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

Optimal. Leaf size=61 \[ \frac{4 a^2 (a+b x)^{m+3}}{b (m+3)}-\frac{4 a (a+b x)^{m+4}}{b (m+4)}+\frac{(a+b x)^{m+5}}{b (m+5)} \]

[Out]

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

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Rubi [A]  time = 0.0857887, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{4 a^2 (a+b x)^{m+3}}{b (m+3)}-\frac{4 a (a+b x)^{m+4}}{b (m+4)}+\frac{(a+b x)^{m+5}}{b (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.2657, size = 48, normalized size = 0.79 \[ \frac{4 a^{2} \left (a + b x\right )^{m + 3}}{b \left (m + 3\right )} - \frac{4 a \left (a + b x\right )^{m + 4}}{b \left (m + 4\right )} + \frac{\left (a + b x\right )^{m + 5}}{b \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0710967, size = 69, normalized size = 1.13 \[ \frac{(a+b x)^{m+3} \left (a^2 \left (m^2+11 m+32\right )-2 a b \left (m^2+9 m+18\right ) x+b^2 \left (m^2+7 m+12\right ) x^2\right )}{b (m+3) (m+4) (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(3 + m)*(a^2*(32 + 11*m + m^2) - 2*a*b*(18 + 9*m + m^2)*x + b^2*(12 +
7*m + m^2)*x^2))/(b*(3 + m)*(4 + m)*(5 + m))

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Maple [A]  time = 0.013, size = 94, normalized size = 1.5 \[{\frac{ \left ( bx+a \right ) ^{3+m} \left ({b}^{2}{m}^{2}{x}^{2}-2\,ab{m}^{2}x+7\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}-18\,abmx+12\,{b}^{2}{x}^{2}+11\,m{a}^{2}-36\,abx+32\,{a}^{2} \right ) }{b \left ({m}^{3}+12\,{m}^{2}+47\,m+60 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x+a)^(3+m)*(b^2*m^2*x^2-2*a*b*m^2*x+7*b^2*m*x^2+a^2*m^2-18*a*b*m*x+12*b^2*x^2
+11*a^2*m-36*a*b*x+32*a^2)/b/(m^3+12*m^2+47*m+60)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228221, size = 234, normalized size = 3.84 \[ \frac{{\left (a^{5} m^{2} + 11 \, a^{5} m +{\left (b^{5} m^{2} + 7 \, b^{5} m + 12 \, b^{5}\right )} x^{5} + 32 \, a^{5} +{\left (a b^{4} m^{2} + 3 \, a b^{4} m\right )} x^{4} - 2 \,{\left (a^{2} b^{3} m^{2} + 11 \, a^{2} b^{3} m + 20 \, a^{2} b^{3}\right )} x^{3} - 2 \,{\left (a^{3} b^{2} m^{2} + 7 \, a^{3} b^{2} m\right )} x^{2} +{\left (a^{4} b m^{2} + 15 \, a^{4} b m + 60 \, a^{4} b\right )} x\right )}{\left (b x + a\right )}^{m}}{b m^{3} + 12 \, b m^{2} + 47 \, b m + 60 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^5*m^2 + 11*a^5*m + (b^5*m^2 + 7*b^5*m + 12*b^5)*x^5 + 32*a^5 + (a*b^4*m^2 + 3
*a*b^4*m)*x^4 - 2*(a^2*b^3*m^2 + 11*a^2*b^3*m + 20*a^2*b^3)*x^3 - 2*(a^3*b^2*m^2
+ 7*a^3*b^2*m)*x^2 + (a^4*b*m^2 + 15*a^4*b*m + 60*a^4*b)*x)*(b*x + a)^m/(b*m^3
+ 12*b*m^2 + 47*b*m + 60*b)

