3.3108 \(\int (a+b x)^m (c+d x)^{2-m} (e+f x) \, dx\)

Optimal. Leaf size=147 \[ \frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{4 b d}-\frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f (3-m)-b (4 d e-c f (m+1))) \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{4 b^4 d (m+1)} \]

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(4*b*d) - ((b*c - a*d)^2*(a*d*f*(3 - m)
- b*(4*d*e - c*f*(1 + m)))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hyper
geometric2F1[-2 + m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(4*b^4*d*(1 +
m)*(c + d*x)^m)

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Rubi [A]  time = 0.24438, antiderivative size = 146, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (-a d f (3-m)-b c f (m+1)+4 b d e) \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{4 b^4 d (m+1)}+\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{4 b d} \]

Antiderivative was successfully verified.

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

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(4*b*d) + ((b*c - a*d)^2*(4*b*d*e - a*d*
f*(3 - m) - b*c*f*(1 + m))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hyper
geometric2F1[-2 + m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(4*b^4*d*(1 +
m)*(c + d*x)^m)

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Rubi in Sympy [A]  time = 27.7612, size = 116, normalized size = 0.79 \[ \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 3}}{4 b d} - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (a d - b c\right )^{2} \left (- 4 b d e + f \left (a d \left (- m + 3\right ) + b c \left (m + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} m - 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{4 b^{4} d \left (m + 1\right )} \]

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]

f*(a + b*x)**(m + 1)*(c + d*x)**(-m + 3)/(4*b*d) - (b*(-c - d*x)/(a*d - b*c))**m
*(a + b*x)**(m + 1)*(c + d*x)**(-m)*(a*d - b*c)**2*(-4*b*d*e + f*(a*d*(-m + 3) +
b*c*(m + 1)))*hyper((m - 2, m + 1), (m + 2,), d*(a + b*x)/(a*d - b*c))/(4*b**4*
d*(m + 1))

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Mathematica [C]  time = 5.43263, size = 509, normalized size = 3.46 \[ c (a+b x)^m (c+d x)^{-m} \left (\frac{5 a d^2 f x^4 F_1\left (4;-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{20 a c F_1\left (4;-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )+4 b c m x F_1\left (5;1-m,m;6;-\frac{b x}{a},-\frac{d x}{c}\right )-4 a d m x F_1\left (5;-m,m+1;6;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{4 a d x^3 (2 c f+d e) F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 \left (4 a c F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )-a d m x F_1\left (4;-m,m+1;5;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}+\frac{3 a c x^2 (c f+2 d e) F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 m x \left (b c F_1\left (3;1-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}-\frac{c^2 e \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-1)}-\frac{c e x \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{m-1}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(c*(a + b*x)^m*((3*a*c*(2*d*e + 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*Appell
F1[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*(d*e + 2*c*f)*x^3*AppellF1[3, -m, m, 4, -((b*x)/a)
, -((d*x)/c)])/(3*(4*a*c*AppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)] + b*c*m*x
*AppellF1[4, 1 - m, m, 5, -((b*x)/a), -((d*x)/c)] - a*d*m*x*AppellF1[4, -m, 1 +
m, 5, -((b*x)/a), -((d*x)/c)])) + (5*a*d^2*f*x^4*AppellF1[4, -m, m, 5, -((b*x)/a
), -((d*x)/c)])/(20*a*c*AppellF1[4, -m, m, 5, -((b*x)/a), -((d*x)/c)] + 4*b*c*m*
x*AppellF1[5, 1 - m, m, 6, -((b*x)/a), -((d*x)/c)] - 4*a*d*m*x*AppellF1[5, -m, 1
+ m, 6, -((b*x)/a), -((d*x)/c)]) - (c^2*e*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*e*
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)))/(c + d*x)^m

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Maple [F]  time = 0.088, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m} \left ( fx+e \right ) \, dx \]

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. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((f*x + e)*(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)*(f*x+e),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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