3.3109 \(\int (a+b x)^m (c+d x)^{2-m} \, dx\)

Optimal. Leaf size=84 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^3 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0875489, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^3 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.0035, size = 66, normalized size = 0.79 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (a d - b c\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} m - 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b^{3} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**(2-m),x)

[Out]

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

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Mathematica [C]  time = 0.77389, size = 319, normalized size = 3.8 \[ \frac{c (a+b x)^m (c+d x)^{-m} \left (\frac{4 a d^3 x^3 F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{12 a c F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )+3 b c m x F_1\left (4;1-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )-3 a d m x F_1\left (4;-m,m+1;5;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{3 a c d^2 x^2 F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 a c F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )+m x \left (b c F_1\left (3;1-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}-\frac{c (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{m-1}\right )}{d} \]

Warning: Unable to verify antiderivative.

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

[Out]

(c*(a + b*x)^m*((3*a*c*d^2*x^2*AppellF1[2, -m, m, 3, -((b*x)/a), -((d*x)/c)])/(3
*a*c*AppellF1[2, -m, m, 3, -((b*x)/a), -((d*x)/c)] + 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)])) + (4*a*d^3*x^3*AppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)])/(12*a*c*A
ppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)] + 3*b*c*m*x*AppellF1[4, 1 - m, m, 5
, -((b*x)/a), -((d*x)/c)] - 3*a*d*m*x*AppellF1[4, -m, 1 + m, 5, -((b*x)/a), -((d
*x)/c)]) - (c*(c + d*x)*Hypergeometric2F1[1 - m, -m, 2 - m, (b*(c + d*x))/(b*c -
a*d)])/((-1 + m)*((d*(a + b*x))/(-(b*c) + a*d))^m)))/(d*(c + d*x)^m)

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Maple [F]  time = 0.113, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(2-m),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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