3.8.28 \(\int \frac {(a+c x^4)^2}{x^{3/2}} \, dx\) [728]

Optimal. Leaf size=34 \[ -\frac {2 a^2}{\sqrt {x}}+\frac {4}{7} a c x^{7/2}+\frac {2}{15} c^2 x^{15/2} \]

[Out]

4/7*a*c*x^(7/2)+2/15*c^2*x^(15/2)-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*} -\frac {2 a^2}{\sqrt {x}}+\frac {4}{7} a c x^{7/2}+\frac {2}{15} c^2 x^{15/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^2/x^(3/2),x]

[Out]

(-2*a^2)/Sqrt[x] + (4*a*c*x^(7/2))/7 + (2*c^2*x^(15/2))/15

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^{3/2}} \, dx &=\int \left (\frac {a^2}{x^{3/2}}+2 a c x^{5/2}+c^2 x^{13/2}\right ) \, dx\\ &=-\frac {2 a^2}{\sqrt {x}}+\frac {4}{7} a c x^{7/2}+\frac {2}{15} c^2 x^{15/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.88 \begin {gather*} -\frac {2 \left (105 a^2-30 a c x^4-7 c^2 x^8\right )}{105 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^2/x^(3/2),x]

[Out]

(-2*(105*a^2 - 30*a*c*x^4 - 7*c^2*x^8))/(105*Sqrt[x])

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Maple [A]
time = 0.14, size = 25, normalized size = 0.74

method	result	size
derivativedivides	\(\frac {4 a c \,x^{\frac {7}{2}}}{7}+\frac {2 c^{2} x^{\frac {15}{2}}}{15}-\frac {2 a^{2}}{\sqrt {x}}\)	\(25\)
default	\(\frac {4 a c \,x^{\frac {7}{2}}}{7}+\frac {2 c^{2} x^{\frac {15}{2}}}{15}-\frac {2 a^{2}}{\sqrt {x}}\)	\(25\)
gosper	\(-\frac {2 \left (-7 c^{2} x^{8}-30 a c \,x^{4}+105 a^{2}\right )}{105 \sqrt {x}}\)	\(27\)
trager	\(-\frac {2 \left (-7 c^{2} x^{8}-30 a c \,x^{4}+105 a^{2}\right )}{105 \sqrt {x}}\)	\(27\)
risch	\(-\frac {2 \left (-7 c^{2} x^{8}-30 a c \,x^{4}+105 a^{2}\right )}{105 \sqrt {x}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*a*c*x^(7/2)+2/15*c^2*x^(15/2)-2*a^2/x^(1/2)

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Maxima [A]
time = 0.30, size = 24, normalized size = 0.71 \begin {gather*} \frac {2}{15} \, c^{2} x^{\frac {15}{2}} + \frac {4}{7} \, a c x^{\frac {7}{2}} - \frac {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^(3/2),x, algorithm="maxima")

[Out]

2/15*c^2*x^(15/2) + 4/7*a*c*x^(7/2) - 2*a^2/sqrt(x)

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Fricas [A]
time = 0.35, size = 26, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (7 \, c^{2} x^{8} + 30 \, a c x^{4} - 105 \, a^{2}\right )}}{105 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(7*c^2*x^8 + 30*a*c*x^4 - 105*a^2)/sqrt(x)

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Sympy [A]
time = 0.63, size = 32, normalized size = 0.94 \begin {gather*} - \frac {2 a^{2}}{\sqrt {x}} + \frac {4 a c x^{\frac {7}{2}}}{7} + \frac {2 c^{2} x^{\frac {15}{2}}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2/sqrt(x) + 4*a*c*x**(7/2)/7 + 2*c**2*x**(15/2)/15

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Giac [A]
time = 0.50, size = 24, normalized size = 0.71 \begin {gather*} \frac {2}{15} \, c^{2} x^{\frac {15}{2}} + \frac {4}{7} \, a c x^{\frac {7}{2}} - \frac {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^(3/2),x, algorithm="giac")

[Out]

2/15*c^2*x^(15/2) + 4/7*a*c*x^(7/2) - 2*a^2/sqrt(x)

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Mupad [B]
time = 0.04, size = 25, normalized size = 0.74 \begin {gather*} \frac {-2\,a^2+\frac {4\,a\,c\,x^4}{7}+\frac {2\,c^2\,x^8}{15}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^2/x^(3/2),x)

[Out]

((2*c^2*x^8)/15 - 2*a^2 + (4*a*c*x^4)/7)/x^(1/2)

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