3.8.29 \(\int \frac {(a+c x^4)^2}{x^{5/2}} \, dx\) [729]

Optimal. Leaf size=36 \[ -\frac {2 a^2}{3 x^{3/2}}+\frac {4}{5} a c x^{5/2}+\frac {2}{13} c^2 x^{13/2} \]

[Out]

-2/3*a^2/x^(3/2)+4/5*a*c*x^(5/2)+2/13*c^2*x^(13/2)

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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 a^2}{3 x^{3/2}}+\frac {4}{5} a c x^{5/2}+\frac {2}{13} c^2 x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^2/x^(5/2),x]

[Out]

(-2*a^2)/(3*x^(3/2)) + (4*a*c*x^(5/2))/5 + (2*c^2*x^(13/2))/13

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}{x^{5/2}} \, dx &=\int \left (\frac {a^2}{x^{5/2}}+2 a c x^{3/2}+c^2 x^{11/2}\right ) \, dx\\ &=-\frac {2 a^2}{3 x^{3/2}}+\frac {4}{5} a c x^{5/2}+\frac {2}{13} c^2 x^{13/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.83 \begin {gather*} -\frac {2 \left (65 a^2-78 a c x^4-15 c^2 x^8\right )}{195 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^2/x^(5/2),x]

[Out]

(-2*(65*a^2 - 78*a*c*x^4 - 15*c^2*x^8))/(195*x^(3/2))

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

method	result	size
derivativedivides	\(-\frac {2 a^{2}}{3 x^{\frac {3}{2}}}+\frac {4 a c \,x^{\frac {5}{2}}}{5}+\frac {2 c^{2} x^{\frac {13}{2}}}{13}\)	\(25\)
default	\(-\frac {2 a^{2}}{3 x^{\frac {3}{2}}}+\frac {4 a c \,x^{\frac {5}{2}}}{5}+\frac {2 c^{2} x^{\frac {13}{2}}}{13}\)	\(25\)
gosper	\(-\frac {2 \left (-15 c^{2} x^{8}-78 a c \,x^{4}+65 a^{2}\right )}{195 x^{\frac {3}{2}}}\)	\(27\)
trager	\(-\frac {2 \left (-15 c^{2} x^{8}-78 a c \,x^{4}+65 a^{2}\right )}{195 x^{\frac {3}{2}}}\)	\(27\)
risch	\(-\frac {2 \left (-15 c^{2} x^{8}-78 a c \,x^{4}+65 a^{2}\right )}{195 x^{\frac {3}{2}}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*a^2/x^(3/2)+4/5*a*c*x^(5/2)+2/13*c^2*x^(13/2)

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Maxima [A]
time = 0.30, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{5} \, a c x^{\frac {5}{2}} - \frac {2 \, a^{2}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*c^2*x^(13/2) + 4/5*a*c*x^(5/2) - 2/3*a^2/x^(3/2)

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Fricas [A]
time = 0.36, size = 26, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (15 \, c^{2} x^{8} + 78 \, a c x^{4} - 65 \, a^{2}\right )}}{195 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/195*(15*c^2*x^8 + 78*a*c*x^4 - 65*a^2)/x^(3/2)

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Sympy [A]
time = 0.68, size = 34, normalized size = 0.94 \begin {gather*} - \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + \frac {4 a c x^{\frac {5}{2}}}{5} + \frac {2 c^{2} x^{\frac {13}{2}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2/(3*x**(3/2)) + 4*a*c*x**(5/2)/5 + 2*c**2*x**(13/2)/13

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Giac [A]
time = 0.47, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{5} \, a c x^{\frac {5}{2}} - \frac {2 \, a^{2}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*c^2*x^(13/2) + 4/5*a*c*x^(5/2) - 2/3*a^2/x^(3/2)

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Mupad [B]
time = 0.04, size = 26, normalized size = 0.72 \begin {gather*} \frac {-130\,a^2+156\,a\,c\,x^4+30\,c^2\,x^8}{195\,x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^2/x^(5/2),x)

[Out]

(30*c^2*x^8 - 130*a^2 + 156*a*c*x^4)/(195*x^(3/2))

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