3.1850 \(\int \frac{(a+b x)^m}{(c+d x)^2} \, dx\)

Optimal. Leaf size=52 \[ \frac{b (a+b x)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0346737, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{b (a+b x)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m/(c + d*x)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.1777, size = 39, normalized size = 0.75 \[ \frac{b \left (a + b x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{\left (m + 1\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0469105, size = 0, normalized size = 0. \[ \int \frac{(a+b x)^m}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(a + b*x)^m/(c + d*x)^2, x]

_______________________________________________________________________________________

Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m/(d*x+c)^2,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^m/(d*x + c)^2, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^m/(d*x + c)^2, x)