3.733 \(\int \sqrt{x} (a+c x^4)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{6}{11} a^2 c x^{11/2}+\frac{2}{3} a^3 x^{3/2}+\frac{6}{19} a c^2 x^{19/2}+\frac{2}{27} c^3 x^{27/2} \]

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*c*x^(11/2))/11 + (6*a*c^2*x^(19/2))/19 + (2*c^3*x^(27/2))/27

________________________________________________________________________________________

Rubi [A]  time = 0.0124144, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ \frac{6}{11} a^2 c x^{11/2}+\frac{2}{3} a^3 x^{3/2}+\frac{6}{19} a c^2 x^{19/2}+\frac{2}{27} c^3 x^{27/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + c*x^4)^3,x]

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*c*x^(11/2))/11 + (6*a*c^2*x^(19/2))/19 + (2*c^3*x^(27/2))/27

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \sqrt{x} \left (a+c x^4\right )^3 \, dx &=\int \left (a^3 \sqrt{x}+3 a^2 c x^{9/2}+3 a c^2 x^{17/2}+c^3 x^{25/2}\right ) \, dx\\ &=\frac{2}{3} a^3 x^{3/2}+\frac{6}{11} a^2 c x^{11/2}+\frac{6}{19} a c^2 x^{19/2}+\frac{2}{27} c^3 x^{27/2}\\ \end{align*}

Mathematica [A]  time = 0.0100755, size = 41, normalized size = 0.8 \[ \frac{2 x^{3/2} \left (1539 a^2 c x^4+1881 a^3+891 a c^2 x^8+209 c^3 x^{12}\right )}{5643} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + c*x^4)^3,x]

[Out]

(2*x^(3/2)*(1881*a^3 + 1539*a^2*c*x^4 + 891*a*c^2*x^8 + 209*c^3*x^12))/5643

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 38, normalized size = 0.8 \begin{align*}{\frac{418\,{c}^{3}{x}^{12}+1782\,a{c}^{2}{x}^{8}+3078\,{a}^{2}c{x}^{4}+3762\,{a}^{3}}{5643}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5643*x^(3/2)*(209*c^3*x^12+891*a*c^2*x^8+1539*a^2*c*x^4+1881*a^3)

________________________________________________________________________________________

Maxima [A]  time = 0.995213, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{27} \, c^{3} x^{\frac{27}{2}} + \frac{6}{19} \, a c^{2} x^{\frac{19}{2}} + \frac{6}{11} \, a^{2} c x^{\frac{11}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*c^3*x^(27/2) + 6/19*a*c^2*x^(19/2) + 6/11*a^2*c*x^(11/2) + 2/3*a^3*x^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.44339, size = 104, normalized size = 2.04 \begin{align*} \frac{2}{5643} \,{\left (209 \, c^{3} x^{13} + 891 \, a c^{2} x^{9} + 1539 \, a^{2} c x^{5} + 1881 \, a^{3} x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5643*(209*c^3*x^13 + 891*a*c^2*x^9 + 1539*a^2*c*x^5 + 1881*a^3*x)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 8.30902, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{3}{2}}}{3} + \frac{6 a^{2} c x^{\frac{11}{2}}}{11} + \frac{6 a c^{2} x^{\frac{19}{2}}}{19} + \frac{2 c^{3} x^{\frac{27}{2}}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**3*x**(3/2)/3 + 6*a**2*c*x**(11/2)/11 + 6*a*c**2*x**(19/2)/19 + 2*c**3*x**(27/2)/27

________________________________________________________________________________________

Giac [A]  time = 1.10533, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{27} \, c^{3} x^{\frac{27}{2}} + \frac{6}{19} \, a c^{2} x^{\frac{19}{2}} + \frac{6}{11} \, a^{2} c x^{\frac{11}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*c^3*x^(27/2) + 6/19*a*c^2*x^(19/2) + 6/11*a^2*c*x^(11/2) + 2/3*a^3*x^(3/2)