3.2164 \(\int \frac{(A+B x) (a c+b c x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=64 \[ -\frac{c^5 (A b-a B) (a c+b c x)^{m-5}}{b^2 (5-m)}-\frac{B c^4 (a c+b c x)^{m-4}}{b^2 (4-m)} \]

[Out]

-(((A*b - a*B)*c^5*(a*c + b*c*x)^(-5 + m))/(b^2*(5 - m))) - (B*c^4*(a*c + b*c*x)
^(-4 + m))/(b^2*(4 - m))

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Rubi [A]  time = 0.0994098, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088 \[ -\frac{c^5 (A b-a B) (a c+b c x)^{m-5}}{b^2 (5-m)}-\frac{B c^4 (a c+b c x)^{m-4}}{b^2 (4-m)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a*c + b*c*x)^m)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(((A*b - a*B)*c^5*(a*c + b*c*x)^(-5 + m))/(b^2*(5 - m))) - (B*c^4*(a*c + b*c*x)
^(-4 + m))/(b^2*(4 - m))

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Rubi in Sympy [A]  time = 37.1799, size = 53, normalized size = 0.83 \[ - \frac{B c^{4} \left (a c + b c x\right )^{m - 4}}{b^{2} \left (- m + 4\right )} - \frac{c^{5} \left (A b - B a\right ) \left (a c + b c x\right )^{m - 5}}{b^{2} \left (- m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*c*x+a*c)**m/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-B*c**4*(a*c + b*c*x)**(m - 4)/(b**2*(-m + 4)) - c**5*(A*b - B*a)*(a*c + b*c*x)*
*(m - 5)/(b**2*(-m + 5))

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Mathematica [A]  time = 0.0597421, size = 48, normalized size = 0.75 \[ \frac{(c (a+b x))^m (-a B+A b (m-4)+b B (m-5) x)}{b^2 (m-5) (m-4) (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a*c + b*c*x)^m)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((c*(a + b*x))^m*(-(a*B) + A*b*(-4 + m) + b*B*(-5 + m)*x))/(b^2*(-5 + m)*(-4 + m
)*(a + b*x)^5)

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Maple [A]  time = 0.004, size = 73, normalized size = 1.1 \[{\frac{ \left ( Bbmx+Abm-5\,Bbx-4\,Ab-Ba \right ) \left ( bxc+ac \right ) ^{m}}{ \left ( bx+a \right ) \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{2}{b}^{2} \left ({m}^{2}-9\,m+20 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*c*x+a*c)^m/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

(B*b*m*x+A*b*m-5*B*b*x-4*A*b-B*a)*(b*c*x+a*c)^m/(b*x+a)/(b^2*x^2+2*a*b*x+a^2)^2/
b^2/(m^2-9*m+20)

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Maxima [A]  time = 0.751635, size = 292, normalized size = 4.56 \[ \frac{{\left (b c^{m}{\left (m - 5\right )} x - a c^{m}\right )}{\left (b x + a\right )}^{m} B}{{\left (m^{2} - 9 \, m + 20\right )} b^{7} x^{5} + 5 \,{\left (m^{2} - 9 \, m + 20\right )} a b^{6} x^{4} + 10 \,{\left (m^{2} - 9 \, m + 20\right )} a^{2} b^{5} x^{3} + 10 \,{\left (m^{2} - 9 \, m + 20\right )} a^{3} b^{4} x^{2} + 5 \,{\left (m^{2} - 9 \, m + 20\right )} a^{4} b^{3} x +{\left (m^{2} - 9 \, m + 20\right )} a^{5} b^{2}} + \frac{{\left (b x + a\right )}^{m} A c^{m}}{b^{6}{\left (m - 5\right )} x^{5} + 5 \, a b^{5}{\left (m - 5\right )} x^{4} + 10 \, a^{2} b^{4}{\left (m - 5\right )} x^{3} + 10 \, a^{3} b^{3}{\left (m - 5\right )} x^{2} + 5 \, a^{4} b^{2}{\left (m - 5\right )} x + a^{5} b{\left (m - 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*c*x + a*c)^m/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

(b*c^m*(m - 5)*x - a*c^m)*(b*x + a)^m*B/((m^2 - 9*m + 20)*b^7*x^5 + 5*(m^2 - 9*m
+ 20)*a*b^6*x^4 + 10*(m^2 - 9*m + 20)*a^2*b^5*x^3 + 10*(m^2 - 9*m + 20)*a^3*b^4
*x^2 + 5*(m^2 - 9*m + 20)*a^4*b^3*x + (m^2 - 9*m + 20)*a^5*b^2) + (b*x + a)^m*A*
c^m/(b^6*(m - 5)*x^5 + 5*a*b^5*(m - 5)*x^4 + 10*a^2*b^4*(m - 5)*x^3 + 10*a^3*b^3
*(m - 5)*x^2 + 5*a^4*b^2*(m - 5)*x + a^5*b*(m - 5))

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Fricas [A]  time = 0.302228, size = 286, normalized size = 4.47 \[ \frac{{\left (A b m - B a - 4 \, A b +{\left (B b m - 5 \, B b\right )} x\right )}{\left (b c x + a c\right )}^{m}}{a^{5} b^{2} m^{2} - 9 \, a^{5} b^{2} m + 20 \, a^{5} b^{2} +{\left (b^{7} m^{2} - 9 \, b^{7} m + 20 \, b^{7}\right )} x^{5} + 5 \,{\left (a b^{6} m^{2} - 9 \, a b^{6} m + 20 \, a b^{6}\right )} x^{4} + 10 \,{\left (a^{2} b^{5} m^{2} - 9 \, a^{2} b^{5} m + 20 \, a^{2} b^{5}\right )} x^{3} + 10 \,{\left (a^{3} b^{4} m^{2} - 9 \, a^{3} b^{4} m + 20 \, a^{3} b^{4}\right )} x^{2} + 5 \,{\left (a^{4} b^{3} m^{2} - 9 \, a^{4} b^{3} m + 20 \, a^{4} b^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*c*x + a*c)^m/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

