3.3056 \(\int (a+b x)^m (c+d x)^{-1-m} (e+f x) \, dx\)

Optimal. Leaf size=152 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m}}{d m (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f m+b (d e-c f (m+1))) \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b d m (m+1) (b c-a d)} \]

[Out]

((d*e - c*f)*(a + b*x)^(1 + m))/(d*(b*c - a*d)*m*(c + d*x)^m) - ((a*d*f*m + b*(d
*e - c*f*(1 + m)))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometri
c2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b*d*(b*c - a*d)*m*(1 + m)*
(c + d*x)^m)

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Rubi [A]  time = 0.253092, antiderivative size = 151, 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 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m}}{d m (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f m-b c f (m+1)+b d e) \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b d m (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

((d*e - c*f)*(a + b*x)^(1 + m))/(d*(b*c - a*d)*m*(c + d*x)^m) - ((b*d*e + a*d*f*
m - b*c*f*(1 + m))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometri
c2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b*d*(b*c - a*d)*m*(1 + m)*
(c + d*x)^m)

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Rubi in Sympy [A]  time = 33.5743, size = 116, normalized size = 0.76 \[ \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (c f - d e\right )}{d m \left (a d - b c\right )} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (b d e + f \left (a d m - b c \left (m + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} m, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b d m \left (m + 1\right ) \left (a d - b c\right )} \]

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]

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

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Mathematica [A]  time = 0.165928, size = 131, normalized size = 0.86 \[ -\frac{(a+b x)^m (c+d x)^{-m} \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \left ((m-1) (d e-c f) \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )+f m (c+d x) \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )\right )}{d^2 (m-1) m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.076, 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)