3.2141 \(\int (a+b \sqrt{x})^5 x^4 \, dx\)

Optimal. Leaf size=75 \[ \frac{20}{13} a^2 b^3 x^{13/2}+\frac{5}{3} a^3 b^2 x^6+\frac{10}{11} a^4 b x^{11/2}+\frac{a^5 x^5}{5}+\frac{5}{7} a b^4 x^7+\frac{2}{15} b^5 x^{15/2} \]

[Out]

(a^5*x^5)/5 + (10*a^4*b*x^(11/2))/11 + (5*a^3*b^2*x^6)/3 + (20*a^2*b^3*x^(13/2))/13 + (5*a*b^4*x^7)/7 + (2*b^5
*x^(15/2))/15

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Rubi [A]  time = 0.0453066, antiderivative size = 75, 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} \[ \frac{20}{13} a^2 b^3 x^{13/2}+\frac{5}{3} a^3 b^2 x^6+\frac{10}{11} a^4 b x^{11/2}+\frac{a^5 x^5}{5}+\frac{5}{7} a b^4 x^7+\frac{2}{15} b^5 x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5*x^4,x]

[Out]

(a^5*x^5)/5 + (10*a^4*b*x^(11/2))/11 + (5*a^3*b^2*x^6)/3 + (20*a^2*b^3*x^(13/2))/13 + (5*a*b^4*x^7)/7 + (2*b^5
*x^(15/2))/15

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 )^5 x^4 \, dx &=2 \operatorname{Subst}\left (\int x^9 (a+b x)^5 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^5 x^9+5 a^4 b x^{10}+10 a^3 b^2 x^{11}+10 a^2 b^3 x^{12}+5 a b^4 x^{13}+b^5 x^{14}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^5 x^5}{5}+\frac{10}{11} a^4 b x^{11/2}+\frac{5}{3} a^3 b^2 x^6+\frac{20}{13} a^2 b^3 x^{13/2}+\frac{5}{7} a b^4 x^7+\frac{2}{15} b^5 x^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0332377, size = 75, normalized size = 1. \[ \frac{20}{13} a^2 b^3 x^{13/2}+\frac{5}{3} a^3 b^2 x^6+\frac{10}{11} a^4 b x^{11/2}+\frac{a^5 x^5}{5}+\frac{5}{7} a b^4 x^7+\frac{2}{15} b^5 x^{15/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5*x^4,x]

[Out]

(a^5*x^5)/5 + (10*a^4*b*x^(11/2))/11 + (5*a^3*b^2*x^6)/3 + (20*a^2*b^3*x^(13/2))/13 + (5*a*b^4*x^7)/7 + (2*b^5
*x^(15/2))/15

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Maple [A]  time = 0.003, size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{5}}{5}}+{\frac{10\,{a}^{4}b}{11}{x}^{{\frac{11}{2}}}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{6}}{3}}+{\frac{20\,{a}^{2}{b}^{3}}{13}{x}^{{\frac{13}{2}}}}+{\frac{5\,a{b}^{4}{x}^{7}}{7}}+{\frac{2\,{b}^{5}}{15}{x}^{{\frac{15}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*x^(1/2))^5,x)

[Out]

1/5*a^5*x^5+10/11*a^4*b*x^(11/2)+5/3*a^3*b^2*x^6+20/13*a^2*b^3*x^(13/2)+5/7*a*b^4*x^7+2/15*b^5*x^(15/2)

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Maxima [B]  time = 1.01085, size = 224, normalized size = 2.99 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{15}}{15 \, b^{10}} - \frac{9 \,{\left (b \sqrt{x} + a\right )}^{14} a}{7 \, b^{10}} + \frac{72 \,{\left (b \sqrt{x} + a\right )}^{13} a^{2}}{13 \, b^{10}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{12} a^{3}}{b^{10}} + \frac{252 \,{\left (b \sqrt{x} + a\right )}^{11} a^{4}}{11 \, b^{10}} - \frac{126 \,{\left (b \sqrt{x} + a\right )}^{10} a^{5}}{5 \, b^{10}} + \frac{56 \,{\left (b \sqrt{x} + a\right )}^{9} a^{6}}{3 \, b^{10}} - \frac{9 \,{\left (b \sqrt{x} + a\right )}^{8} a^{7}}{b^{10}} + \frac{18 \,{\left (b \sqrt{x} + a\right )}^{7} a^{8}}{7 \, b^{10}} - \frac{{\left (b \sqrt{x} + a\right )}^{6} a^{9}}{3 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

2/15*(b*sqrt(x) + a)^15/b^10 - 9/7*(b*sqrt(x) + a)^14*a/b^10 + 72/13*(b*sqrt(x) + a)^13*a^2/b^10 - 14*(b*sqrt(
x) + a)^12*a^3/b^10 + 252/11*(b*sqrt(x) + a)^11*a^4/b^10 - 126/5*(b*sqrt(x) + a)^10*a^5/b^10 + 56/3*(b*sqrt(x)
+ a)^9*a^6/b^10 - 9*(b*sqrt(x) + a)^8*a^7/b^10 + 18/7*(b*sqrt(x) + a)^7*a^8/b^10 - 1/3*(b*sqrt(x) + a)^6*a^9/
b^10

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Fricas [A]  time = 1.42887, size = 153, normalized size = 2.04 \begin{align*} \frac{5}{7} \, a b^{4} x^{7} + \frac{5}{3} \, a^{3} b^{2} x^{6} + \frac{1}{5} \, a^{5} x^{5} + \frac{2}{2145} \,{\left (143 \, b^{5} x^{7} + 1650 \, a^{2} b^{3} x^{6} + 975 \, a^{4} 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))^5,x, algorithm="fricas")

[Out]

5/7*a*b^4*x^7 + 5/3*a^3*b^2*x^6 + 1/5*a^5*x^5 + 2/2145*(143*b^5*x^7 + 1650*a^2*b^3*x^6 + 975*a^4*b*x^5)*sqrt(x
)

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Sympy [A]  time = 3.41645, size = 73, normalized size = 0.97 \begin{align*} \frac{a^{5} x^{5}}{5} + \frac{10 a^{4} b x^{\frac{11}{2}}}{11} + \frac{5 a^{3} b^{2} x^{6}}{3} + \frac{20 a^{2} b^{3} x^{\frac{13}{2}}}{13} + \frac{5 a b^{4} x^{7}}{7} + \frac{2 b^{5} x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*x**(1/2))**5,x)

[Out]

a**5*x**5/5 + 10*a**4*b*x**(11/2)/11 + 5*a**3*b**2*x**6/3 + 20*a**2*b**3*x**(13/2)/13 + 5*a*b**4*x**7/7 + 2*b*
*5*x**(15/2)/15

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Giac [A]  time = 1.09436, size = 77, normalized size = 1.03 \begin{align*} \frac{2}{15} \, b^{5} x^{\frac{15}{2}} + \frac{5}{7} \, a b^{4} x^{7} + \frac{20}{13} \, a^{2} b^{3} x^{\frac{13}{2}} + \frac{5}{3} \, a^{3} b^{2} x^{6} + \frac{10}{11} \, a^{4} b x^{\frac{11}{2}} + \frac{1}{5} \, a^{5} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

2/15*b^5*x^(15/2) + 5/7*a*b^4*x^7 + 20/13*a^2*b^3*x^(13/2) + 5/3*a^3*b^2*x^6 + 10/11*a^4*b*x^(11/2) + 1/5*a^5*
x^5