3.8.21 \(\int \frac {a+c x^4}{x^{3/2}} \, dx\) [721]

Optimal. Leaf size=19 \[ -\frac {2 a}{\sqrt {x}}+\frac {2}{7} c x^{7/2} \]

[Out]

2/7*c*x^(7/2)-2*a/x^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \begin {gather*} \frac {2}{7} c x^{7/2}-\frac {2 a}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + (2*c*x^(7/2))/7

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int \frac {a+c x^4}{x^{3/2}} \, dx &=\int \left (\frac {a}{x^{3/2}}+c x^{5/2}\right ) \, dx\\ &=-\frac {2 a}{\sqrt {x}}+\frac {2}{7} c x^{7/2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} -\frac {2 \left (7 a-c x^4\right )}{7 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)/x^(3/2),x]

[Out]

(-2*(7*a - c*x^4))/(7*Sqrt[x])

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Maple [A]
time = 0.04, size = 14, normalized size = 0.74

method	result	size
derivativedivides	\(\frac {2 c \,x^{\frac {7}{2}}}{7}-\frac {2 a}{\sqrt {x}}\)	\(14\)
default	\(\frac {2 c \,x^{\frac {7}{2}}}{7}-\frac {2 a}{\sqrt {x}}\)	\(14\)
gosper	\(-\frac {2 \left (-x^{4} c +7 a \right )}{7 \sqrt {x}}\)	\(16\)
trager	\(-\frac {2 \left (-x^{4} c +7 a \right )}{7 \sqrt {x}}\)	\(16\)
risch	\(-\frac {2 \left (-x^{4} c +7 a \right )}{7 \sqrt {x}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*c*x^(7/2)-2*a/x^(1/2)

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{7} \, c x^{\frac {7}{2}} - \frac {2 \, a}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*c*x^(7/2) - 2*a/sqrt(x)

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Fricas [A]
time = 0.36, size = 14, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (c x^{4} - 7 \, a\right )}}{7 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(c*x^4 - 7*a)/sqrt(x)

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Sympy [A]
time = 0.23, size = 17, normalized size = 0.89 \begin {gather*} - \frac {2 a}{\sqrt {x}} + \frac {2 c x^{\frac {7}{2}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/sqrt(x) + 2*c*x**(7/2)/7

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Giac [A]
time = 0.69, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{7} \, c x^{\frac {7}{2}} - \frac {2 \, a}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*c*x^(7/2) - 2*a/sqrt(x)

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Mupad [B]
time = 0.03, size = 15, normalized size = 0.79 \begin {gather*} -\frac {14\,a-2\,c\,x^4}{7\,\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)/x^(3/2),x)

[Out]

-(14*a - 2*c*x^4)/(7*x^(1/2))

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