3.3047 \(\int (a+b x)^m (c+d x)^{-m} \, dx\)

Optimal. Leaf size=72 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.076857, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m/(c + d*x)^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 14.82, size = 54, normalized size = 0.75 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} m, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0670995, size = 80, normalized size = 1.11 \[ -\frac{(a+b x)^m (c+d x)^{1-m} \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.106, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{m}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m/((d*x+c)^m),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^m/(d*x + c)^m, x)

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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/((d*x+c)**m),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^m/(d*x + c)^m, x)