3.8.34 \(\int \frac {(a+c x^4)^3}{\sqrt {x}} \, dx\) [734]

Optimal. Leaf size=49 \[ 2 a^3 \sqrt {x}+\frac {2}{3} a^2 c x^{9/2}+\frac {6}{17} a c^2 x^{17/2}+\frac {2}{25} c^3 x^{25/2} \]

[Out]

2/3*a^2*c*x^(9/2)+6/17*a*c^2*x^(17/2)+2/25*c^3*x^(25/2)+2*a^3*x^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 49, 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^3 \sqrt {x}+\frac {2}{3} a^2 c x^{9/2}+\frac {6}{17} a c^2 x^{17/2}+\frac {2}{25} c^3 x^{25/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^3/Sqrt[x],x]

[Out]

2*a^3*Sqrt[x] + (2*a^2*c*x^(9/2))/3 + (6*a*c^2*x^(17/2))/17 + (2*c^3*x^(25/2))/25

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 )^3}{\sqrt {x}} \, dx &=\int \left (\frac {a^3}{\sqrt {x}}+3 a^2 c x^{7/2}+3 a c^2 x^{15/2}+c^3 x^{23/2}\right ) \, dx\\ &=2 a^3 \sqrt {x}+\frac {2}{3} a^2 c x^{9/2}+\frac {6}{17} a c^2 x^{17/2}+\frac {2}{25} c^3 x^{25/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.84 \begin {gather*} \frac {2 \sqrt {x} \left (1275 a^3+425 a^2 c x^4+225 a c^2 x^8+51 c^3 x^{12}\right )}{1275} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^3/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(1275*a^3 + 425*a^2*c*x^4 + 225*a*c^2*x^8 + 51*c^3*x^12))/1275

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Maple [A]
time = 0.13, size = 36, normalized size = 0.73

method	result	size
derivativedivides	\(\frac {2 a^{2} c \,x^{\frac {9}{2}}}{3}+\frac {6 a \,c^{2} x^{\frac {17}{2}}}{17}+\frac {2 c^{3} x^{\frac {25}{2}}}{25}+2 a^{3} \sqrt {x}\)	\(36\)
default	\(\frac {2 a^{2} c \,x^{\frac {9}{2}}}{3}+\frac {6 a \,c^{2} x^{\frac {17}{2}}}{17}+\frac {2 c^{3} x^{\frac {25}{2}}}{25}+2 a^{3} \sqrt {x}\)	\(36\)
trager	\(\left (\frac {2}{25} c^{3} x^{12}+\frac {6}{17} a \,c^{2} x^{8}+\frac {2}{3} a^{2} c \,x^{4}+2 a^{3}\right ) \sqrt {x}\)	\(37\)
gosper	\(\frac {2 \sqrt {x}\, \left (51 c^{3} x^{12}+225 a \,c^{2} x^{8}+425 a^{2} c \,x^{4}+1275 a^{3}\right )}{1275}\)	\(38\)
risch	\(\frac {2 \sqrt {x}\, \left (51 c^{3} x^{12}+225 a \,c^{2} x^{8}+425 a^{2} c \,x^{4}+1275 a^{3}\right )}{1275}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*a^2*c*x^(9/2)+6/17*a*c^2*x^(17/2)+2/25*c^3*x^(25/2)+2*a^3*x^(1/2)

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Maxima [A]
time = 0.29, size = 35, normalized size = 0.71 \begin {gather*} \frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {6}{17} \, a c^{2} x^{\frac {17}{2}} + \frac {2}{3} \, a^{2} c x^{\frac {9}{2}} + 2 \, a^{3} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/25*c^3*x^(25/2) + 6/17*a*c^2*x^(17/2) + 2/3*a^2*c*x^(9/2) + 2*a^3*sqrt(x)

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Fricas [A]
time = 0.36, size = 37, normalized size = 0.76 \begin {gather*} \frac {2}{1275} \, {\left (51 \, c^{3} x^{12} + 225 \, a c^{2} x^{8} + 425 \, a^{2} c x^{4} + 1275 \, a^{3}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1275*(51*c^3*x^12 + 225*a*c^2*x^8 + 425*a^2*c*x^4 + 1275*a^3)*sqrt(x)

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Sympy [A]
time = 1.15, size = 48, normalized size = 0.98 \begin {gather*} 2 a^{3} \sqrt {x} + \frac {2 a^{2} c x^{\frac {9}{2}}}{3} + \frac {6 a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 c^{3} x^{\frac {25}{2}}}{25} \end {gather*}

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*sqrt(x) + 2*a**2*c*x**(9/2)/3 + 6*a*c**2*x**(17/2)/17 + 2*c**3*x**(25/2)/25

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Giac [A]
time = 1.35, size = 35, normalized size = 0.71 \begin {gather*} \frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {6}{17} \, a c^{2} x^{\frac {17}{2}} + \frac {2}{3} \, a^{2} c x^{\frac {9}{2}} + 2 \, a^{3} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/25*c^3*x^(25/2) + 6/17*a*c^2*x^(17/2) + 2/3*a^2*c*x^(9/2) + 2*a^3*sqrt(x)

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Mupad [B]
time = 0.04, size = 35, normalized size = 0.71 \begin {gather*} 2\,a^3\,\sqrt {x}+\frac {2\,c^3\,x^{25/2}}{25}+\frac {2\,a^2\,c\,x^{9/2}}{3}+\frac {6\,a\,c^2\,x^{17/2}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^3/x^(1/2),x)

[Out]

2*a^3*x^(1/2) + (2*c^3*x^(25/2))/25 + (2*a^2*c*x^(9/2))/3 + (6*a*c^2*x^(17/2))/17

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