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

Optimal. Leaf size=51 \[ \frac{2}{7} b^2 c x^{21/2}+\frac{2}{17} b^3 x^{17/2}+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2} \]

[Out]

(2*b^3*x^(17/2))/17 + (2*b^2*c*x^(21/2))/7 + (6*b*c^2*x^(25/2))/25 + (2*c^3*x^(29/2))/29

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Rubi [A]  time = 0.019643, 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} \[ \frac{2}{7} b^2 c x^{21/2}+\frac{2}{17} b^3 x^{17/2}+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(b*x^2 + c*x^4)^3,x]

[Out]

(2*b^3*x^(17/2))/17 + (2*b^2*c*x^(21/2))/7 + (6*b*c^2*x^(25/2))/25 + (2*c^3*x^(29/2))/29

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^{3/2} \left (b x^2+c x^4\right )^3 \, dx &=\int x^{15/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{15/2}+3 b^2 c x^{19/2}+3 b c^2 x^{23/2}+c^3 x^{27/2}\right ) \, dx\\ &=\frac{2}{17} b^3 x^{17/2}+\frac{2}{7} b^2 c x^{21/2}+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2}\\ \end{align*}

Mathematica [A]  time = 0.0114702, size = 51, normalized size = 1. \[ \frac{2}{7} b^2 c x^{21/2}+\frac{2}{17} b^3 x^{17/2}+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^3*x^(17/2))/17 + (2*b^2*c*x^(21/2))/7 + (6*b*c^2*x^(25/2))/25 + (2*c^3*x^(29/2))/29

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Maple [A]  time = 0.046, size = 38, normalized size = 0.8 \begin{align*}{\frac{5950\,{c}^{3}{x}^{6}+20706\,b{c}^{2}{x}^{4}+24650\,{b}^{2}c{x}^{2}+10150\,{b}^{3}}{86275}{x}^{{\frac{17}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/86275*x^(17/2)*(2975*c^3*x^6+10353*b*c^2*x^4+12325*b^2*c*x^2+5075*b^3)

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Maxima [A]  time = 0.989323, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{29} \, c^{3} x^{\frac{29}{2}} + \frac{6}{25} \, b c^{2} x^{\frac{25}{2}} + \frac{2}{7} \, b^{2} c x^{\frac{21}{2}} + \frac{2}{17} \, b^{3} x^{\frac{17}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/29*c^3*x^(29/2) + 6/25*b*c^2*x^(25/2) + 2/7*b^2*c*x^(21/2) + 2/17*b^3*x^(17/2)

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Fricas [A]  time = 1.21739, size = 116, normalized size = 2.27 \begin{align*} \frac{2}{86275} \,{\left (2975 \, c^{3} x^{14} + 10353 \, b c^{2} x^{12} + 12325 \, b^{2} c x^{10} + 5075 \, b^{3} x^{8}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/86275*(2975*c^3*x^14 + 10353*b*c^2*x^12 + 12325*b^2*c*x^10 + 5075*b^3*x^8)*sqrt(x)

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Sympy [A]  time = 41.7851, size = 49, normalized size = 0.96 \begin{align*} \frac{2 b^{3} x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c x^{\frac{21}{2}}}{7} + \frac{6 b c^{2} x^{\frac{25}{2}}}{25} + \frac{2 c^{3} x^{\frac{29}{2}}}{29} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**3*x**(17/2)/17 + 2*b**2*c*x**(21/2)/7 + 6*b*c**2*x**(25/2)/25 + 2*c**3*x**(29/2)/29

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Giac [A]  time = 1.15085, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{29} \, c^{3} x^{\frac{29}{2}} + \frac{6}{25} \, b c^{2} x^{\frac{25}{2}} + \frac{2}{7} \, b^{2} c x^{\frac{21}{2}} + \frac{2}{17} \, b^{3} x^{\frac{17}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/29*c^3*x^(29/2) + 6/25*b*c^2*x^(25/2) + 2/7*b^2*c*x^(21/2) + 2/17*b^3*x^(17/2)