3.2111 \(\int (a+b \sqrt{x}) x^4 \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^5}{5}+\frac{2}{11} b x^{11/2} \]

[Out]

(a*x^5)/5 + (2*b*x^(11/2))/11

________________________________________________________________________________________

Rubi [A]  time = 0.0053891, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{a x^5}{5}+\frac{2}{11} b x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])*x^4,x]

[Out]

(a*x^5)/5 + (2*b*x^(11/2))/11

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \left (a+b \sqrt{x}\right ) x^4 \, dx &=\int \left (a x^4+b x^{9/2}\right ) \, dx\\ &=\frac{a x^5}{5}+\frac{2}{11} b x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0069905, size = 19, normalized size = 1. \[ \frac{a x^5}{5}+\frac{2}{11} b x^{11/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])*x^4,x]

[Out]

(a*x^5)/5 + (2*b*x^(11/2))/11

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \begin{align*}{\frac{a{x}^{5}}{5}}+{\frac{2\,b}{11}{x}^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 0.960809, size = 224, normalized size = 11.79 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11}}{11 \, b^{10}} - \frac{9 \,{\left (b \sqrt{x} + a\right )}^{10} a}{5 \, b^{10}} + \frac{8 \,{\left (b \sqrt{x} + a\right )}^{9} a^{2}}{b^{10}} - \frac{21 \,{\left (b \sqrt{x} + a\right )}^{8} a^{3}}{b^{10}} + \frac{36 \,{\left (b \sqrt{x} + a\right )}^{7} a^{4}}{b^{10}} - \frac{42 \,{\left (b \sqrt{x} + a\right )}^{6} a^{5}}{b^{10}} + \frac{168 \,{\left (b \sqrt{x} + a\right )}^{5} a^{6}}{5 \, b^{10}} - \frac{18 \,{\left (b \sqrt{x} + a\right )}^{4} a^{7}}{b^{10}} + \frac{6 \,{\left (b \sqrt{x} + a\right )}^{3} a^{8}}{b^{10}} - \frac{{\left (b \sqrt{x} + a\right )}^{2} a^{9}}{b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*(b*sqrt(x) + a)^11/b^10 - 9/5*(b*sqrt(x) + a)^10*a/b^10 + 8*(b*sqrt(x) + a)^9*a^2/b^10 - 21*(b*sqrt(x) +
a)^8*a^3/b^10 + 36*(b*sqrt(x) + a)^7*a^4/b^10 - 42*(b*sqrt(x) + a)^6*a^5/b^10 + 168/5*(b*sqrt(x) + a)^5*a^6/b^
10 - 18*(b*sqrt(x) + a)^4*a^7/b^10 + 6*(b*sqrt(x) + a)^3*a^8/b^10 - (b*sqrt(x) + a)^2*a^9/b^10

________________________________________________________________________________________

Fricas [A]  time = 1.4452, size = 39, normalized size = 2.05 \begin{align*} \frac{2}{11} \, b x^{\frac{11}{2}} + \frac{1}{5} \, a x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*b*x^(11/2) + 1/5*a*x^5

________________________________________________________________________________________

Sympy [A]  time = 1.21596, size = 15, normalized size = 0.79 \begin{align*} \frac{a x^{5}}{5} + \frac{2 b x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**5/5 + 2*b*x**(11/2)/11

________________________________________________________________________________________

Giac [A]  time = 1.12158, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{11} \, b x^{\frac{11}{2}} + \frac{1}{5} \, a x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*b*x^(11/2) + 1/5*a*x^5