3.404 \(\int (c x)^m (b x^2+c x^4)^2 \, dx\)

Optimal. Leaf size=52 \[ \frac{b^2 x^5 (c x)^m}{m+5}+\frac{2 b c x^7 (c x)^m}{m+7}+\frac{c^2 x^9 (c x)^m}{m+9} \]

[Out]

(b^2*x^5*(c*x)^m)/(5 + m) + (2*b*c*x^7*(c*x)^m)/(7 + m) + (c^2*x^9*(c*x)^m)/(9 + m)

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Rubi [A]  time = 0.0394327, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1142, 1584, 270} \[ \frac{b^2 x^5 (c x)^m}{m+5}+\frac{2 b c x^7 (c x)^m}{m+7}+\frac{c^2 x^9 (c x)^m}{m+9} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(b*x^2 + c*x^4)^2,x]

[Out]

(b^2*x^5*(c*x)^m)/(5 + m) + (2*b*c*x^7*(c*x)^m)/(7 + m) + (c^2*x^9*(c*x)^m)/(9 + m)

Rule 1142

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

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 (c x)^m \left (b x^2+c x^4\right )^2 \, dx &=\left (x^{-m} (c x)^m\right ) \operatorname{Subst}\left (\int x^m \left (b x^2+c x^4\right )^2 \, dx,x,x\right )\\ &=\left (x^{-m} (c x)^m\right ) \operatorname{Subst}\left (\int x^{4+m} \left (b+c x^2\right )^2 \, dx,x,x\right )\\ &=\left (x^{-m} (c x)^m\right ) \operatorname{Subst}\left (\int \left (b^2 x^{4+m}+2 b c x^{6+m}+c^2 x^{8+m}\right ) \, dx,x,x\right )\\ &=\frac{b^2 x^5 (c x)^m}{5+m}+\frac{2 b c x^7 (c x)^m}{7+m}+\frac{c^2 x^9 (c x)^m}{9+m}\\ \end{align*}

Mathematica [A]  time = 0.034236, size = 43, normalized size = 0.83 \[ x^5 (c x)^m \left (\frac{b^2}{m+5}+\frac{2 b c x^2}{m+7}+\frac{c^2 x^4}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^5*(c*x)^m*(b^2/(5 + m) + (2*b*c*x^2)/(7 + m) + (c^2*x^4)/(9 + m))

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Maple [A]  time = 0.049, size = 96, normalized size = 1.9 \begin{align*}{\frac{ \left ( cx \right ) ^{m} \left ({c}^{2}{m}^{2}{x}^{4}+12\,{c}^{2}m{x}^{4}+2\,bc{m}^{2}{x}^{2}+35\,{c}^{2}{x}^{4}+28\,bcm{x}^{2}+{b}^{2}{m}^{2}+90\,bc{x}^{2}+16\,{b}^{2}m+63\,{b}^{2} \right ){x}^{5}}{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x)^m*(c^2*m^2*x^4+12*c^2*m*x^4+2*b*c*m^2*x^2+35*c^2*x^4+28*b*c*m*x^2+b^2*m^2+90*b*c*x^2+16*b^2*m+63*b^2)*x^
5/(9+m)/(7+m)/(5+m)

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Maxima [A]  time = 1.00875, size = 74, normalized size = 1.42 \begin{align*} \frac{c^{m + 2} x^{9} x^{m}}{m + 9} + \frac{2 \, b c^{m + 1} x^{7} x^{m}}{m + 7} + \frac{b^{2} c^{m} x^{5} x^{m}}{m + 5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^(m + 2)*x^9*x^m/(m + 9) + 2*b*c^(m + 1)*x^7*x^m/(m + 7) + b^2*c^m*x^5*x^m/(m + 5)

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Fricas [A]  time = 1.5866, size = 200, normalized size = 3.85 \begin{align*} \frac{{\left ({\left (c^{2} m^{2} + 12 \, c^{2} m + 35 \, c^{2}\right )} x^{9} + 2 \,{\left (b c m^{2} + 14 \, b c m + 45 \, b c\right )} x^{7} +{\left (b^{2} m^{2} + 16 \, b^{2} m + 63 \, b^{2}\right )} x^{5}\right )} \left (c x\right )^{m}}{m^{3} + 21 \, m^{2} + 143 \, m + 315} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c^2*m^2 + 12*c^2*m + 35*c^2)*x^9 + 2*(b*c*m^2 + 14*b*c*m + 45*b*c)*x^7 + (b^2*m^2 + 16*b^2*m + 63*b^2)*x^5)*
(c*x)^m/(m^3 + 21*m^2 + 143*m + 315)

