3.8.26 \(\int \sqrt {x} (a+c x^4)^2 \, dx\) [726]

Optimal. Leaf size=36 \[ \frac {2}{3} a^2 x^{3/2}+\frac {4}{11} a c x^{11/2}+\frac {2}{19} c^2 x^{19/2} \]

[Out]

2/3*a^2*x^(3/2)+4/11*a*c*x^(11/2)+2/19*c^2*x^(19/2)

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Rubi [A]
time = 0.01, antiderivative size = 36, 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 = {276} \begin {gather*} \frac {2}{3} a^2 x^{3/2}+\frac {4}{11} a c x^{11/2}+\frac {2}{19} c^2 x^{19/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + c*x^4)^2,x]

[Out]

(2*a^2*x^(3/2))/3 + (4*a*c*x^(11/2))/11 + (2*c^2*x^(19/2))/19

Rule 276

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

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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.83 \begin {gather*} \frac {2}{627} x^{3/2} \left (209 a^2+114 a c x^4+33 c^2 x^8\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + c*x^4)^2,x]

[Out]

(2*x^(3/2)*(209*a^2 + 114*a*c*x^4 + 33*c^2*x^8))/627

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Maple [A]
time = 0.13, size = 25, normalized size = 0.69

method	result	size
derivativedivides	\(\frac {2 a^{2} x^{\frac {3}{2}}}{3}+\frac {4 a c \,x^{\frac {11}{2}}}{11}+\frac {2 c^{2} x^{\frac {19}{2}}}{19}\)	\(25\)
default	\(\frac {2 a^{2} x^{\frac {3}{2}}}{3}+\frac {4 a c \,x^{\frac {11}{2}}}{11}+\frac {2 c^{2} x^{\frac {19}{2}}}{19}\)	\(25\)
gosper	\(\frac {2 x^{\frac {3}{2}} \left (33 c^{2} x^{8}+114 a c \,x^{4}+209 a^{2}\right )}{627}\)	\(27\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (33 c^{2} x^{8}+114 a c \,x^{4}+209 a^{2}\right )}{627}\)	\(27\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (33 c^{2} x^{8}+114 a c \,x^{4}+209 a^{2}\right )}{627}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^2*x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*a^2*x^(3/2)+4/11*a*c*x^(11/2)+2/19*c^2*x^(19/2)

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Maxima [A]
time = 0.29, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{19} \, c^{2} x^{\frac {19}{2}} + \frac {4}{11} \, a c x^{\frac {11}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*c^2*x^(19/2) + 4/11*a*c*x^(11/2) + 2/3*a^2*x^(3/2)

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Fricas [A]
time = 0.35, size = 27, normalized size = 0.75 \begin {gather*} \frac {2}{627} \, {\left (33 \, c^{2} x^{9} + 114 \, a c x^{5} + 209 \, a^{2} x\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/627*(33*c^2*x^9 + 114*a*c*x^5 + 209*a^2*x)*sqrt(x)

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Sympy [A]
time = 1.14, size = 34, normalized size = 0.94 \begin {gather*} \frac {2 a^{2} x^{\frac {3}{2}}}{3} + \frac {4 a c x^{\frac {11}{2}}}{11} + \frac {2 c^{2} x^{\frac {19}{2}}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*x**(3/2)/3 + 4*a*c*x**(11/2)/11 + 2*c**2*x**(19/2)/19

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Giac [A]
time = 0.50, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{19} \, c^{2} x^{\frac {19}{2}} + \frac {4}{11} \, a c x^{\frac {11}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*c^2*x^(19/2) + 4/11*a*c*x^(11/2) + 2/3*a^2*x^(3/2)

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Mupad [B]
time = 1.06, size = 26, normalized size = 0.72 \begin {gather*} \frac {2\,x^{3/2}\,\left (209\,a^2+114\,a\,c\,x^4+33\,c^2\,x^8\right )}{627} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + c*x^4)^2,x)

[Out]

(2*x^(3/2)*(209*a^2 + 33*c^2*x^8 + 114*a*c*x^4))/627

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