3.8.36 \(\int \frac {(a+c x^4)^3}{x^{5/2}} \, dx\) [736]

Optimal. Leaf size=51 \[ -\frac {2 a^3}{3 x^{3/2}}+\frac {6}{5} a^2 c x^{5/2}+\frac {6}{13} a c^2 x^{13/2}+\frac {2}{21} c^3 x^{21/2} \]

[Out]

-2/3*a^3/x^(3/2)+6/5*a^2*c*x^(5/2)+6/13*a*c^2*x^(13/2)+2/21*c^3*x^(21/2)

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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 = {276} \begin {gather*} -\frac {2 a^3}{3 x^{3/2}}+\frac {6}{5} a^2 c x^{5/2}+\frac {6}{13} a c^2 x^{13/2}+\frac {2}{21} c^3 x^{21/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^3/x^(5/2),x]

[Out]

(-2*a^3)/(3*x^(3/2)) + (6*a^2*c*x^(5/2))/5 + (6*a*c^2*x^(13/2))/13 + (2*c^3*x^(21/2))/21

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}{x^{5/2}} \, dx &=\int \left (\frac {a^3}{x^{5/2}}+3 a^2 c x^{3/2}+3 a c^2 x^{11/2}+c^3 x^{19/2}\right ) \, dx\\ &=-\frac {2 a^3}{3 x^{3/2}}+\frac {6}{5} a^2 c x^{5/2}+\frac {6}{13} a c^2 x^{13/2}+\frac {2}{21} c^3 x^{21/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.80 \begin {gather*} -\frac {2 \left (455 a^3-819 a^2 c x^4-315 a c^2 x^8-65 c^3 x^{12}\right )}{1365 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^3/x^(5/2),x]

[Out]

(-2*(455*a^3 - 819*a^2*c*x^4 - 315*a*c^2*x^8 - 65*c^3*x^12))/(1365*x^(3/2))

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Maple [A]
time = 0.13, size = 36, normalized size = 0.71

method	result	size
derivativedivides	\(-\frac {2 a^{3}}{3 x^{\frac {3}{2}}}+\frac {6 a^{2} c \,x^{\frac {5}{2}}}{5}+\frac {6 a \,c^{2} x^{\frac {13}{2}}}{13}+\frac {2 c^{3} x^{\frac {21}{2}}}{21}\)	\(36\)
default	\(-\frac {2 a^{3}}{3 x^{\frac {3}{2}}}+\frac {6 a^{2} c \,x^{\frac {5}{2}}}{5}+\frac {6 a \,c^{2} x^{\frac {13}{2}}}{13}+\frac {2 c^{3} x^{\frac {21}{2}}}{21}\)	\(36\)
gosper	\(-\frac {2 \left (-65 c^{3} x^{12}-315 a \,c^{2} x^{8}-819 a^{2} c \,x^{4}+455 a^{3}\right )}{1365 x^{\frac {3}{2}}}\)	\(38\)
trager	\(-\frac {2 \left (-65 c^{3} x^{12}-315 a \,c^{2} x^{8}-819 a^{2} c \,x^{4}+455 a^{3}\right )}{1365 x^{\frac {3}{2}}}\)	\(38\)
risch	\(-\frac {2 \left (-65 c^{3} x^{12}-315 a \,c^{2} x^{8}-819 a^{2} c \,x^{4}+455 a^{3}\right )}{1365 x^{\frac {3}{2}}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^3/x^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*a^3/x^(3/2)+6/5*a^2*c*x^(5/2)+6/13*a*c^2*x^(13/2)+2/21*c^3*x^(21/2)

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Maxima [A]
time = 0.29, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{21} \, c^{3} x^{\frac {21}{2}} + \frac {6}{13} \, a c^{2} x^{\frac {13}{2}} + \frac {6}{5} \, a^{2} c x^{\frac {5}{2}} - \frac {2 \, a^{3}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^3/x^(5/2),x, algorithm="maxima")

[Out]

2/21*c^3*x^(21/2) + 6/13*a*c^2*x^(13/2) + 6/5*a^2*c*x^(5/2) - 2/3*a^3/x^(3/2)

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Fricas [A]
time = 0.38, size = 37, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (65 \, c^{3} x^{12} + 315 \, a c^{2} x^{8} + 819 \, a^{2} c x^{4} - 455 \, a^{3}\right )}}{1365 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^3/x^(5/2),x, algorithm="fricas")

[Out]

2/1365*(65*c^3*x^12 + 315*a*c^2*x^8 + 819*a^2*c*x^4 - 455*a^3)/x^(3/2)

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Sympy [A]
time = 1.57, size = 49, normalized size = 0.96 \begin {gather*} - \frac {2 a^{3}}{3 x^{\frac {3}{2}}} + \frac {6 a^{2} c x^{\frac {5}{2}}}{5} + \frac {6 a c^{2} x^{\frac {13}{2}}}{13} + \frac {2 c^{3} x^{\frac {21}{2}}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**3/x**(5/2),x)

[Out]

-2*a**3/(3*x**(3/2)) + 6*a**2*c*x**(5/2)/5 + 6*a*c**2*x**(13/2)/13 + 2*c**3*x**(21/2)/21

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Giac [A]
time = 1.12, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{21} \, c^{3} x^{\frac {21}{2}} + \frac {6}{13} \, a c^{2} x^{\frac {13}{2}} + \frac {6}{5} \, a^{2} c x^{\frac {5}{2}} - \frac {2 \, a^{3}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^3/x^(5/2),x, algorithm="giac")

[Out]

2/21*c^3*x^(21/2) + 6/13*a*c^2*x^(13/2) + 6/5*a^2*c*x^(5/2) - 2/3*a^3/x^(3/2)

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Mupad [B]
time = 0.04, size = 35, normalized size = 0.69 \begin {gather*} \frac {2\,c^3\,x^{21/2}}{21}-\frac {2\,a^3}{3\,x^{3/2}}+\frac {6\,a^2\,c\,x^{5/2}}{5}+\frac {6\,a\,c^2\,x^{13/2}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^3/x^(5/2),x)

[Out]

(2*c^3*x^(21/2))/21 - (2*a^3)/(3*x^(3/2)) + (6*a^2*c*x^(5/2))/5 + (6*a*c^2*x^(13/2))/13

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