3.3110 \(\int \frac{(a+b x)^m (c+d x)^{2-m}}{e+f x} \, dx\)

Optimal. Leaf size=319 \[ \frac{d (b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^2 f (m+1)}-\frac{(a+b x)^m (d e-c f)^2 (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^3 m}+\frac{(a+b x)^m (d e-c f)^2 (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{f^3 m}-\frac{d (a+b x)^{m+1} (d e-c f) (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b f^2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.61728, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{d (b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^2 f (m+1)}-\frac{(a+b x)^m (d e-c f)^2 (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^3 m}+\frac{(a+b x)^m (d e-c f)^2 (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{f^3 m}-\frac{d (a+b x)^{m+1} (d e-c f) (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b f^2 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 82.9083, size = 258, normalized size = 0.81 \[ \frac{\left (a + b x\right )^{m} \left (c + d x\right )^{- m} \left (c f - d e\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} - m, 1 \\ - m + 1 \end{matrix}\middle |{\frac{\left (- c - d x\right ) \left (- a f + b e\right )}{\left (a + b x\right ) \left (c f - d e\right )}} \right )}}{f^{3} m} - \frac{\left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m} \left (c f - d e\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{f^{3} m} + \frac{d \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (c f - d e\right ){{}_{2}F_{1}\left (\begin{matrix} m, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b f^{2} \left (m + 1\right )} - \frac{d \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} m - 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b^{2} f \left (m + 1\right )} \]

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),x)

[Out]

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

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Mathematica [C]  time = 1.79532, size = 303, normalized size = 0.95 \[ -\frac{(m+2) (b c-a d) (b e-a f)^2 (a+b x)^{m+1} (c+d x)^{2-m} F_1\left (m+1;m-2,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) (e+f x) (a f-b e) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m-2,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+(a+b x) \left ((a d f-b c f) F_1\left (m+2;m-2,2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-d (m-2) (b e-a f) F_1\left (m+2;m-1,1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(((b*c - a*d)*(b*e - a*f)^2*(2 + m)*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*AppellF
1[1 + m, -2 + m, 1, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) +
a*f)])/(b*(-(b*e) + a*f)*(1 + m)*(e + f*x)*((b*c - a*d)*(b*e - a*f)*(2 + m)*App
ellF1[1 + m, -2 + m, 1, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*
e) + a*f)] + (a + b*x)*((-(b*c*f) + a*d*f)*AppellF1[2 + m, -2 + m, 2, 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, 1, 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 = 0.091, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}}{fx+e}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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