3.2756 \(\int (c x)^m (a+b x^3)^2 \, dx\)

Optimal. Leaf size=58 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b (c x)^{m+4}}{c^4 (m+4)}+\frac{b^2 (c x)^{m+7}}{c^7 (m+7)} \]

[Out]

(a^2*(c*x)^(1 + m))/(c*(1 + m)) + (2*a*b*(c*x)^(4 + m))/(c^4*(4 + m)) + (b^2*(c*x)^(7 + m))/(c^7*(7 + m))

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Rubi [A]  time = 0.0298515, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b (c x)^{m+4}}{c^4 (m+4)}+\frac{b^2 (c x)^{m+7}}{c^7 (m+7)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(a + b*x^3)^2,x]

[Out]

(a^2*(c*x)^(1 + m))/(c*(1 + m)) + (2*a*b*(c*x)^(4 + m))/(c^4*(4 + m)) + (b^2*(c*x)^(7 + m))/(c^7*(7 + m))

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 (a+b x^3\right )^2 \, dx &=\int \left (a^2 (c x)^m+\frac{2 a b (c x)^{3+m}}{c^3}+\frac{b^2 (c x)^{6+m}}{c^6}\right ) \, dx\\ &=\frac{a^2 (c x)^{1+m}}{c (1+m)}+\frac{2 a b (c x)^{4+m}}{c^4 (4+m)}+\frac{b^2 (c x)^{7+m}}{c^7 (7+m)}\\ \end{align*}

Mathematica [A]  time = 0.0283439, size = 41, normalized size = 0.71 \[ x (c x)^m \left (\frac{a^2}{m+1}+\frac{2 a b x^3}{m+4}+\frac{b^2 x^6}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(a + b*x^3)^2,x]

[Out]

x*(c*x)^m*(a^2/(1 + m) + (2*a*b*x^3)/(4 + m) + (b^2*x^6)/(7 + m))

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Maple [A]  time = 0.006, size = 94, normalized size = 1.6 \begin{align*}{\frac{ \left ({b}^{2}{m}^{2}{x}^{6}+5\,{b}^{2}m{x}^{6}+4\,{b}^{2}{x}^{6}+2\,ab{m}^{2}{x}^{3}+16\,abm{x}^{3}+14\,{x}^{3}ab+{a}^{2}{m}^{2}+11\,m{a}^{2}+28\,{a}^{2} \right ) x \left ( cx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(b*x^3+a)^2,x)

[Out]

x*(b^2*m^2*x^6+5*b^2*m*x^6+4*b^2*x^6+2*a*b*m^2*x^3+16*a*b*m*x^3+14*a*b*x^3+a^2*m^2+11*a^2*m+28*a^2)*(c*x)^m/(7
+m)/(4+m)/(1+m)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2982, size = 189, normalized size = 3.26 \begin{align*} \frac{{\left ({\left (b^{2} m^{2} + 5 \, b^{2} m + 4 \, b^{2}\right )} x^{7} + 2 \,{\left (a b m^{2} + 8 \, a b m + 7 \, a b\right )} x^{4} +{\left (a^{2} m^{2} + 11 \, a^{2} m + 28 \, a^{2}\right )} x\right )} \left (c x\right )^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^2*m^2 + 5*b^2*m + 4*b^2)*x^7 + 2*(a*b*m^2 + 8*a*b*m + 7*a*b)*x^4 + (a^2*m^2 + 11*a^2*m + 28*a^2)*x)*(c*x)^
m/(m^3 + 12*m^2 + 39*m + 28)

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Sympy [A]  time = 1.31026, size = 352, normalized size = 6.07 \begin{align*} \begin{cases} \frac{- \frac{a^{2}}{6 x^{6}} - \frac{2 a b}{3 x^{3}} + b^{2} \log{\left (x \right )}}{c^{7}} & \text{for}\: m = -7 \\\frac{- \frac{a^{2}}{3 x^{3}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{3}}{3}}{c^{4}} & \text{for}\: m = -4 \\\frac{a^{2} \log{\left (x \right )} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{6}}{6}}{c} & \text{for}\: m = -1 \\\frac{a^{2} c^{m} m^{2} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{11 a^{2} c^{m} m x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{28 a^{2} c^{m} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{2 a b c^{m} m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{16 a b c^{m} m x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{14 a b c^{m} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{b^{2} c^{m} m^{2} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{5 b^{2} c^{m} m x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{4 b^{2} c^{m} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(b*x**3+a)**2,x)

[Out]

Piecewise(((-a**2/(6*x**6) - 2*a*b/(3*x**3) + b**2*log(x))/c**7, Eq(m, -7)), ((-a**2/(3*x**3) + 2*a*b*log(x) +
b**2*x**3/3)/c**4, Eq(m, -4)), ((a**2*log(x) + 2*a*b*x**3/3 + b**2*x**6/6)/c, Eq(m, -1)), (a**2*c**m*m**2*x*x
**m/(m**3 + 12*m**2 + 39*m + 28) + 11*a**2*c**m*m*x*x**m/(m**3 + 12*m**2 + 39*m + 28) + 28*a**2*c**m*x*x**m/(m
**3 + 12*m**2 + 39*m + 28) + 2*a*b*c**m*m**2*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 16*a*b*c**m*m*x**4*x**m/
(m**3 + 12*m**2 + 39*m + 28) + 14*a*b*c**m*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + b**2*c**m*m**2*x**7*x**m/(
m**3 + 12*m**2 + 39*m + 28) + 5*b**2*c**m*m*x**7*x**m/(m**3 + 12*m**2 + 39*m + 28) + 4*b**2*c**m*x**7*x**m/(m*
*3 + 12*m**2 + 39*m + 28), True))

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Giac [B]  time = 1.14797, size = 182, normalized size = 3.14 \begin{align*} \frac{\left (c x\right )^{m} b^{2} m^{2} x^{7} + 5 \, \left (c x\right )^{m} b^{2} m x^{7} + 4 \, \left (c x\right )^{m} b^{2} x^{7} + 2 \, \left (c x\right )^{m} a b m^{2} x^{4} + 16 \, \left (c x\right )^{m} a b m x^{4} + 14 \, \left (c x\right )^{m} a b x^{4} + \left (c x\right )^{m} a^{2} m^{2} x + 11 \, \left (c x\right )^{m} a^{2} m x + 28 \, \left (c x\right )^{m} a^{2} x}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*x)^m*b^2*m^2*x^7 + 5*(c*x)^m*b^2*m*x^7 + 4*(c*x)^m*b^2*x^7 + 2*(c*x)^m*a*b*m^2*x^4 + 16*(c*x)^m*a*b*m*x^4
+ 14*(c*x)^m*a*b*x^4 + (c*x)^m*a^2*m^2*x + 11*(c*x)^m*a^2*m*x + 28*(c*x)^m*a^2*x)/(m^3 + 12*m^2 + 39*m + 28)