3.3073 \(\int (a+b x)^m (c+d x)^{-3-m} (e+f x)^2 \, dx\)

Optimal. Leaf size=205 \[ -\frac{f^2 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^3 m}+\frac{(a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^2 (m+2) (b c-a d)}-\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (2 a d f (m+2)-b (c f (2 m+3)+d e))}{d^2 (m+1) (m+2) (b c-a d)^2} \]

[Out]

((d*e - c*f)^2*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^2*(b*c - a*d)*(2 + m)) -
((d*e - c*f)*(2*a*d*f*(2 + m) - b*(d*e + c*f*(3 + 2*m)))*(a + b*x)^(1 + m)*(c +
d*x)^(-1 - m))/(d^2*(b*c - a*d)^2*(1 + m)*(2 + m)) - (f^2*(a + b*x)^m*Hypergeom
etric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^3*m*(-((d*(a + b*x))/(b*c
- a*d)))^m*(c + d*x)^m)

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Rubi [A]  time = 0.542107, antiderivative size = 202, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{f^2 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^3 m}+\frac{(a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^2 (m+2) (b c-a d)}+\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (-2 a d f (m+2)+b c f (2 m+3)+b d e)}{d^2 (m+1) (m+2) (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

((d*e - c*f)^2*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^2*(b*c - a*d)*(2 + m)) +
((d*e - c*f)*(b*d*e - 2*a*d*f*(2 + m) + b*c*f*(3 + 2*m))*(a + b*x)^(1 + m)*(c +
d*x)^(-1 - m))/(d^2*(b*c - a*d)^2*(1 + m)*(2 + m)) - (f^2*(a + b*x)^m*Hypergeom
etric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^3*m*(-((d*(a + b*x))/(b*c
- a*d)))^m*(c + d*x)^m)

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Rubi in Sympy [A]  time = 93.0442, size = 178, normalized size = 0.87 \[ - \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2} \left (c f - d e\right )^{2}}{d^{2} \left (m + 2\right ) \left (a d - b c\right )} + \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (c f - d e\right ) \left (2 a d f m + 4 a d f - 2 b c f m - 3 b c f - b d e\right )}{d^{2} \left (m + 1\right ) \left (m + 2\right ) \left (a d - b c\right )^{2}} - \frac{f^{2} \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d^{3} m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**(m + 1)*(c + d*x)**(-m - 2)*(c*f - d*e)**2/(d**2*(m + 2)*(a*d - b*c)
) + (a + b*x)**(m + 1)*(c + d*x)**(-m - 1)*(c*f - d*e)*(2*a*d*f*m + 4*a*d*f - 2*
b*c*f*m - 3*b*c*f - b*d*e)/(d**2*(m + 1)*(m + 2)*(a*d - b*c)**2) - f**2*(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**3*m)

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Mathematica [C]  time = 2.69463, size = 426, normalized size = 2.08 \[ \frac{1}{3} (a+b x)^m (c+d x)^{-m-3} \left (\frac{6 e f (c+d x) \left (\frac{c (a+b x)}{a (c+d x)}\right )^{-m} \left (a^2 \left (c^2 \left (-\left (\left (\frac{c (a+b x)}{a (c+d x)}\right )^m-1\right )\right )-c d x \left (m \left (\frac{c (a+b x)}{a (c+d x)}\right )^m+2 \left (\frac{c (a+b x)}{a (c+d x)}\right )^m-2\right )+d^2 x^2\right )+b^2 c^2 (m+1) x^2 \left (\frac{c (a+b x)}{a (c+d x)}\right )^m-a b c x (d (m+2) x-c m) \left (\frac{c (a+b x)}{a (c+d x)}\right )^m\right )}{c (m+1) (m+2) (b c-a d)^2}-\frac{4 a c f^2 x^3 F_1\left (3;-m,m+3;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{-4 a c F_1\left (3;-m,m+3;4;-\frac{b x}{a},-\frac{d x}{c}\right )-b c m x F_1\left (4;1-m,m+3;5;-\frac{b x}{a},-\frac{d x}{c}\right )+a d (m+3) x F_1\left (4;-m,m+4;5;-\frac{b x}{a},-\frac{d x}{c}\right )}-\frac{3 e^2 (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m-2,-m;-m-1;\frac{b (c+d x)}{b c-a d}\right )}{d (m+2)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*(c + d*x)^(-3 - m)*((6*e*f*(c + d*x)*(b^2*c^2*(1 + m)*x^2*((c*(a +
b*x))/(a*(c + d*x)))^m - a*b*c*x*((c*(a + b*x))/(a*(c + d*x)))^m*(-(c*m) + d*(2
+ m)*x) + a^2*(d^2*x^2 - c^2*(-1 + ((c*(a + b*x))/(a*(c + d*x)))^m) - c*d*x*(-2
+ 2*((c*(a + b*x))/(a*(c + d*x)))^m + m*((c*(a + b*x))/(a*(c + d*x)))^m))))/(c*(
b*c - a*d)^2*(1 + m)*(2 + m)*((c*(a + b*x))/(a*(c + d*x)))^m) - (4*a*c*f^2*x^3*A
ppellF1[3, -m, 3 + m, 4, -((b*x)/a), -((d*x)/c)])/(-4*a*c*AppellF1[3, -m, 3 + m,
4, -((b*x)/a), -((d*x)/c)] - b*c*m*x*AppellF1[4, 1 - m, 3 + m, 5, -((b*x)/a), -
((d*x)/c)] + a*d*(3 + m)*x*AppellF1[4, -m, 4 + m, 5, -((b*x)/a), -((d*x)/c)]) -
(3*e^2*(c + d*x)*Hypergeometric2F1[-2 - m, -m, -1 - m, (b*(c + d*x))/(b*c - a*d)
])/(d*(2 + m)*((d*(a + b*x))/(-(b*c) + a*d))^m)))/3

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-3-m} \left ( fx+e \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m - 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((f^2*x^2 + 2*e*f*x + e^2)*(b*x + a)^m*(d*x + c)^(-m - 3), 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)**(-3-m)*(f*x+e)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m - 3), x)