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Sympy [A]  time = 2.1886, size = 352, normalized size = 6.77 \begin{align*} \begin{cases} \frac{- \frac{b^{2}}{4 x^{4}} - \frac{b c}{x^{2}} + c^{2} \log{\left (x \right )}}{c^{9}} & \text{for}\: m = -9 \\\frac{- \frac{b^{2}}{2 x^{2}} + 2 b c \log{\left (x \right )} + \frac{c^{2} x^{2}}{2}}{c^{7}} & \text{for}\: m = -7 \\\frac{b^{2} \log{\left (x \right )} + b c x^{2} + \frac{c^{2} x^{4}}{4}}{c^{5}} & \text{for}\: m = -5 \\\frac{b^{2} c^{m} m^{2} x^{5} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{16 b^{2} c^{m} m x^{5} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{63 b^{2} c^{m} x^{5} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{2 b c c^{m} m^{2} x^{7} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{28 b c c^{m} m x^{7} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{90 b c c^{m} x^{7} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{c^{2} c^{m} m^{2} x^{9} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{12 c^{2} c^{m} m x^{9} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{35 c^{2} c^{m} x^{9} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(c*x**4+b*x**2)**2,x)

[Out]

Piecewise(((-b**2/(4*x**4) - b*c/x**2 + c**2*log(x))/c**9, Eq(m, -9)), ((-b**2/(2*x**2) + 2*b*c*log(x) + c**2*
x**2/2)/c**7, Eq(m, -7)), ((b**2*log(x) + b*c*x**2 + c**2*x**4/4)/c**5, Eq(m, -5)), (b**2*c**m*m**2*x**5*x**m/
(m**3 + 21*m**2 + 143*m + 315) + 16*b**2*c**m*m*x**5*x**m/(m**3 + 21*m**2 + 143*m + 315) + 63*b**2*c**m*x**5*x
**m/(m**3 + 21*m**2 + 143*m + 315) + 2*b*c*c**m*m**2*x**7*x**m/(m**3 + 21*m**2 + 143*m + 315) + 28*b*c*c**m*m*
x**7*x**m/(m**3 + 21*m**2 + 143*m + 315) + 90*b*c*c**m*x**7*x**m/(m**3 + 21*m**2 + 143*m + 315) + c**2*c**m*m*
*2*x**9*x**m/(m**3 + 21*m**2 + 143*m + 315) + 12*c**2*c**m*m*x**9*x**m/(m**3 + 21*m**2 + 143*m + 315) + 35*c**
2*c**m*x**9*x**m/(m**3 + 21*m**2 + 143*m + 315), True))

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Giac [B]  time = 1.15981, size = 190, normalized size = 3.65 \begin{align*} \frac{\left (c x\right )^{m} c^{2} m^{2} x^{9} + 12 \, \left (c x\right )^{m} c^{2} m x^{9} + 2 \, \left (c x\right )^{m} b c m^{2} x^{7} + 35 \, \left (c x\right )^{m} c^{2} x^{9} + 28 \, \left (c x\right )^{m} b c m x^{7} + \left (c x\right )^{m} b^{2} m^{2} x^{5} + 90 \, \left (c x\right )^{m} b c x^{7} + 16 \, \left (c x\right )^{m} b^{2} m x^{5} + 63 \, \left (c x\right )^{m} b^{2} x^{5}}{m^{3} + 21 \, m^{2} + 143 \, m + 315} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*x)^m*c^2*m^2*x^9 + 12*(c*x)^m*c^2*m*x^9 + 2*(c*x)^m*b*c*m^2*x^7 + 35*(c*x)^m*c^2*x^9 + 28*(c*x)^m*b*c*m*x^
7 + (c*x)^m*b^2*m^2*x^5 + 90*(c*x)^m*b*c*x^7 + 16*(c*x)^m*b^2*m*x^5 + 63*(c*x)^m*b^2*x^5)/(m^3 + 21*m^2 + 143*
m + 315)