(A*b*m - B*a - 4*A*b + (B*b*m - 5*B*b)*x)*(b*c*x + a*c)^m/(a^5*b^2*m^2 - 9*a^5*b
^2*m + 20*a^5*b^2 + (b^7*m^2 - 9*b^7*m + 20*b^7)*x^5 + 5*(a*b^6*m^2 - 9*a*b^6*m
+ 20*a*b^6)*x^4 + 10*(a^2*b^5*m^2 - 9*a^2*b^5*m + 20*a^2*b^5)*x^3 + 10*(a^3*b^4*
m^2 - 9*a^3*b^4*m + 20*a^3*b^4)*x^2 + 5*(a^4*b^3*m^2 - 9*a^4*b^3*m + 20*a^4*b^3)
*x)

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Sympy [A]  time = 15.9542, size = 1268, normalized size = 19.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b*c*x+a*c)**m/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Piecewise(((a*c)**m*(A*x + B*x**2/2)/a**6, Eq(b, 0)), (-A*b*c**4/(a*b**2 + b**3*
x) + B*a*c**4*log(a/b + x)/(a*b**2 + b**3*x) + B*a*c**4/(a*b**2 + b**3*x) + B*b*
c**4*x*log(a/b + x)/(a*b**2 + b**3*x), Eq(m, 4)), (A*c**5*log(a/b + x)/b - B*a*c
**5*log(a/b + x)/b**2 + B*c**5*x/b, Eq(m, 5)), (A*b*m*(a*c + b*c*x)**m/(a**5*b**
2*m**2 - 9*a**5*b**2*m + 20*a**5*b**2 + 5*a**4*b**3*m**2*x - 45*a**4*b**3*m*x +
100*a**4*b**3*x + 10*a**3*b**4*m**2*x**2 - 90*a**3*b**4*m*x**2 + 200*a**3*b**4*x
**2 + 10*a**2*b**5*m**2*x**3 - 90*a**2*b**5*m*x**3 + 200*a**2*b**5*x**3 + 5*a*b*
*6*m**2*x**4 - 45*a*b**6*m*x**4 + 100*a*b**6*x**4 + b**7*m**2*x**5 - 9*b**7*m*x*
*5 + 20*b**7*x**5) - 4*A*b*(a*c + b*c*x)**m/(a**5*b**2*m**2 - 9*a**5*b**2*m + 20
*a**5*b**2 + 5*a**4*b**3*m**2*x - 45*a**4*b**3*m*x + 100*a**4*b**3*x + 10*a**3*b
**4*m**2*x**2 - 90*a**3*b**4*m*x**2 + 200*a**3*b**4*x**2 + 10*a**2*b**5*m**2*x**
3 - 90*a**2*b**5*m*x**3 + 200*a**2*b**5*x**3 + 5*a*b**6*m**2*x**4 - 45*a*b**6*m*
x**4 + 100*a*b**6*x**4 + b**7*m**2*x**5 - 9*b**7*m*x**5 + 20*b**7*x**5) - B*a*(a
*c + b*c*x)**m/(a**5*b**2*m**2 - 9*a**5*b**2*m + 20*a**5*b**2 + 5*a**4*b**3*m**2
*x - 45*a**4*b**3*m*x + 100*a**4*b**3*x + 10*a**3*b**4*m**2*x**2 - 90*a**3*b**4*
m*x**2 + 200*a**3*b**4*x**2 + 10*a**2*b**5*m**2*x**3 - 90*a**2*b**5*m*x**3 + 200
*a**2*b**5*x**3 + 5*a*b**6*m**2*x**4 - 45*a*b**6*m*x**4 + 100*a*b**6*x**4 + b**7
*m**2*x**5 - 9*b**7*m*x**5 + 20*b**7*x**5) + B*b*m*x*(a*c + b*c*x)**m/(a**5*b**2
*m**2 - 9*a**5*b**2*m + 20*a**5*b**2 + 5*a**4*b**3*m**2*x - 45*a**4*b**3*m*x + 1
00*a**4*b**3*x + 10*a**3*b**4*m**2*x**2 - 90*a**3*b**4*m*x**2 + 200*a**3*b**4*x*
*2 + 10*a**2*b**5*m**2*x**3 - 90*a**2*b**5*m*x**3 + 200*a**2*b**5*x**3 + 5*a*b**
6*m**2*x**4 - 45*a*b**6*m*x**4 + 100*a*b**6*x**4 + b**7*m**2*x**5 - 9*b**7*m*x**
5 + 20*b**7*x**5) - 5*B*b*x*(a*c + b*c*x)**m/(a**5*b**2*m**2 - 9*a**5*b**2*m + 2
0*a**5*b**2 + 5*a**4*b**3*m**2*x - 45*a**4*b**3*m*x + 100*a**4*b**3*x + 10*a**3*
b**4*m**2*x**2 - 90*a**3*b**4*m*x**2 + 200*a**3*b**4*x**2 + 10*a**2*b**5*m**2*x*
*3 - 90*a**2*b**5*m*x**3 + 200*a**2*b**5*x**3 + 5*a*b**6*m**2*x**4 - 45*a*b**6*m
*x**4 + 100*a*b**6*x**4 + b**7*m**2*x**5 - 9*b**7*m*x**5 + 20*b**7*x**5), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (b c x + a c\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*c*x + a*c)^m/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

integrate((B*x + A)*(b*c*x + a*c)^m/(b^2*x^2 + 2*a*b*x + a^2)^3, x)