3.3136 \(\int \frac{(a+b x)^m (c+d x)^n}{(e+f x)^2} \, dx\)

Optimal. Leaf size=101 \[ \frac{b (a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2} \]

[Out]

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

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Rubi [A]  time = 0.170328, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{b (a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.5091, size = 76, normalized size = 0.75 \[ \frac{b \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,2,- n,m + 2,\frac{f \left (a + b x\right )}{a f - b e},\frac{d \left (a + b x\right )}{a d - b c} \right )}}{\left (m + 1\right ) \left (a f - b e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 1.07068, size = 286, normalized size = 2.83 \[ \frac{(m+2) (b c-a d) (b e-a f) (a+b x)^{m+1} (c+d x)^n F_1\left (m+1;-n,2;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) (e+f x)^2 \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;-n,2;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (d n (a f-b e) F_1\left (m+2;1-n,2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+2 f (b c-a d) F_1\left (m+2;-n,3;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

((b*c - a*d)*(b*e - a*f)*(2 + m)*(a + b*x)^(1 + m)*(c + d*x)^n*AppellF1[1 + m, -
n, 2, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])/(b*(1
+ m)*(e + f*x)^2*((b*c - a*d)*(b*e - a*f)*(2 + m)*AppellF1[1 + m, -n, 2, 2 + m,
(d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] - (a + b*x)*(d*(-(b*
e) + a*f)*n*AppellF1[2 + m, 1 - n, 2, 3 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a
+ b*x))/(-(b*e) + a*f)] + 2*(b*c - a*d)*f*AppellF1[2 + m, -n, 3, 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 = 0.11, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n}}{ \left ( fx+e \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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