3.309 \(\int x^{5/2} (b x^2+c x^4)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{6}{23} b^2 c x^{23/2}+\frac{2}{19} b^3 x^{19/2}+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2} \]

[Out]

(2*b^3*x^(19/2))/19 + (6*b^2*c*x^(23/2))/23 + (2*b*c^2*x^(27/2))/9 + (2*c^3*x^(31/2))/31

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Rubi [A]  time = 0.0204168, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1584, 270} \[ \frac{6}{23} b^2 c x^{23/2}+\frac{2}{19} b^3 x^{19/2}+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(b*x^2 + c*x^4)^3,x]

[Out]

(2*b^3*x^(19/2))/19 + (6*b^2*c*x^(23/2))/23 + (2*b*c^2*x^(27/2))/9 + (2*c^3*x^(31/2))/31

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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^{5/2} \left (b x^2+c x^4\right )^3 \, dx &=\int x^{17/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{17/2}+3 b^2 c x^{21/2}+3 b c^2 x^{25/2}+c^3 x^{29/2}\right ) \, dx\\ &=\frac{2}{19} b^3 x^{19/2}+\frac{6}{23} b^2 c x^{23/2}+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2}\\ \end{align*}

Mathematica [A]  time = 0.0126824, size = 51, normalized size = 1. \[ \frac{6}{23} b^2 c x^{23/2}+\frac{2}{19} b^3 x^{19/2}+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(b*x^2 + c*x^4)^3,x]

[Out]

(2*b^3*x^(19/2))/19 + (6*b^2*c*x^(23/2))/23 + (2*b*c^2*x^(27/2))/9 + (2*c^3*x^(31/2))/31

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Maple [A]  time = 0.047, size = 38, normalized size = 0.8 \begin{align*}{\frac{7866\,{c}^{3}{x}^{6}+27094\,b{c}^{2}{x}^{4}+31806\,{b}^{2}c{x}^{2}+12834\,{b}^{3}}{121923}{x}^{{\frac{19}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(c*x^4+b*x^2)^3,x)

[Out]

2/121923*x^(19/2)*(3933*c^3*x^6+13547*b*c^2*x^4+15903*b^2*c*x^2+6417*b^3)

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Maxima [A]  time = 1.06052, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{31} \, c^{3} x^{\frac{31}{2}} + \frac{2}{9} \, b c^{2} x^{\frac{27}{2}} + \frac{6}{23} \, b^{2} c x^{\frac{23}{2}} + \frac{2}{19} \, b^{3} x^{\frac{19}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/31*c^3*x^(31/2) + 2/9*b*c^2*x^(27/2) + 6/23*b^2*c*x^(23/2) + 2/19*b^3*x^(19/2)

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Fricas [A]  time = 1.21276, size = 117, normalized size = 2.29 \begin{align*} \frac{2}{121923} \,{\left (3933 \, c^{3} x^{15} + 13547 \, b c^{2} x^{13} + 15903 \, b^{2} c x^{11} + 6417 \, b^{3} x^{9}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/121923*(3933*c^3*x^15 + 13547*b*c^2*x^13 + 15903*b^2*c*x^11 + 6417*b^3*x^9)*sqrt(x)

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Sympy [A]  time = 70.7733, size = 49, normalized size = 0.96 \begin{align*} \frac{2 b^{3} x^{\frac{19}{2}}}{19} + \frac{6 b^{2} c x^{\frac{23}{2}}}{23} + \frac{2 b c^{2} x^{\frac{27}{2}}}{9} + \frac{2 c^{3} x^{\frac{31}{2}}}{31} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**3*x**(19/2)/19 + 6*b**2*c*x**(23/2)/23 + 2*b*c**2*x**(27/2)/9 + 2*c**3*x**(31/2)/31

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Giac [A]  time = 1.12223, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{31} \, c^{3} x^{\frac{31}{2}} + \frac{2}{9} \, b c^{2} x^{\frac{27}{2}} + \frac{6}{23} \, b^{2} c x^{\frac{23}{2}} + \frac{2}{19} \, b^{3} x^{\frac{19}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/31*c^3*x^(31/2) + 2/9*b*c^2*x^(27/2) + 6/23*b^2*c*x^(23/2) + 2/19*b^3*x^(19/2)