3.1882 \(\int (a+b x)^m (a c (1+m)+b c (2+m) x)^{-3-m} \, dx\)

Optimal. Leaf size=95 \[ \frac{(a+b x)^{m+1} (a c (m+1)+b c (m+2) x)^{-m-1}}{a^2 b c^2 (m+1) (m+2)}-\frac{(a+b x)^{m+1} (a c (m+1)+b c (m+2) x)^{-m-2}}{a b c (m+2)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.112567, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{(a+b x)^{m+1} (a c (m+1)+b c (m+2) x)^{-m-1}}{a^2 b c^2 (m+1) (m+2)}-\frac{(a+b x)^{m+1} (a c (m+1)+b c (m+2) x)^{-m-2}}{a b c (m+2)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(a*c*(1 + m) + b*c*(2 + m)*x)^(-3 - m),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.543, size = 80, normalized size = 0.84 \[ - \frac{\left (a + b x\right )^{m + 1} \left (a c \left (m + 1\right ) + b c x \left (m + 2\right )\right )^{- m - 2}}{a b c \left (m + 2\right )} + \frac{\left (a + b x\right )^{m + 1} \left (a c \left (m + 1\right ) + b c x \left (m + 2\right )\right )^{- m - 1}}{a^{2} b c^{2} \left (m + 1\right ) \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(a*c*(1+m)+b*c*(2+m)*x)**(-3-m),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.250216, size = 82, normalized size = 0.86 \[ -\frac{(a+b x)^{m+1} \left (-\frac{b (m+2) x}{a}-m-1\right )^m (c (a (m+1)+b (m+2) x))^{-m} \, _2F_1\left (m+1,m+3;m+2;\frac{(m+2) (a+b x)}{a}\right )}{a^3 b c^3 (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^m*(a*c*(1 + m) + b*c*(2 + m)*x)^(-3 - m),x]

[Out]

-(((a + b*x)^(1 + m)*(-1 - m - (b*(2 + m)*x)/a)^m*Hypergeometric2F1[1 + m, 3 + m
, 2 + m, ((2 + m)*(a + b*x))/a])/(a^3*b*c^3*(1 + m)*(c*(a*(1 + m) + b*(2 + m)*x)
)^m))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 57, normalized size = 0.6 \[{\frac{ \left ( bx+a \right ) ^{1+m} \left ( bxm+am+2\,bx+a \right ) x \left ( bcxm+acm+2\,bcx+ac \right ) ^{-3-m}}{{a}^{2} \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(a*c*(1+m)+b*c*(2+m)*x)^(-3-m),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b c{\left (m + 2\right )} x + a c{\left (m + 1\right )}\right )}^{-m - 3}{\left (b x + a\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*(m + 2)*x + a*c*(m + 1))^(-m - 3)*(b*x + a)^m,x, algorithm="maxima")

[Out]

integrate((b*c*(m + 2)*x + a*c*(m + 1))^(-m - 3)*(b*x + a)^m, x)

_______________________________________________________________________________________

Fricas [A]  time = 0.242855, size = 115, normalized size = 1.21 \[ \frac{{\left ({\left (b^{2} m + 2 \, b^{2}\right )} x^{3} +{\left (2 \, a b m + 3 \, a b\right )} x^{2} +{\left (a^{2} m + a^{2}\right )} x\right )}{\left (a c m + a c +{\left (b c m + 2 \, b c\right )} x\right )}^{-m - 3}{\left (b x + a\right )}^{m}}{a^{2} m + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*(m + 2)*x + a*c*(m + 1))^(-m - 3)*(b*x + a)^m,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**m*(a*c*(1+m)+b*c*(2+m)*x)**(-3-m),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b c{\left (m + 2\right )} x + a c{\left (m + 1\right )}\right )}^{-m - 3}{\left (b x + a\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*(m + 2)*x + a*c*(m + 1))^(-m - 3)*(b*x + a)^m,x, algorithm="giac")

[Out]

integrate((b*c*(m + 2)*x + a*c*(m + 1))^(-m - 3)*(b*x + a)^m, x)