3.3105 \(\int (a+b x)^m (c+d x)^{2-m} (e+f x)^p \, dx\)

Optimal. Leaf size=133 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;m-2,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^3 (m+1)} \]

[Out]

((b*c - a*d)^2*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*(e + f*x)^p*Appel
lF1[1 + m, -2 + m, -p, 2 + m, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e
- a*f))])/(b^3*(1 + m)*(c + d*x)^m*((b*(e + f*x))/(b*e - a*f))^p)

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Rubi [A]  time = 0.355204, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;m-2,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^3 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*(e + f*x)^p*Appel
lF1[1 + m, -2 + m, -p, 2 + m, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e
- a*f))])/(b^3*(1 + m)*(c + d*x)^m*((b*(e + f*x))/(b*e - a*f))^p)

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Rubi in Sympy [A]  time = 75.6544, size = 104, normalized size = 0.78 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- p} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (e + f x\right )^{p} \left (a d - b c\right )^{2} \operatorname{appellf_{1}}{\left (m + 1,- p,m - 2,m + 2,\frac{f \left (a + b x\right )}{a f - b e},\frac{d \left (a + b x\right )}{a d - b c} \right )}}{b^{3} \left (m + 1\right )} \]

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)**p,x)

[Out]

(b*(-c - d*x)/(a*d - b*c))**m*(b*(-e - f*x)/(a*f - b*e))**(-p)*(a + b*x)**(m + 1
)*(c + d*x)**(-m)*(e + f*x)**p*(a*d - b*c)**2*appellf1(m + 1, -p, m - 2, m + 2,
f*(a + b*x)/(a*f - b*e), d*(a + b*x)/(a*d - b*c))/(b**3*(m + 1))

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Mathematica [B]  time = 1.88256, size = 300, normalized size = 2.26 \[ \frac{(m+2) (b c-a d) (b e-a f) (a+b x)^{m+1} (c+d x)^{2-m} (e+f x)^p F_1\left (m+1;m-2,-p;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m-2,-p;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (f p (a d-b c) F_1\left (m+2;m-2,1-p;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (m-2) (b e-a f) F_1\left (m+2;m-1,-p;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

((b*c - a*d)*(b*e - a*f)*(2 + m)*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*(e + f*x)^p
*AppellF1[1 + m, -2 + m, -p, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/
(-(b*e) + a*f)])/(b*(1 + m)*((b*c - a*d)*(b*e - a*f)*(2 + m)*AppellF1[1 + m, -2
+ m, -p, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] - (a
+ b*x)*((-(b*c) + a*d)*f*p*AppellF1[2 + m, -2 + m, 1 - p, 3 + m, (d*(a + b*x))/
(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] + d*(b*e - a*f)*(-2 + m)*AppellF1[
2 + m, -1 + m, -p, 3 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) +
a*f)])))

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Maple [F]  time = 0.228, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m} \left ( fx+e \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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