3.3076 \(\int \frac{(a+b x)^m (c+d x)^{-3-m}}{e+f x} \, dx\)

Optimal. Leaf size=209 \[ \frac{f^2 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f) (d e-c f)^2}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (b c-a d) (d e-c f)}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+2)+b (d e-c f (m+3)))}{(m+1) (m+2) (b c-a d)^2 (d e-c f)^2} \]

[Out]

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

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Rubi [A]  time = 0.602126, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{f^2 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f) (d e-c f)^2}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (b c-a d) (d e-c f)}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+2)-b c f (m+3)+b d e)}{(m+1) (m+2) (b c-a d)^2 (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(-3 - m))/(e + f*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 122.084, size = 177, normalized size = 0.85 \[ \frac{d \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2}}{\left (m + 2\right ) \left (a d - b c\right ) \left (c f - d e\right )} + \frac{d \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (- b c f \left (m + 2\right ) - b c f + d \left (a f \left (m + 2\right ) + b e\right )\right )}{\left (m + 1\right ) \left (m + 2\right ) \left (a d - b c\right )^{2} \left (c f - d e\right )^{2}} - \frac{f^{2} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1}{{}_{2}F_{1}\left (\begin{matrix} m + 1, 1 \\ m + 2 \end{matrix}\middle |{\frac{\left (- a - b x\right ) \left (- c f + d e\right )}{\left (c + d x\right ) \left (a f - b e\right )}} \right )}}{\left (m + 1\right ) \left (a f - b e\right ) \left (c f - d e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**(-3-m)/(f*x+e),x)

[Out]

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

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Mathematica [C]  time = 27.4659, size = 12578, normalized size = 60.18 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(-3 - m))/(e + f*x),x]

[Out]

Result too large to show

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-3-m}}{fx+e}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-3-m)/(f*x+e),x)

[Out]

int((b*x+a)^m*(d*x+c)^(-3-m)/(f*x+e),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}}{f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}}{f x + e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e),x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m - 3)/(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)**(-3-m)/(f*x+e),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}}{f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e), x)