3.8.31 \(\int x^{5/2} (a+c x^4)^3 \, dx\) [731]

Optimal. Leaf size=51 \[ \frac {2}{7} a^3 x^{7/2}+\frac {2}{5} a^2 c x^{15/2}+\frac {6}{23} a c^2 x^{23/2}+\frac {2}{31} c^3 x^{31/2} \]

[Out]

2/7*a^3*x^(7/2)+2/5*a^2*c*x^(15/2)+6/23*a*c^2*x^(23/2)+2/31*c^3*x^(31/2)

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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}{7} a^3 x^{7/2}+\frac {2}{5} a^2 c x^{15/2}+\frac {6}{23} a c^2 x^{23/2}+\frac {2}{31} c^3 x^{31/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + c*x^4)^3,x]

[Out]

(2*a^3*x^(7/2))/7 + (2*a^2*c*x^(15/2))/5 + (6*a*c^2*x^(23/2))/23 + (2*c^3*x^(31/2))/31

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 x^{5/2} \left (a+c x^4\right )^3 \, dx &=\int \left (a^3 x^{5/2}+3 a^2 c x^{13/2}+3 a c^2 x^{21/2}+c^3 x^{29/2}\right ) \, dx\\ &=\frac {2}{7} a^3 x^{7/2}+\frac {2}{5} a^2 c x^{15/2}+\frac {6}{23} a c^2 x^{23/2}+\frac {2}{31} c^3 x^{31/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 47, normalized size = 0.92 \begin {gather*} \frac {2 \left (3565 a^3 x^{7/2}+4991 a^2 c x^{15/2}+3255 a c^2 x^{23/2}+805 c^3 x^{31/2}\right )}{24955} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + c*x^4)^3,x]

[Out]

(2*(3565*a^3*x^(7/2) + 4991*a^2*c*x^(15/2) + 3255*a*c^2*x^(23/2) + 805*c^3*x^(31/2)))/24955

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

method	result	size
derivativedivides	\(\frac {2 a^{3} x^{\frac {7}{2}}}{7}+\frac {2 a^{2} c \,x^{\frac {15}{2}}}{5}+\frac {6 a \,c^{2} x^{\frac {23}{2}}}{23}+\frac {2 c^{3} x^{\frac {31}{2}}}{31}\)	\(36\)
default	\(\frac {2 a^{3} x^{\frac {7}{2}}}{7}+\frac {2 a^{2} c \,x^{\frac {15}{2}}}{5}+\frac {6 a \,c^{2} x^{\frac {23}{2}}}{23}+\frac {2 c^{3} x^{\frac {31}{2}}}{31}\)	\(36\)
gosper	\(\frac {2 x^{\frac {7}{2}} \left (805 c^{3} x^{12}+3255 a \,c^{2} x^{8}+4991 a^{2} c \,x^{4}+3565 a^{3}\right )}{24955}\)	\(38\)
trager	\(\frac {2 x^{\frac {7}{2}} \left (805 c^{3} x^{12}+3255 a \,c^{2} x^{8}+4991 a^{2} c \,x^{4}+3565 a^{3}\right )}{24955}\)	\(38\)
risch	\(\frac {2 x^{\frac {7}{2}} \left (805 c^{3} x^{12}+3255 a \,c^{2} x^{8}+4991 a^{2} c \,x^{4}+3565 a^{3}\right )}{24955}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*a^3*x^(7/2)+2/5*a^2*c*x^(15/2)+6/23*a*c^2*x^(23/2)+2/31*c^3*x^(31/2)

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Maxima [A]
time = 0.29, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{31} \, c^{3} x^{\frac {31}{2}} + \frac {6}{23} \, a c^{2} x^{\frac {23}{2}} + \frac {2}{5} \, a^{2} c x^{\frac {15}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/31*c^3*x^(31/2) + 6/23*a*c^2*x^(23/2) + 2/5*a^2*c*x^(15/2) + 2/7*a^3*x^(7/2)

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Fricas [A]
time = 0.38, size = 40, normalized size = 0.78 \begin {gather*} \frac {2}{24955} \, {\left (805 \, c^{3} x^{15} + 3255 \, a c^{2} x^{11} + 4991 \, a^{2} c x^{7} + 3565 \, a^{3} x^{3}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/24955*(805*c^3*x^15 + 3255*a*c^2*x^11 + 4991*a^2*c*x^7 + 3565*a^3*x^3)*sqrt(x)

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Sympy [A]
time = 2.11, size = 49, normalized size = 0.96 \begin {gather*} \frac {2 a^{3} x^{\frac {7}{2}}}{7} + \frac {2 a^{2} c x^{\frac {15}{2}}}{5} + \frac {6 a c^{2} x^{\frac {23}{2}}}{23} + \frac {2 c^{3} x^{\frac {31}{2}}}{31} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**3*x**(7/2)/7 + 2*a**2*c*x**(15/2)/5 + 6*a*c**2*x**(23/2)/23 + 2*c**3*x**(31/2)/31

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Giac [A]
time = 0.62, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{31} \, c^{3} x^{\frac {31}{2}} + \frac {6}{23} \, a c^{2} x^{\frac {23}{2}} + \frac {2}{5} \, a^{2} c x^{\frac {15}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/31*c^3*x^(31/2) + 6/23*a*c^2*x^(23/2) + 2/5*a^2*c*x^(15/2) + 2/7*a^3*x^(7/2)

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Mupad [B]
time = 0.04, size = 35, normalized size = 0.69 \begin {gather*} \frac {2\,a^3\,x^{7/2}}{7}+\frac {2\,c^3\,x^{31/2}}{31}+\frac {2\,a^2\,c\,x^{15/2}}{5}+\frac {6\,a\,c^2\,x^{23/2}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(a + c*x^4)^3,x)

[Out]

(2*a^3*x^(7/2))/7 + (2*c^3*x^(31/2))/31 + (2*a^2*c*x^(15/2))/5 + (6*a*c^2*x^(23/2))/23

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