3.2751 \(\int (c x)^m (a+b x)^2 \, dx\)

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 = 0.0698293, 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 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b (c x)^{m+2}}{c^2 (m+2)}+\frac{b^2 (c x)^{m+3}}{c^3 (m+3)} \]

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))

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Rubi in Sympy [A]  time = 12.7376, size = 49, normalized size = 0.84 \[ \frac{a^{2} \left (c x\right )^{m + 1}}{c \left (m + 1\right )} + \frac{2 a b \left (c x\right )^{m + 2}}{c^{2} \left (m + 2\right )} + \frac{b^{2} \left (c x\right )^{m + 3}}{c^{3} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**m*(b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0284321, size = 41, normalized size = 0.71 \[ (c x)^m \left (\frac{a^2 x}{m+1}+\frac{2 a b x^2}{m+2}+\frac{b^2 x^3}{m+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 88, normalized size = 1.5 \[{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x+3\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}+8\,abmx+2\,{b}^{2}{x}^{2}+5\,m{a}^{2}+6\,abx+6\,{a}^{2} \right ) x \left ( cx \right ) ^{m}}{ \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228486, size = 117, normalized size = 2.02 \[ \frac{{\left ({\left (b^{2} m^{2} + 3 \, b^{2} m + 2 \, b^{2}\right )} x^{3} + 2 \,{\left (a b m^{2} + 4 \, a b m + 3 \, a b\right )} x^{2} +{\left (a^{2} m^{2} + 5 \, a^{2} m + 6 \, a^{2}\right )} x\right )} \left (c x\right )^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*(c*x)^m,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 = 1.46635, size = 338, normalized size = 5.83 \[ \begin{cases} \frac{- \frac{a^{2}}{2 x^{2}} - \frac{2 a b}{x} + b^{2} \log{\left (x \right )}}{c^{3}} & \text{for}\: m = -3 \\\frac{- \frac{a^{2}}{x} + 2 a b \log{\left (x \right )} + b^{2} x}{c^{2}} & \text{for}\: m = -2 \\\frac{a^{2} \log{\left (x \right )} + 2 a b x + \frac{b^{2} x^{2}}{2}}{c} & \text{for}\: m = -1 \\\frac{a^{2} c^{m} m^{2} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{5 a^{2} c^{m} m x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{6 a^{2} c^{m} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{2 a b c^{m} m^{2} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{8 a b c^{m} m x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{6 a b c^{m} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{b^{2} c^{m} m^{2} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{3 b^{2} c^{m} m x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{2 b^{2} c^{m} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.22003, size = 207, normalized size = 3.57 \[ \frac{b^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (c x\right )\right )} + 2 \, a b m^{2} x^{2} e^{\left (m{\rm ln}\left (c x\right )\right )} + 3 \, b^{2} m x^{3} e^{\left (m{\rm ln}\left (c x\right )\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (c x\right )\right )} + 8 \, a b m x^{2} e^{\left (m{\rm ln}\left (c x\right )\right )} + 2 \, b^{2} x^{3} e^{\left (m{\rm ln}\left (c x\right )\right )} + 5 \, a^{2} m x e^{\left (m{\rm ln}\left (c x\right )\right )} + 6 \, a b x^{2} e^{\left (m{\rm ln}\left (c x\right )\right )} + 6 \, a^{2} x e^{\left (m{\rm ln}\left (c x\right )\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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