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Sympy [A]  time = 6.50245, size = 894, normalized size = 14.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((a**4*a**m*x, Eq(b, 0)), (a**2*log(a/b + x)/(a**2*b + 2*a*b**2*x + b**
3*x**2) + 2*a**2/(a**2*b + 2*a*b**2*x + b**3*x**2) + 2*a*b*x*log(a/b + x)/(a**2*
b + 2*a*b**2*x + b**3*x**2) + 4*a*b*x/(a**2*b + 2*a*b**2*x + b**3*x**2) + b**2*x
**2*log(a/b + x)/(a**2*b + 2*a*b**2*x + b**3*x**2), Eq(m, -5)), (-12*a**2*log(a/
b + x)/(3*a*b + 3*b**2*x) - 5*a**2/(3*a*b + 3*b**2*x) - 12*a*b*x*log(a/b + x)/(3
*a*b + 3*b**2*x) + 10*a*b*x/(3*a*b + 3*b**2*x) + 3*b**2*x**2/(3*a*b + 3*b**2*x),
Eq(m, -4)), (4*a**2*log(a/b + x)/b - 3*a*x + b*x**2/2, Eq(m, -3)), (a**5*m**2*(
a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + 11*a**5*m*(a + b*x)**m/(b*m**
3 + 12*b*m**2 + 47*b*m + 60*b) + 32*a**5*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b
*m + 60*b) + a**4*b*m**2*x*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + 1
5*a**4*b*m*x*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + 60*a**4*b*x*(a
+ b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) - 2*a**3*b**2*m**2*x**2*(a + b*x)
**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) - 14*a**3*b**2*m*x**2*(a + b*x)**m/(b*m
**3 + 12*b*m**2 + 47*b*m + 60*b) - 2*a**2*b**3*m**2*x**3*(a + b*x)**m/(b*m**3 +
12*b*m**2 + 47*b*m + 60*b) - 22*a**2*b**3*m*x**3*(a + b*x)**m/(b*m**3 + 12*b*m**
2 + 47*b*m + 60*b) - 40*a**2*b**3*x**3*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m
+ 60*b) + a*b**4*m**2*x**4*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) +
3*a*b**4*m*x**4*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + b**5*m**2*x*
*5*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + 7*b**5*m*x**5*(a + b*x)**
m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + 12*b**5*x**5*(a + b*x)**m/(b*m**3 + 12*
b*m**2 + 47*b*m + 60*b), True))

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GIAC/XCAS [A]  time = 0.213981, size = 432, normalized size = 7.08 \[ \frac{b^{5} m^{2} x^{5} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + a b^{4} m^{2} x^{4} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + 7 \, b^{5} m x^{5} e^{\left (m{\rm ln}\left (b x + a\right )\right )} - 2 \, a^{2} b^{3} m^{2} x^{3} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + 3 \, a b^{4} m x^{4} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + 12 \, b^{5} x^{5} e^{\left (m{\rm ln}\left (b x + a\right )\right )} - 2 \, a^{3} b^{2} m^{2} x^{2} e^{\left (m{\rm ln}\left (b x + a\right )\right )} - 22 \, a^{2} b^{3} m x^{3} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + a^{4} b m^{2} x e^{\left (m{\rm ln}\left (b x + a\right )\right )} - 14 \, a^{3} b^{2} m x^{2} e^{\left (m{\rm ln}\left (b x + a\right )\right )} - 40 \, a^{2} b^{3} x^{3} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + a^{5} m^{2} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + 15 \, a^{4} b m x e^{\left (m{\rm ln}\left (b x + a\right )\right )} + 11 \, a^{5} m e^{\left (m{\rm ln}\left (b x + a\right )\right )} + 60 \, a^{4} b x e^{\left (m{\rm ln}\left (b x + a\right )\right )} + 32 \, a^{5} e^{\left (m{\rm ln}\left (b x + a\right )\right )}}{b m^{3} + 12 \, b m^{2} + 47 \, b m + 60 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^5*m^2*x^5*e^(m*ln(b*x + a)) + a*b^4*m^2*x^4*e^(m*ln(b*x + a)) + 7*b^5*m*x^5*e
^(m*ln(b*x + a)) - 2*a^2*b^3*m^2*x^3*e^(m*ln(b*x + a)) + 3*a*b^4*m*x^4*e^(m*ln(b
*x + a)) + 12*b^5*x^5*e^(m*ln(b*x + a)) - 2*a^3*b^2*m^2*x^2*e^(m*ln(b*x + a)) -
22*a^2*b^3*m*x^3*e^(m*ln(b*x + a)) + a^4*b*m^2*x*e^(m*ln(b*x + a)) - 14*a^3*b^2*
m*x^2*e^(m*ln(b*x + a)) - 40*a^2*b^3*x^3*e^(m*ln(b*x + a)) + a^5*m^2*e^(m*ln(b*x
+ a)) + 15*a^4*b*m*x*e^(m*ln(b*x + a)) + 11*a^5*m*e^(m*ln(b*x + a)) + 60*a^4*b*
x*e^(m*ln(b*x + a)) + 32*a^5*e^(m*ln(b*x + a)))/(b*m^3 + 12*b*m^2 + 47*b*m + 60*
b)