3.2130 \(\int (a+b \sqrt{x})^3 x^4 \, dx\)

Optimal. Leaf size=47 \[ \frac{6}{11} a^2 b x^{11/2}+\frac{a^3 x^5}{5}+\frac{1}{2} a b^2 x^6+\frac{2}{13} b^3 x^{13/2} \]

[Out]

(a^3*x^5)/5 + (6*a^2*b*x^(11/2))/11 + (a*b^2*x^6)/2 + (2*b^3*x^(13/2))/13

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Rubi [A]  time = 0.0301109, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{6}{11} a^2 b x^{11/2}+\frac{a^3 x^5}{5}+\frac{1}{2} a b^2 x^6+\frac{2}{13} b^3 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3*x^4,x]

[Out]

(a^3*x^5)/5 + (6*a^2*b*x^(11/2))/11 + (a*b^2*x^6)/2 + (2*b^3*x^(13/2))/13

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 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 )^3 x^4 \, dx &=2 \operatorname{Subst}\left (\int x^9 (a+b x)^3 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^3 x^9+3 a^2 b x^{10}+3 a b^2 x^{11}+b^3 x^{12}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^3 x^5}{5}+\frac{6}{11} a^2 b x^{11/2}+\frac{1}{2} a b^2 x^6+\frac{2}{13} b^3 x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0225418, size = 47, normalized size = 1. \[ \frac{6}{11} a^2 b x^{11/2}+\frac{a^3 x^5}{5}+\frac{1}{2} a b^2 x^6+\frac{2}{13} b^3 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3*x^4,x]

[Out]

(a^3*x^5)/5 + (6*a^2*b*x^(11/2))/11 + (a*b^2*x^6)/2 + (2*b^3*x^(13/2))/13

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{5}}{5}}+{\frac{6\,b{a}^{2}}{11}{x}^{{\frac{11}{2}}}}+{\frac{a{b}^{2}{x}^{6}}{2}}+{\frac{2\,{b}^{3}}{13}{x}^{{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*x^(1/2))^3,x)

[Out]

1/5*a^3*x^5+6/11*a^2*b*x^(11/2)+1/2*a*b^2*x^6+2/13*b^3*x^(13/2)

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Maxima [B]  time = 1.00509, size = 224, normalized size = 4.77 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{13}}{13 \, b^{10}} - \frac{3 \,{\left (b \sqrt{x} + a\right )}^{12} a}{2 \, b^{10}} + \frac{72 \,{\left (b \sqrt{x} + a\right )}^{11} a^{2}}{11 \, b^{10}} - \frac{84 \,{\left (b \sqrt{x} + a\right )}^{10} a^{3}}{5 \, b^{10}} + \frac{28 \,{\left (b \sqrt{x} + a\right )}^{9} a^{4}}{b^{10}} - \frac{63 \,{\left (b \sqrt{x} + a\right )}^{8} a^{5}}{2 \, b^{10}} + \frac{24 \,{\left (b \sqrt{x} + a\right )}^{7} a^{6}}{b^{10}} - \frac{12 \,{\left (b \sqrt{x} + a\right )}^{6} a^{7}}{b^{10}} + \frac{18 \,{\left (b \sqrt{x} + a\right )}^{5} a^{8}}{5 \, b^{10}} - \frac{{\left (b \sqrt{x} + a\right )}^{4} a^{9}}{2 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^3,x, algorithm="maxima")

[Out]

2/13*(b*sqrt(x) + a)^13/b^10 - 3/2*(b*sqrt(x) + a)^12*a/b^10 + 72/11*(b*sqrt(x) + a)^11*a^2/b^10 - 84/5*(b*sqr
t(x) + a)^10*a^3/b^10 + 28*(b*sqrt(x) + a)^9*a^4/b^10 - 63/2*(b*sqrt(x) + a)^8*a^5/b^10 + 24*(b*sqrt(x) + a)^7
*a^6/b^10 - 12*(b*sqrt(x) + a)^6*a^7/b^10 + 18/5*(b*sqrt(x) + a)^5*a^8/b^10 - 1/2*(b*sqrt(x) + a)^4*a^9/b^10

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Fricas [A]  time = 1.47283, size = 99, normalized size = 2.11 \begin{align*} \frac{1}{2} \, a b^{2} x^{6} + \frac{1}{5} \, a^{3} x^{5} + \frac{2}{143} \,{\left (11 \, b^{3} x^{6} + 39 \, a^{2} b x^{5}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^3,x, algorithm="fricas")

[Out]

1/2*a*b^2*x^6 + 1/5*a^3*x^5 + 2/143*(11*b^3*x^6 + 39*a^2*b*x^5)*sqrt(x)

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Sympy [A]  time = 1.78542, size = 42, normalized size = 0.89 \begin{align*} \frac{a^{3} x^{5}}{5} + \frac{6 a^{2} b x^{\frac{11}{2}}}{11} + \frac{a b^{2} x^{6}}{2} + \frac{2 b^{3} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*x**(1/2))**3,x)

[Out]

a**3*x**5/5 + 6*a**2*b*x**(11/2)/11 + a*b**2*x**6/2 + 2*b**3*x**(13/2)/13

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Giac [A]  time = 1.12035, size = 47, normalized size = 1. \begin{align*} \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{1}{2} \, a b^{2} x^{6} + \frac{6}{11} \, a^{2} b x^{\frac{11}{2}} + \frac{1}{5} \, a^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^3,x, algorithm="giac")

[Out]

2/13*b^3*x^(13/2) + 1/2*a*b^2*x^6 + 6/11*a^2*b*x^(11/2) + 1/5*a^3*x^5