3.3120 \(\int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-1+n} \, dx\)

Optimal. Leaf size=138 \[ \frac{(a+b x)^{m+1} (e+f x)^n (c+d x)^{-m-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{m+n} F_1\left (m+1;m+n,1-n;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)} \]

[Out]

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

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Rubi [A]  time = 0.333286, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(a+b x)^{m+1} (e+f x)^n (c+d x)^{-m-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{m+n} F_1\left (m+1;m+n,1-n;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(-m - n)*(e + f*x)^(-1 + n),x]

[Out]

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

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Rubi in Sympy [A]  time = 77.4004, size = 105, normalized size = 0.76 \[ - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m + n} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - n} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,m + n,- n + 1,m + 2,\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{\left (m + 1\right ) \left (a f - b e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**(-m-n)*(f*x+e)**(-1+n),x)

[Out]

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

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Mathematica [B]  time = 1.55274, size = 315, normalized size = 2.28 \[ -\frac{(m+2) (b c-a d) (b e-a f) (a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} F_1\left (m+1;m+n,1-n;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) \left ((a+b x) \left (d (m+n) (b e-a f) F_1\left (m+2;m+n+1,1-n;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-f (n-1) (b c-a d) F_1\left (m+2;m+n,2-n;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )-(m+2) (b c-a d) (b e-a f) F_1\left (m+1;m+n,1-n;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^(-m - n)*(e + f*x)^(-1 + n),x]

[Out]

-(((b*c - a*d)*(b*e - a*f)*(2 + m)*(a + b*x)^(1 + m)*(c + d*x)^(-m - n)*(e + f*x
)^(-1 + n)*AppellF1[1 + m, m + n, 1 - n, 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)*Appel
lF1[1 + m, m + n, 1 - n, 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*(-1 + n)*AppellF1[2 + m, m + n, 2 - n,
3 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)]) + d*(b*e -
a*f)*(m + n)*AppellF1[2 + m, 1 + m + n, 1 - n, 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 = 0.237, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-m-n} \left ( fx+e \right ) ^{-1+n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(-1+n),x)

[Out]

int((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(-1+n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 1),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 1),x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 1), 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-n)*(f*x+e)**(-1+n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 1),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 1), x)