3.2145 \(\int (a+b \sqrt{x})^5 \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \left (a+b \sqrt{x}\right )^7}{7 b^2}-\frac{a \left (a+b \sqrt{x}\right )^6}{3 b^2} \]

[Out]

-(a*(a + b*Sqrt[x])^6)/(3*b^2) + (2*(a + b*Sqrt[x])^7)/(7*b^2)

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Rubi [A]  time = 0.0133622, 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} \[ \frac{2 \left (a+b \sqrt{x}\right )^7}{7 b^2}-\frac{a \left (a+b \sqrt{x}\right )^6}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5,x]

[Out]

-(a*(a + b*Sqrt[x])^6)/(3*b^2) + (2*(a + b*Sqrt[x])^7)/(7*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 )^5 \, dx &=2 \operatorname{Subst}\left (\int x (a+b x)^5 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^5}{b}+\frac{(a+b x)^6}{b}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a \left (a+b \sqrt{x}\right )^6}{3 b^2}+\frac{2 \left (a+b \sqrt{x}\right )^7}{7 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0197621, size = 28, normalized size = 0.74 \[ -\frac{\left (a-6 b \sqrt{x}\right ) \left (a+b \sqrt{x}\right )^6}{21 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5,x]

[Out]

-((a - 6*b*Sqrt[x])*(a + b*Sqrt[x])^6)/(21*b^2)

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Maple [A]  time = 0.002, size = 55, normalized size = 1.5 \begin{align*}{\frac{2\,{b}^{5}}{7}{x}^{{\frac{7}{2}}}}+{\frac{5\,a{b}^{4}{x}^{3}}{3}}+4\,{x}^{5/2}{a}^{2}{b}^{3}+5\,{a}^{3}{b}^{2}{x}^{2}+{\frac{10\,{a}^{4}b}{3}{x}^{{\frac{3}{2}}}}+x{a}^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^5,x)

[Out]

2/7*x^(7/2)*b^5+5/3*a*b^4*x^3+4*x^(5/2)*a^2*b^3+5*a^3*b^2*x^2+10/3*x^(3/2)*a^4*b+x*a^5

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Maxima [A]  time = 0.94596, size = 73, normalized size = 1.92 \begin{align*} \frac{2}{7} \, b^{5} x^{\frac{7}{2}} + \frac{5}{3} \, a b^{4} x^{3} + 4 \, a^{2} b^{3} x^{\frac{5}{2}} + 5 \, a^{3} b^{2} x^{2} + \frac{10}{3} \, a^{4} b x^{\frac{3}{2}} + a^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

2/7*b^5*x^(7/2) + 5/3*a*b^4*x^3 + 4*a^2*b^3*x^(5/2) + 5*a^3*b^2*x^2 + 10/3*a^4*b*x^(3/2) + a^5*x

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Fricas [A]  time = 1.44519, size = 130, normalized size = 3.42 \begin{align*} \frac{5}{3} \, a b^{4} x^{3} + 5 \, a^{3} b^{2} x^{2} + a^{5} x + \frac{2}{21} \,{\left (3 \, b^{5} x^{3} + 42 \, a^{2} b^{3} x^{2} + 35 \, a^{4} b x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

5/3*a*b^4*x^3 + 5*a^3*b^2*x^2 + a^5*x + 2/21*(3*b^5*x^3 + 42*a^2*b^3*x^2 + 35*a^4*b*x)*sqrt(x)

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Sympy [B]  time = 2.1036, size = 66, normalized size = 1.74 \begin{align*} a^{5} x + \frac{10 a^{4} b x^{\frac{3}{2}}}{3} + 5 a^{3} b^{2} x^{2} + 4 a^{2} b^{3} x^{\frac{5}{2}} + \frac{5 a b^{4} x^{3}}{3} + \frac{2 b^{5} x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**5,x)

[Out]

a**5*x + 10*a**4*b*x**(3/2)/3 + 5*a**3*b**2*x**2 + 4*a**2*b**3*x**(5/2) + 5*a*b**4*x**3/3 + 2*b**5*x**(7/2)/7

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Giac [A]  time = 1.11843, size = 73, normalized size = 1.92 \begin{align*} \frac{2}{7} \, b^{5} x^{\frac{7}{2}} + \frac{5}{3} \, a b^{4} x^{3} + 4 \, a^{2} b^{3} x^{\frac{5}{2}} + 5 \, a^{3} b^{2} x^{2} + \frac{10}{3} \, a^{4} b x^{\frac{3}{2}} + a^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

2/7*b^5*x^(7/2) + 5/3*a*b^4*x^3 + 4*a^2*b^3*x^(5/2) + 5*a^3*b^2*x^2 + 10/3*a^4*b*x^(3/2) + a^5*x