3.2758 \(\int (c x)^m (a+b x)^2 \, dx\)

Optimal. Leaf size=58 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b (c x)^{m+2}}{c^2 (m+2)}+\frac{b^2 (c x)^{m+3}}{c^3 (m+3)} \]

[Out]

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

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Rubi [A]  time = 0.0230505, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b (c x)^{m+2}}{c^2 (m+2)}+\frac{b^2 (c x)^{m+3}}{c^3 (m+3)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(a + b*x)^2,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0280721, size = 39, normalized size = 0.67 \[ x (c x)^m \left (\frac{a^2}{m+1}+\frac{2 a b x}{m+2}+\frac{b^2 x^2}{m+3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(a + b*x)^2,x]

[Out]

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

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Maple [A]  time = 0.004, size = 88, normalized size = 1.5 \begin{align*}{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x+3\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}+8\,abmx+2\,{b}^{2}{x}^{2}+5\,m{a}^{2}+6\,xab+6\,{a}^{2} \right ) x \left ( cx \right ) ^{m}}{ \left ( 3+m \right ) \left ( 2+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+a)^2,x)

[Out]

x*(b^2*m^2*x^2+2*a*b*m^2*x+3*b^2*m*x^2+a^2*m^2+8*a*b*m*x+2*b^2*x^2+5*a^2*m+6*a*b*x+6*a^2)*(c*x)^m/(3+m)/(2+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+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33879, size = 184, normalized size = 3.17 \begin{align*} \frac{{\left ({\left (b^{2} m^{2} + 3 \, b^{2} m + 2 \, b^{2}\right )} x^{3} + 2 \,{\left (a b m^{2} + 4 \, a b m + 3 \, a b\right )} x^{2} +{\left (a^{2} m^{2} + 5 \, a^{2} m + 6 \, a^{2}\right )} x\right )} \left (c x\right )^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^2*m^2 + 3*b^2*m + 2*b^2)*x^3 + 2*(a*b*m^2 + 4*a*b*m + 3*a*b)*x^2 + (a^2*m^2 + 5*a^2*m + 6*a^2)*x)*(c*x)^m/
(m^3 + 6*m^2 + 11*m + 6)

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Sympy [A]  time = 0.476777, size = 338, normalized size = 5.83 \begin{align*} \begin{cases} \frac{- \frac{a^{2}}{2 x^{2}} - \frac{2 a b}{x} + b^{2} \log{\left (x \right )}}{c^{3}} & \text{for}\: m = -3 \\\frac{- \frac{a^{2}}{x} + 2 a b \log{\left (x \right )} + b^{2} x}{c^{2}} & \text{for}\: m = -2 \\\frac{a^{2} \log{\left (x \right )} + 2 a b x + \frac{b^{2} x^{2}}{2}}{c} & \text{for}\: m = -1 \\\frac{a^{2} c^{m} m^{2} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{5 a^{2} c^{m} m x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{6 a^{2} c^{m} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{2 a b c^{m} m^{2} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{8 a b c^{m} m x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{6 a b c^{m} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{b^{2} c^{m} m^{2} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{3 b^{2} c^{m} m x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{2 b^{2} c^{m} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-a**2/(2*x**2) - 2*a*b/x + b**2*log(x))/c**3, Eq(m, -3)), ((-a**2/x + 2*a*b*log(x) + b**2*x)/c**2,
Eq(m, -2)), ((a**2*log(x) + 2*a*b*x + b**2*x**2/2)/c, Eq(m, -1)), (a**2*c**m*m**2*x*x**m/(m**3 + 6*m**2 + 11*
m + 6) + 5*a**2*c**m*m*x*x**m/(m**3 + 6*m**2 + 11*m + 6) + 6*a**2*c**m*x*x**m/(m**3 + 6*m**2 + 11*m + 6) + 2*a
*b*c**m*m**2*x**2*x**m/(m**3 + 6*m**2 + 11*m + 6) + 8*a*b*c**m*m*x**2*x**m/(m**3 + 6*m**2 + 11*m + 6) + 6*a*b*
c**m*x**2*x**m/(m**3 + 6*m**2 + 11*m + 6) + b**2*c**m*m**2*x**3*x**m/(m**3 + 6*m**2 + 11*m + 6) + 3*b**2*c**m*
m*x**3*x**m/(m**3 + 6*m**2 + 11*m + 6) + 2*b**2*c**m*x**3*x**m/(m**3 + 6*m**2 + 11*m + 6), True))

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Giac [B]  time = 1.14637, size = 182, normalized size = 3.14 \begin{align*} \frac{\left (c x\right )^{m} b^{2} m^{2} x^{3} + 2 \, \left (c x\right )^{m} a b m^{2} x^{2} + 3 \, \left (c x\right )^{m} b^{2} m x^{3} + \left (c x\right )^{m} a^{2} m^{2} x + 8 \, \left (c x\right )^{m} a b m x^{2} + 2 \, \left (c x\right )^{m} b^{2} x^{3} + 5 \, \left (c x\right )^{m} a^{2} m x + 6 \, \left (c x\right )^{m} a b x^{2} + 6 \, \left (c x\right )^{m} a^{2} x}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*x)^m*b^2*m^2*x^3 + 2*(c*x)^m*a*b*m^2*x^2 + 3*(c*x)^m*b^2*m*x^3 + (c*x)^m*a^2*m^2*x + 8*(c*x)^m*a*b*m*x^2 +
2*(c*x)^m*b^2*x^3 + 5*(c*x)^m*a^2*m*x + 6*(c*x)^m*a*b*x^2 + 6*(c*x)^m*a^2*x)/(m^3 + 6*m^2 + 11*m + 6)