3.427 \(\int \left (a+\frac{b}{x}\right )^m (c+d x)^2 \, dx\)

Optimal. Leaf size=138 \[ \frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1} (6 a c-b d (2-m))}{6 a^2}-\frac{b \left (a+\frac{b}{x}\right )^{m+1} \left (6 a^2 c^2-6 a b c d (1-m)+b^2 d^2 \left (m^2-3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{6 a^4 (m+1)}+\frac{d^2 x^3 \left (a+\frac{b}{x}\right )^{m+1}}{3 a} \]

[Out]

(d*(6*a*c - b*d*(2 - m))*(a + b/x)^(1 + m)*x^2)/(6*a^2) + (d^2*(a + b/x)^(1 + m)
*x^3)/(3*a) - (b*(6*a^2*c^2 - 6*a*b*c*d*(1 - m) + b^2*d^2*(2 - 3*m + m^2))*(a +
b/x)^(1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m, 1 + b/(a*x)])/(6*a^4*(1 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.28323, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1} (6 a c-b d (2-m))}{6 a^2}-\frac{b \left (a+\frac{b}{x}\right )^{m+1} \left (6 a^2 c^2-6 a b c d (1-m)+b^2 d^2 \left (m^2-3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{6 a^4 (m+1)}+\frac{d^2 x^3 \left (a+\frac{b}{x}\right )^{m+1}}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^m*(c + d*x)^2,x]

[Out]

(d*(6*a*c - b*d*(2 - m))*(a + b/x)^(1 + m)*x^2)/(6*a^2) + (d^2*(a + b/x)^(1 + m)
*x^3)/(3*a) - (b*(6*a^2*c^2 - 6*a*b*c*d*(1 - m) + b^2*d^2*(2 - 3*m + m^2))*(a +
b/x)^(1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m, 1 + b/(a*x)])/(6*a^4*(1 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.7217, size = 105, normalized size = 0.76 \[ \frac{d^{2} x^{3} \left (a + \frac{b}{x}\right )^{m + 1}}{3 a} + \frac{d x^{2} \left (a + \frac{b}{x}\right )^{m + 1} \left (6 a c - b d \left (- m + 2\right )\right )}{6 a^{2}} - \frac{b \left (a + \frac{b}{x}\right )^{m + 1} \left (6 a^{2} c^{2} - b d \left (- m + 1\right ) \left (6 a c - b d \left (- m + 2\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{1 + \frac{b}{a x}} \right )}}{6 a^{4} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*x**3*(a + b/x)**(m + 1)/(3*a) + d*x**2*(a + b/x)**(m + 1)*(6*a*c - b*d*(-m
+ 2))/(6*a**2) - b*(a + b/x)**(m + 1)*(6*a**2*c**2 - b*d*(-m + 1)*(6*a*c - b*d*(
-m + 2)))*hyper((2, m + 1), (m + 2,), 1 + b/(a*x))/(6*a**4*(m + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.131228, size = 134, normalized size = 0.97 \[ -\frac{x \left (a+\frac{b}{x}\right )^m \left (\frac{a x}{b}+1\right )^{-m} \left (c^2 \left (m^2-5 m+6\right ) \, _2F_1\left (1-m,-m;2-m;-\frac{a x}{b}\right )+d (m-1) x \left (2 c (m-3) \, _2F_1\left (2-m,-m;3-m;-\frac{a x}{b}\right )+d (m-2) x \, _2F_1\left (3-m,-m;4-m;-\frac{a x}{b}\right )\right )\right )}{(m-3) (m-2) (m-1)} \]

Antiderivative was successfully verified.

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

[Out]

-(((a + b/x)^m*x*(c^2*(6 - 5*m + m^2)*Hypergeometric2F1[1 - m, -m, 2 - m, -((a*x
)/b)] + d*(-1 + m)*x*(2*c*(-3 + m)*Hypergeometric2F1[2 - m, -m, 3 - m, -((a*x)/b
)] + d*(-2 + m)*x*Hypergeometric2F1[3 - m, -m, 4 - m, -((a*x)/b)])))/((-3 + m)*(
-2 + m)*(-1 + m)*(1 + (a*x)/b)^m))

_______________________________________________________________________________________

Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{x}} \right ) ^{m} \left ( dx+c \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^m*(d*x+c)^2,x)

[Out]

int((a+b/x)^m*(d*x+c)^2,x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^2*(a + b/x)^m, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (\frac{a x + b}{x}\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d^2*x^2 + 2*c*d*x + c^2)*((a*x + b)/x)^m, x)

_______________________________________________________________________________________

Sympy [A]  time = 16.921, size = 121, normalized size = 0.88 \[ \frac{b^{m} c^{2} x x^{- m} \Gamma \left (- m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 1 \\ - m + 2 \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (- m + 2\right )} + \frac{2 b^{m} c d x^{2} x^{- m} \Gamma \left (- m + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 2 \\ - m + 3 \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (- m + 3\right )} + \frac{b^{m} d^{2} x^{3} x^{- m} \Gamma \left (- m + 3\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 3 \\ - m + 4 \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (- m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**m*c**2*x*x**(-m)*gamma(-m + 1)*hyper((-m, -m + 1), (-m + 2,), a*x*exp_polar(I
*pi)/b)/gamma(-m + 2) + 2*b**m*c*d*x**2*x**(-m)*gamma(-m + 2)*hyper((-m, -m + 2)
, (-m + 3,), a*x*exp_polar(I*pi)/b)/gamma(-m + 3) + b**m*d**2*x**3*x**(-m)*gamma
(-m + 3)*hyper((-m, -m + 3), (-m + 4,), a*x*exp_polar(I*pi)/b)/gamma(-m + 4)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^2*(a + b/x)^m, x)