3.3046 \(\int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx\)

Optimal. Leaf size=135 \[ \frac{f (a+b x)^{m+1} (c+d x)^{1-m}}{2 b d}-\frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f (1-m)-b (2 d e-c f (m+1))) \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^2 d (m+1)} \]

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(1 - m))/(2*b*d) - ((a*d*f*(1 - m) - b*(2*d*e - c
*f*(1 + m)))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[m
, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(2*b^2*d*(1 + m)*(c + d*x)^m)

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Rubi [A]  time = 0.187967, antiderivative size = 134, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (-a d f (1-m)-b c f (m+1)+2 b d e) \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^2 d (m+1)}+\frac{f (a+b x)^{m+1} (c+d x)^{1-m}}{2 b d} \]

Antiderivative was successfully verified.

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

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(1 - m))/(2*b*d) + ((2*b*d*e - a*d*f*(1 - m) - b*
c*f*(1 + m))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[m
, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(2*b^2*d*(1 + m)*(c + d*x)^m)

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Rubi in Sympy [A]  time = 21.9422, size = 105, normalized size = 0.78 \[ \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 1}}{2 b d} - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (- 2 b d e + f \left (a d \left (- m + 1\right ) + b c \left (m + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} m, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{2 b^{2} d \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.368438, size = 201, normalized size = 1.49 \[ (a+b x)^m (c+d x)^{-m} \left (\frac{3 a c f x^2 F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 m x \left (b c F_1\left (3;1-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}-\frac{e (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-1)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( fx+e \right ) }{ \left ( dx+c \right ) ^{m}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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