} + \frac{b^{10} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.09573, size = 147, normalized size = 3.87 \begin{align*} \frac{1}{6} \, b^{10} x^{6} + \frac{20}{11} \, a b^{9} x^{\frac{11}{2}} + 9 \, a^{2} b^{8} x^{5} + \frac{80}{3} \, a^{3} b^{7} x^{\frac{9}{2}} + \frac{105}{2} \, a^{4} b^{6} x^{4} + 72 \, a^{5} b^{5} x^{\frac{7}{2}} + 70 \, a^{6} b^{4} x^{3} + 48 \, a^{7} b^{3} x^{\frac{5}{2}} + \frac{45}{2} \, a^{8} b^{2} x^{2} + \frac{20}{3} \, a^{9} b x^{\frac{3}{2}} + a^{10} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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