3.308 \(\int x^{7/2} (b x^2+c x^4)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{6}{25} b^2 c x^{25/2}+\frac{2}{21} b^3 x^{21/2}+\frac{6}{29} b c^2 x^{29/2}+\frac{2}{33} c^3 x^{33/2} \]

[Out]

(2*b^3*x^(21/2))/21 + (6*b^2*c*x^(25/2))/25 + (6*b*c^2*x^(29/2))/29 + (2*c^3*x^(33/2))/33

________________________________________________________________________________________

Rubi [A]  time = 0.0212143, 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} \[ \frac{6}{25} b^2 c x^{25/2}+\frac{2}{21} b^3 x^{21/2}+\frac{6}{29} b c^2 x^{29/2}+\frac{2}{33} c^3 x^{33/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(b*x^2 + c*x^4)^3,x]

[Out]

(2*b^3*x^(21/2))/21 + (6*b^2*c*x^(25/2))/25 + (6*b*c^2*x^(29/2))/29 + (2*c^3*x^(33/2))/33

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^{7/2} \left (b x^2+c x^4\right )^3 \, dx &=\int x^{19/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{19/2}+3 b^2 c x^{23/2}+3 b c^2 x^{27/2}+c^3 x^{31/2}\right ) \, dx\\ &=\frac{2}{21} b^3 x^{21/2}+\frac{6}{25} b^2 c x^{25/2}+\frac{6}{29} b c^2 x^{29/2}+\frac{2}{33} c^3 x^{33/2}\\ \end{align*}

Mathematica [A]  time = 0.0138796, size = 51, normalized size = 1. \[ \frac{6}{25} b^2 c x^{25/2}+\frac{2}{21} b^3 x^{21/2}+\frac{6}{29} b c^2 x^{29/2}+\frac{2}{33} c^3 x^{33/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(b*x^2 + c*x^4)^3,x]

[Out]

(2*b^3*x^(21/2))/21 + (6*b^2*c*x^(25/2))/25 + (6*b*c^2*x^(29/2))/29 + (2*c^3*x^(33/2))/33

________________________________________________________________________________________

Maple [A]  time = 0.047, size = 38, normalized size = 0.8 \begin{align*}{\frac{10150\,{c}^{3}{x}^{6}+34650\,b{c}^{2}{x}^{4}+40194\,{b}^{2}c{x}^{2}+15950\,{b}^{3}}{167475}{x}^{{\frac{21}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(c*x^4+b*x^2)^3,x)

[Out]

2/167475*x^(21/2)*(5075*c^3*x^6+17325*b*c^2*x^4+20097*b^2*c*x^2+7975*b^3)

________________________________________________________________________________________

Maxima [A]  time = 0.968687, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{33} \, c^{3} x^{\frac{33}{2}} + \frac{6}{29} \, b c^{2} x^{\frac{29}{2}} + \frac{6}{25} \, b^{2} c x^{\frac{25}{2}} + \frac{2}{21} \, b^{3} x^{\frac{21}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^4+b*x^2)^3,x, algorithm="maxima")

[Out]

2/33*c^3*x^(33/2) + 6/29*b*c^2*x^(29/2) + 6/25*b^2*c*x^(25/2) + 2/21*b^3*x^(21/2)

________________________________________________________________________________________

Fricas [A]  time = 1.26853, size = 119, normalized size = 2.33 \begin{align*} \frac{2}{167475} \,{\left (5075 \, c^{3} x^{16} + 17325 \, b c^{2} x^{14} + 20097 \, b^{2} c x^{12} + 7975 \, b^{3} x^{10}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^4+b*x^2)^3,x, algorithm="fricas")

[Out]

2/167475*(5075*c^3*x^16 + 17325*b*c^2*x^14 + 20097*b^2*c*x^12 + 7975*b^3*x^10)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 112.55, size = 49, normalized size = 0.96 \begin{align*} \frac{2 b^{3} x^{\frac{21}{2}}}{21} + \frac{6 b^{2} c x^{\frac{25}{2}}}{25} + \frac{6 b c^{2} x^{\frac{29}{2}}}{29} + \frac{2 c^{3} x^{\frac{33}{2}}}{33} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(c*x**4+b*x**2)**3,x)

[Out]

2*b**3*x**(21/2)/21 + 6*b**2*c*x**(25/2)/25 + 6*b*c**2*x**(29/2)/29 + 2*c**3*x**(33/2)/33

________________________________________________________________________________________

Giac [A]  time = 1.15837, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{33} \, c^{3} x^{\frac{33}{2}} + \frac{6}{29} \, b c^{2} x^{\frac{29}{2}} + \frac{6}{25} \, b^{2} c x^{\frac{25}{2}} + \frac{2}{21} \, b^{3} x^{\frac{21}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^4+b*x^2)^3,x, algorithm="giac")

[Out]

2/33*c^3*x^(33/2) + 6/29*b*c^2*x^(29/2) + 6/25*b^2*c*x^(25/2) + 2/21*b^3*x^(21/2)