3.936 \(\int (a+b x)^m \left (a^2-b^2 x^2\right ) \, dx\)

Optimal. Leaf size=40 \[ \frac{2 a (a+b x)^{m+2}}{b (m+2)}-\frac{(a+b x)^{m+3}}{b (m+3)} \]

[Out]

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

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Rubi [A]  time = 0.0558223, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 a (a+b x)^{m+2}}{b (m+2)}-\frac{(a+b x)^{m+3}}{b (m+3)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(a^2 - b^2*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.776, size = 29, normalized size = 0.72 \[ \frac{2 a \left (a + b x\right )^{m + 2}}{b \left (m + 2\right )} - \frac{\left (a + b x\right )^{m + 3}}{b \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*(a + b*x)**(m + 2)/(b*(m + 2)) - (a + b*x)**(m + 3)/(b*(m + 3))

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Mathematica [A]  time = 0.0401319, size = 36, normalized size = 0.9 \[ \frac{(a+b x)^{m+2} (a (m+4)-b (m+2) x)}{b (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(2 + m)*(a*(4 + m) - b*(2 + m)*x))/(b*(2 + m)*(3 + m))

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Maple [A]  time = 0.004, size = 40, normalized size = 1. \[{\frac{ \left ( bx+a \right ) ^{2+m} \left ( -bmx+am-2\,bx+4\,a \right ) }{b \left ({m}^{2}+5\,m+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(-b^2*x^2+a^2),x)

[Out]

(b*x+a)^(2+m)*(-b*m*x+a*m-2*b*x+4*a)/b/(m^2+5*m+6)

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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^2*x^2 - a^2)*(b*x + a)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232417, size = 103, normalized size = 2.58 \[ -\frac{{\left (a b^{2} m x^{2} - a^{3} m +{\left (b^{3} m + 2 \, b^{3}\right )} x^{3} - 4 \, a^{3} -{\left (a^{2} b m + 6 \, a^{2} b\right )} x\right )}{\left (b x + a\right )}^{m}}{b m^{2} + 5 \, b m + 6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.5727, size = 275, normalized size = 6.88 \[ \begin{cases} a^{2} a^{m} x & \text{for}\: b = 0 \\- \frac{a \log{\left (\frac{a}{b} + x \right )}}{a b + b^{2} x} - \frac{a}{a b + b^{2} x} - \frac{b x \log{\left (\frac{a}{b} + x \right )}}{a b + b^{2} x} + \frac{b x}{a b + b^{2} x} & \text{for}\: m = -3 \\\frac{2 a \log{\left (\frac{a}{b} + x \right )}}{b} - x & \text{for}\: m = -2 \\\frac{a^{3} m \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} + \frac{4 a^{3} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} + \frac{a^{2} b m x \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} + \frac{6 a^{2} b x \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} - \frac{a b^{2} m x^{2} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} - \frac{b^{3} m x^{3} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} - \frac{2 b^{3} x^{3} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**m*(-b**2*x**2+a**2),x)

[Out]

Piecewise((a**2*a**m*x, Eq(b, 0)), (-a*log(a/b + x)/(a*b + b**2*x) - a/(a*b + b*
*2*x) - b*x*log(a/b + x)/(a*b + b**2*x) + b*x/(a*b + b**2*x), Eq(m, -3)), (2*a*l
og(a/b + x)/b - x, Eq(m, -2)), (a**3*m*(a + b*x)**m/(b*m**2 + 5*b*m + 6*b) + 4*a
**3*(a + b*x)**m/(b*m**2 + 5*b*m + 6*b) + a**2*b*m*x*(a + b*x)**m/(b*m**2 + 5*b*
m + 6*b) + 6*a**2*b*x*(a + b*x)**m/(b*m**2 + 5*b*m + 6*b) - a*b**2*m*x**2*(a + b
*x)**m/(b*m**2 + 5*b*m + 6*b) - b**3*m*x**3*(a + b*x)**m/(b*m**2 + 5*b*m + 6*b)
- 2*b**3*x**3*(a + b*x)**m/(b*m**2 + 5*b*m + 6*b), True))

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GIAC/XCAS [A]  time = 0.212177, size = 178, normalized size = 4.45 \[ -\frac{b^{3} m x^{3} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + a b^{2} m x^{2} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + 2 \, b^{3} x^{3} e^{\left (m{\rm ln}\left (b x + a\right )\right )} - a^{2} b m x e^{\left (m{\rm ln}\left (b x + a\right )\right )} - a^{3} m e^{\left (m{\rm ln}\left (b x + a\right )\right )} - 6 \, a^{2} b x e^{\left (m{\rm ln}\left (b x + a\right )\right )} - 4 \, a^{3} e^{\left (m{\rm ln}\left (b x + a\right )\right )}}{b m^{2} + 5 \, b m + 6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b^3*m*x^3*e^(m*ln(b*x + a)) + a*b^2*m*x^2*e^(m*ln(b*x + a)) + 2*b^3*x^3*e^(m*l
n(b*x + a)) - a^2*b*m*x*e^(m*ln(b*x + a)) - a^3*m*e^(m*ln(b*x + a)) - 6*a^2*b*x*
e^(m*ln(b*x + a)) - 4*a^3*e^(m*ln(b*x + a)))/(b*m^2 + 5*b*m + 6*b)