3.8.27 \(\int \frac {(a+c x^4)^2}{\sqrt {x}} \, dx\) [727]

Optimal. Leaf size=34 \[ 2 a^2 \sqrt {x}+\frac {4}{9} a c x^{9/2}+\frac {2}{17} c^2 x^{17/2} \]

[Out]

4/9*a*c*x^(9/2)+2/17*c^2*x^(17/2)+2*a^2*x^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 34, 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*} 2 a^2 \sqrt {x}+\frac {4}{9} a c x^{9/2}+\frac {2}{17} c^2 x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^2/Sqrt[x],x]

[Out]

2*a^2*Sqrt[x] + (4*a*c*x^(9/2))/9 + (2*c^2*x^(17/2))/17

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 \frac {\left (a+c x^4\right )^2}{\sqrt {x}} \, dx &=\int \left (\frac {a^2}{\sqrt {x}}+2 a c x^{7/2}+c^2 x^{15/2}\right ) \, dx\\ &=2 a^2 \sqrt {x}+\frac {4}{9} a c x^{9/2}+\frac {2}{17} c^2 x^{17/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.88 \begin {gather*} \frac {2}{153} \sqrt {x} \left (153 a^2+34 a c x^4+9 c^2 x^8\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^2/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(153*a^2 + 34*a*c*x^4 + 9*c^2*x^8))/153

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Maple [A]
time = 0.13, size = 25, normalized size = 0.74

method	result	size
derivativedivides	\(\frac {4 a c \,x^{\frac {9}{2}}}{9}+\frac {2 c^{2} x^{\frac {17}{2}}}{17}+2 a^{2} \sqrt {x}\)	\(25\)
default	\(\frac {4 a c \,x^{\frac {9}{2}}}{9}+\frac {2 c^{2} x^{\frac {17}{2}}}{17}+2 a^{2} \sqrt {x}\)	\(25\)
trager	\(\left (\frac {2}{17} c^{2} x^{8}+\frac {4}{9} a c \,x^{4}+2 a^{2}\right ) \sqrt {x}\)	\(26\)
gosper	\(\frac {2 \sqrt {x}\, \left (9 c^{2} x^{8}+34 a c \,x^{4}+153 a^{2}\right )}{153}\)	\(27\)
risch	\(\frac {2 \sqrt {x}\, \left (9 c^{2} x^{8}+34 a c \,x^{4}+153 a^{2}\right )}{153}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/9*a*c*x^(9/2)+2/17*c^2*x^(17/2)+2*a^2*x^(1/2)

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Maxima [A]
time = 0.31, size = 24, normalized size = 0.71 \begin {gather*} \frac {2}{17} \, c^{2} x^{\frac {17}{2}} + \frac {4}{9} \, a c x^{\frac {9}{2}} + 2 \, a^{2} \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="maxima")

[Out]

2/17*c^2*x^(17/2) + 4/9*a*c*x^(9/2) + 2*a^2*sqrt(x)

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Fricas [A]
time = 0.36, size = 26, normalized size = 0.76 \begin {gather*} \frac {2}{153} \, {\left (9 \, c^{2} x^{8} + 34 \, a c x^{4} + 153 \, a^{2}\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]

2/153*(9*c^2*x^8 + 34*a*c*x^4 + 153*a^2)*sqrt(x)

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Sympy [A]
time = 0.46, size = 32, normalized size = 0.94 \begin {gather*} 2 a^{2} \sqrt {x} + \frac {4 a c x^{\frac {9}{2}}}{9} + \frac {2 c^{2} x^{\frac {17}{2}}}{17} \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*sqrt(x) + 4*a*c*x**(9/2)/9 + 2*c**2*x**(17/2)/17

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Giac [A]
time = 0.53, size = 24, normalized size = 0.71 \begin {gather*} \frac {2}{17} \, c^{2} x^{\frac {17}{2}} + \frac {4}{9} \, a c x^{\frac {9}{2}} + 2 \, a^{2} \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="giac")

[Out]

2/17*c^2*x^(17/2) + 4/9*a*c*x^(9/2) + 2*a^2*sqrt(x)

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Mupad [B]
time = 0.04, size = 26, normalized size = 0.76 \begin {gather*} \frac {2\,\sqrt {x}\,\left (153\,a^2+34\,a\,c\,x^4+9\,c^2\,x^8\right )}{153} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^2/x^(1/2),x)

[Out]

(2*x^(1/2)*(153*a^2 + 9*c^2*x^8 + 34*a*c*x^4))/153

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