3.725 \(\int x^{3/2} (a+c x^4)^2 \, dx\)

Optimal. Leaf size=36 \[ \frac {2}{5} a^2 x^{5/2}+\frac {4}{13} a c x^{13/2}+\frac {2}{21} c^2 x^{21/2} \]

[Out]

2/5*a^2*x^(5/2)+4/13*a*c*x^(13/2)+2/21*c^2*x^(21/2)

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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 = {270} \[ \frac {2}{5} a^2 x^{5/2}+\frac {4}{13} a c x^{13/2}+\frac {2}{21} c^2 x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + c*x^4)^2,x]

[Out]

(2*a^2*x^(5/2))/5 + (4*a*c*x^(13/2))/13 + (2*c^2*x^(21/2))/21

Rule 270

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^{3/2} \left (a+c x^4\right )^2 \, dx &=\int \left (a^2 x^{3/2}+2 a c x^{11/2}+c^2 x^{19/2}\right ) \, dx\\ &=\frac {2}{5} a^2 x^{5/2}+\frac {4}{13} a c x^{13/2}+\frac {2}{21} c^2 x^{21/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.83 \[ \frac {2 x^{5/2} \left (273 a^2+210 a c x^4+65 c^2 x^8\right )}{1365} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + c*x^4)^2,x]

[Out]

(2*x^(5/2)*(273*a^2 + 210*a*c*x^4 + 65*c^2*x^8))/1365

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fricas [A]  time = 0.68, size = 29, normalized size = 0.81 \[ \frac {2}{1365} \, {\left (65 \, c^{2} x^{10} + 210 \, a c x^{6} + 273 \, a^{2} x^{2}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1365*(65*c^2*x^10 + 210*a*c*x^6 + 273*a^2*x^2)*sqrt(x)

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giac [A]  time = 0.15, size = 24, normalized size = 0.67 \[ \frac {2}{21} \, c^{2} x^{\frac {21}{2}} + \frac {4}{13} \, a c x^{\frac {13}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*c^2*x^(21/2) + 4/13*a*c*x^(13/2) + 2/5*a^2*x^(5/2)

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maple [A]  time = 0.00, size = 27, normalized size = 0.75 \[ \frac {2 \left (65 c^{2} x^{8}+210 a c \,x^{4}+273 a^{2}\right ) x^{\frac {5}{2}}}{1365} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1365*x^(5/2)*(65*c^2*x^8+210*a*c*x^4+273*a^2)

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maxima [A]  time = 1.31, size = 24, normalized size = 0.67 \[ \frac {2}{21} \, c^{2} x^{\frac {21}{2}} + \frac {4}{13} \, a c x^{\frac {13}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*c^2*x^(21/2) + 4/13*a*c*x^(13/2) + 2/5*a^2*x^(5/2)

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mupad [B]  time = 0.04, size = 26, normalized size = 0.72 \[ \frac {2\,x^{5/2}\,\left (273\,a^2+210\,a\,c\,x^4+65\,c^2\,x^8\right )}{1365} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(a + c*x^4)^2,x)

[Out]

(2*x^(5/2)*(273*a^2 + 65*c^2*x^8 + 210*a*c*x^4))/1365

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sympy [A]  time = 17.04, size = 34, normalized size = 0.94 \[ \frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a c x^{\frac {13}{2}}}{13} + \frac {2 c^{2} x^{\frac {21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*x**(5/2)/5 + 4*a*c*x**(13/2)/13 + 2*c**2*x**(21/2)/21

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