3.3065 \(\int (a+b x)^m (c+d x)^{-2-m} (e+f x) \, dx\)

Optimal. Leaf size=124 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d (m+1) (b c-a d)}-\frac{f (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^2 m} \]

[Out]

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

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Rubi [A]  time = 0.183302, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d (m+1) (b c-a d)}-\frac{f (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^2 m} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(-2 - m)*(e + f*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.0852, size = 94, normalized size = 0.76 \[ \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (c f - d e\right )}{d \left (m + 1\right ) \left (a d - b c\right )} - \frac{f \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d^{2} m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.331011, size = 132, normalized size = 1.06 \[ \frac{(a+b x)^m (c+d x)^{-m-1} \left (f (m+1) (c+d x) (-(b c-a d)) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )-c d f m (a+b x)+d^2 e m (a+b x)\right )}{d^2 m (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-2-m} \left ( fx+e \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-2-m)*(f*x+e),x)

[Out]

int((b*x+a)^m*(d*x+c)^(-2-m)*(f*x+e),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 2), 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)**(-2-m)*(f*x+e),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 2), x)