3.3042 \(\int (a+b x)^m (c+d x)^{-m} (e+f x)^p \, dx\)

Optimal. Leaf size=121 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;m,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.307606, antiderivative size = 121, 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 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;m,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 75.0046, size = 92, normalized size = 0.76 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- p} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (e + f x\right )^{p} \operatorname{appellf_{1}}{\left (m + 1,m,- p,m + 2,\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{b \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(f*x+e)**p/((d*x+c)**m),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*appellf1(m + 1, m, -p, m + 2, d*(a + b*x)/(a*d -
b*c), f*(a + b*x)/(a*f - b*e))/(b*(m + 1))

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Mathematica [B]  time = 0.974467, size = 290, normalized size = 2.4 \[ \frac{(m+2) (b c-a d) (b e-a f) (a+b x)^{m+1} (c+d x)^{-m} (e+f x)^p F_1\left (m+1;m,-p;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m,-p;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (f p (a d-b c) F_1\left (m+2;m,1-p;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d m (b e-a f) F_1\left (m+2;m+1,-p;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

((b*c - a*d)*(b*e - a*f)*(2 + m)*(a + b*x)^(1 + m)*(e + f*x)^p*AppellF1[1 + m, m
, -p, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])/(b*(1
+ m)*(c + d*x)^m*((b*c - a*d)*(b*e - a*f)*(2 + m)*AppellF1[1 + m, 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, 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)*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 = 0.225, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( fx+e \right ) ^{p}}{ \left ( dx+c \right ) ^{m}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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