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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 0.80 \[ \frac {2 x^{3/2} \left (1881 a^3+1539 a^2 c x^4+891 a c^2 x^8+209 c^3 x^{12}\right )}{5643} \]

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

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fricas [A]  time = 0.69, size = 38, normalized size = 0.75 \[ \frac {2}{5643} \, {\left (209 \, c^{3} x^{13} + 891 \, a c^{2} x^{9} + 1539 \, a^{2} c x^{5} + 1881 \, a^{3} x\right )} \sqrt {x} \]

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)

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giac [A]  time = 0.17, size = 35, normalized size = 0.69 \[ \frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{19} \, a c^{2} x^{\frac {19}{2}} + \frac {6}{11} \, a^{2} c x^{\frac {11}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \]

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)

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maple [A]  time = 0.01, size = 38, normalized size = 0.75 \[ \frac {2 \left (209 c^{3} x^{12}+891 a \,c^{2} x^{8}+1539 a^{2} c \,x^{4}+1881 a^{3}\right ) x^{\frac {3}{2}}}{5643} \]

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)

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maxima [A]  time = 1.31, size = 35, normalized size = 0.69 \[ \frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{19} \, a c^{2} x^{\frac {19}{2}} + \frac {6}{11} \, a^{2} c x^{\frac {11}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \]

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)

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mupad [B]  time = 0.04, size = 35, normalized size = 0.69 \[ \frac {2\,a^3\,x^{3/2}}{3}+\frac {2\,c^3\,x^{27/2}}{27}+\frac {6\,a^2\,c\,x^{11/2}}{11}+\frac {6\,a\,c^2\,x^{19/2}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + c*x^4)^3,x)

[Out]

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

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sympy [A]  time = 5.95, size = 49, normalized size = 0.96 \[ \frac {2 a^{3} x^{\frac {3}{2}}}{3} + \frac {6 a^{2} c x^{\frac {11}{2}}}{11} + \frac {6 a c^{2} x^{\frac {19}{2}}}{19} + \frac {2 c^{3} x^{\frac {27}{2}}}{27} \]

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

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