3.8.35 \(\int \frac {(a+c x^4)^3}{x^{3/2}} \, dx\) [735]

Optimal. Leaf size=49 \[ -\frac {2 a^3}{\sqrt {x}}+\frac {6}{7} a^2 c x^{7/2}+\frac {2}{5} a c^2 x^{15/2}+\frac {2}{23} c^3 x^{23/2} \]

[Out]

6/7*a^2*c*x^(7/2)+2/5*a*c^2*x^(15/2)+2/23*c^3*x^(23/2)-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*} -\frac {2 a^3}{\sqrt {x}}+\frac {6}{7} a^2 c x^{7/2}+\frac {2}{5} a c^2 x^{15/2}+\frac {2}{23} c^3 x^{23/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^3/x^(3/2),x]

[Out]

(-2*a^3)/Sqrt[x] + (6*a^2*c*x^(7/2))/7 + (2*a*c^2*x^(15/2))/5 + (2*c^3*x^(23/2))/23

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^{3/2}} \, dx &=\int \left (\frac {a^3}{x^{3/2}}+3 a^2 c x^{5/2}+3 a c^2 x^{13/2}+c^3 x^{21/2}\right ) \, dx\\ &=-\frac {2 a^3}{\sqrt {x}}+\frac {6}{7} a^2 c x^{7/2}+\frac {2}{5} a c^2 x^{15/2}+\frac {2}{23} c^3 x^{23/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.84 \begin {gather*} -\frac {2 \left (805 a^3-345 a^2 c x^4-161 a c^2 x^8-35 c^3 x^{12}\right )}{805 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^3/x^(3/2),x]

[Out]

(-2*(805*a^3 - 345*a^2*c*x^4 - 161*a*c^2*x^8 - 35*c^3*x^12))/(805*Sqrt[x])

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Maple [A]
time = 0.14, size = 36, normalized size = 0.73

method	result	size
derivativedivides	\(\frac {6 a^{2} c \,x^{\frac {7}{2}}}{7}+\frac {2 a \,c^{2} x^{\frac {15}{2}}}{5}+\frac {2 c^{3} x^{\frac {23}{2}}}{23}-\frac {2 a^{3}}{\sqrt {x}}\)	\(36\)
default	\(\frac {6 a^{2} c \,x^{\frac {7}{2}}}{7}+\frac {2 a \,c^{2} x^{\frac {15}{2}}}{5}+\frac {2 c^{3} x^{\frac {23}{2}}}{23}-\frac {2 a^{3}}{\sqrt {x}}\)	\(36\)
gosper	\(-\frac {2 \left (-35 c^{3} x^{12}-161 a \,c^{2} x^{8}-345 a^{2} c \,x^{4}+805 a^{3}\right )}{805 \sqrt {x}}\)	\(38\)
trager	\(-\frac {2 \left (-35 c^{3} x^{12}-161 a \,c^{2} x^{8}-345 a^{2} c \,x^{4}+805 a^{3}\right )}{805 \sqrt {x}}\)	\(38\)
risch	\(-\frac {2 \left (-35 c^{3} x^{12}-161 a \,c^{2} x^{8}-345 a^{2} c \,x^{4}+805 a^{3}\right )}{805 \sqrt {x}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/7*a^2*c*x^(7/2)+2/5*a*c^2*x^(15/2)+2/23*c^3*x^(23/2)-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}{23} \, c^{3} x^{\frac {23}{2}} + \frac {2}{5} \, a c^{2} x^{\frac {15}{2}} + \frac {6}{7} \, a^{2} c x^{\frac {7}{2}} - \frac {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^(3/2),x, algorithm="maxima")

[Out]

2/23*c^3*x^(23/2) + 2/5*a*c^2*x^(15/2) + 6/7*a^2*c*x^(7/2) - 2*a^3/sqrt(x)

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Fricas [A]
time = 0.36, size = 37, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (35 \, c^{3} x^{12} + 161 \, a c^{2} x^{8} + 345 \, a^{2} c x^{4} - 805 \, a^{3}\right )}}{805 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/805*(35*c^3*x^12 + 161*a*c^2*x^8 + 345*a^2*c*x^4 - 805*a^3)/sqrt(x)

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Sympy [A]
time = 1.37, size = 48, normalized size = 0.98 \begin {gather*} - \frac {2 a^{3}}{\sqrt {x}} + \frac {6 a^{2} c x^{\frac {7}{2}}}{7} + \frac {2 a c^{2} x^{\frac {15}{2}}}{5} + \frac {2 c^{3} x^{\frac {23}{2}}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**3/sqrt(x) + 6*a**2*c*x**(7/2)/7 + 2*a*c**2*x**(15/2)/5 + 2*c**3*x**(23/2)/23

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Giac [A]
time = 0.91, size = 35, normalized size = 0.71 \begin {gather*} \frac {2}{23} \, c^{3} x^{\frac {23}{2}} + \frac {2}{5} \, a c^{2} x^{\frac {15}{2}} + \frac {6}{7} \, a^{2} c x^{\frac {7}{2}} - \frac {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^(3/2),x, algorithm="giac")

[Out]

2/23*c^3*x^(23/2) + 2/5*a*c^2*x^(15/2) + 6/7*a^2*c*x^(7/2) - 2*a^3/sqrt(x)

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Mupad [B]
time = 0.05, size = 35, normalized size = 0.71 \begin {gather*} \frac {2\,c^3\,x^{23/2}}{23}-\frac {2\,a^3}{\sqrt {x}}+\frac {6\,a^2\,c\,x^{7/2}}{7}+\frac {2\,a\,c^2\,x^{15/2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^3/x^(3/2),x)

[Out]

(2*c^3*x^(23/2))/23 - (2*a^3)/x^(1/2) + (6*a^2*c*x^(7/2))/7 + (2*a*c^2*x^(15/2))/5

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