3.3098 \(\int (a+b x)^m (c+d x)^{1-m} (e+f x) \, dx\)

Optimal. Leaf size=145 \[ \frac{f (a+b x)^{m+1} (c+d x)^{2-m}}{3 b d}-\frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f (2-m)-b (3 d e-c f (m+1))) \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{3 b^3 d (m+1)} \]

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(3*b*d) - ((b*c - a*d)*(a*d*f*(2 - m) -
b*(3*d*e - c*f*(1 + m)))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hyperge
ometric2F1[-1 + m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(3*b^3*d*(1 + m)
*(c + d*x)^m)

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Rubi [A]  time = 0.240527, antiderivative size = 144, 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 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (-a d f (2-m)-b c f (m+1)+3 b d e) \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{3 b^3 d (m+1)}+\frac{f (a+b x)^{m+1} (c+d x)^{2-m}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(1 - m)*(e + f*x),x]

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(3*b*d) + ((b*c - a*d)*(3*b*d*e - a*d*f*
(2 - m) - b*c*f*(1 + m))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hyperge
ometric2F1[-1 + m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(3*b^3*d*(1 + m)
*(c + d*x)^m)

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Rubi in Sympy [A]  time = 23.8967, size = 114, normalized size = 0.79 \[ \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 2}}{3 b d} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (a d - b c\right ) \left (- 3 b d e + f \left (a d \left (- m + 2\right ) + b c \left (m + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} m - 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{3 b^{3} d \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**(1-m)*(f*x+e),x)

[Out]

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

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Mathematica [C]  time = 1.06977, size = 322, normalized size = 2.22 \[ c (a+b x)^m (c+d x)^{-m} \left (\frac{3 a x^2 (c f+d e) F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 m x \left (b c F_1\left (3;1-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}+\frac{4 a d f x^3 F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{12 a c F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )+3 b c m x F_1\left (4;1-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )-3 a d m x F_1\left (4;-m,m+1;5;-\frac{b x}{a},-\frac{d x}{c}\right )}-\frac{e (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-1)}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^(1 - m)*(e + f*x),x]

[Out]

(c*(a + b*x)^m*((3*a*(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*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*f*x^3*AppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)])/(
12*a*c*AppellF1[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)]) - (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 = 0.087, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{1-m} \left ( fx+e \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(1-m)*(f*x+e),x)

[Out]

int((b*x+a)^m*(d*x+c)^(1-m)*(f*x+e),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="maxima")

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="fricas")

[Out]

integral((f*x + e)*(b*x + a)^m*(d*x + c)^(-m + 1), 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)**(1-m)*(f*x+e),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="giac")

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m + 1), x)