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

Optimal. Leaf size=34 \[ \frac{b (c x)^{m+3}}{c^3 (m+3)}+\frac{(c x)^{m+5}}{c^4 (m+5)} \]

[Out]

(b*(c*x)^(3 + m))/(c^3*(3 + m)) + (c*x)^(5 + m)/(c^4*(5 + m))

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Rubi [A]  time = 0.014332, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {14} \[ \frac{b (c x)^{m+3}}{c^3 (m+3)}+\frac{(c x)^{m+5}}{c^4 (m+5)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(b*x^2 + c*x^4),x]

[Out]

(b*(c*x)^(3 + m))/(c^3*(3 + m)) + (c*x)^(5 + m)/(c^4*(5 + m))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int (c x)^m \left (b x^2+c x^4\right ) \, dx &=\int \left (\frac{b (c x)^{2+m}}{c^2}+\frac{(c x)^{4+m}}{c^3}\right ) \, dx\\ &=\frac{b (c x)^{3+m}}{c^3 (3+m)}+\frac{(c x)^{5+m}}{c^4 (5+m)}\\ \end{align*}

Mathematica [A]  time = 0.016443, size = 27, normalized size = 0.79 \[ x^3 (c x)^m \left (\frac{b}{m+3}+\frac{c x^2}{m+5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(b*x^2 + c*x^4),x]

[Out]

x^3*(c*x)^m*(b/(3 + m) + (c*x^2)/(5 + m))

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Maple [A]  time = 0.044, size = 39, normalized size = 1.2 \begin{align*}{\frac{ \left ( cx \right ) ^{m} \left ( cm{x}^{2}+3\,c{x}^{2}+bm+5\,b \right ){x}^{3}}{ \left ( 5+m \right ) \left ( 3+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),x)

[Out]

(c*x)^m*(c*m*x^2+3*c*x^2+b*m+5*b)*x^3/(5+m)/(3+m)

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Maxima [A]  time = 1.0151, size = 46, normalized size = 1.35 \begin{align*} \frac{c^{m + 1} x^{5} x^{m}}{m + 5} + \frac{b c^{m} x^{3} x^{m}}{m + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^(m + 1)*x^5*x^m/(m + 5) + b*c^m*x^3*x^m/(m + 3)

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Fricas [A]  time = 1.57861, size = 84, normalized size = 2.47 \begin{align*} \frac{{\left ({\left (c m + 3 \, c\right )} x^{5} +{\left (b m + 5 \, b\right )} x^{3}\right )} \left (c x\right )^{m}}{m^{2} + 8 \, m + 15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*m + 3*c)*x^5 + (b*m + 5*b)*x^3)*(c*x)^m/(m^2 + 8*m + 15)

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Sympy [A]  time = 0.711242, size = 119, normalized size = 3.5 \begin{align*} \begin{cases} \frac{- \frac{b}{2 x^{2}} + c \log{\left (x \right )}}{c^{5}} & \text{for}\: m = -5 \\\frac{b \log{\left (x \right )} + \frac{c x^{2}}{2}}{c^{3}} & \text{for}\: m = -3 \\\frac{b c^{m} m x^{3} x^{m}}{m^{2} + 8 m + 15} + \frac{5 b c^{m} x^{3} x^{m}}{m^{2} + 8 m + 15} + \frac{c c^{m} m x^{5} x^{m}}{m^{2} + 8 m + 15} + \frac{3 c c^{m} x^{5} x^{m}}{m^{2} + 8 m + 15} & \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),x)

[Out]

Piecewise(((-b/(2*x**2) + c*log(x))/c**5, Eq(m, -5)), ((b*log(x) + c*x**2/2)/c**3, Eq(m, -3)), (b*c**m*m*x**3*
x**m/(m**2 + 8*m + 15) + 5*b*c**m*x**3*x**m/(m**2 + 8*m + 15) + c*c**m*m*x**5*x**m/(m**2 + 8*m + 15) + 3*c*c**
m*x**5*x**m/(m**2 + 8*m + 15), True))

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Giac [A]  time = 1.21245, size = 76, normalized size = 2.24 \begin{align*} \frac{\left (c x\right )^{m} c m x^{5} + 3 \, \left (c x\right )^{m} c x^{5} + \left (c x\right )^{m} b m x^{3} + 5 \, \left (c x\right )^{m} b x^{3}}{m^{2} + 8 \, m + 15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*x)^m*c*m*x^5 + 3*(c*x)^m*c*x^5 + (c*x)^m*b*m*x^3 + 5*(c*x)^m*b*x^3)/(m^2 + 8*m + 15)