3.3113 \(\int \frac{(a+b x)^m (c+d x)^{2-m}}{(e+f x)^4} \, dx\)

Optimal. Leaf size=85 \[ \frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^4} \]

[Out]

((b*c - a*d)^3*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m,
2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^4*(1 + m))

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Rubi [A]  time = 0.104042, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^4,x]

[Out]

((b*c - a*d)^3*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m,
2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^4*(1 + m))

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Rubi in Sympy [A]  time = 12.0146, size = 66, normalized size = 0.78 \[ \frac{\left (a + b x\right )^{m - 3} \left (c + d x\right )^{- m + 3} \left (a d - b c\right )^{3}{{}_{2}F_{1}\left (\begin{matrix} - m + 3, 4 \\ - m + 4 \end{matrix}\middle |{\frac{\left (- c - d x\right ) \left (- a f + b e\right )}{\left (a + b x\right ) \left (c f - d e\right )}} \right )}}{\left (- m + 3\right ) \left (c f - d e\right )^{4}} \]

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)**4,x)

[Out]

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

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Mathematica [A]  time = 1.85442, size = 122, normalized size = 1.44 \[ \frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+1} \, _2F_1\left (m-2,m+1;m+2;\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^4,x]

[Out]

((b*c - a*d)^3*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*(((b*e - a*f)*(c + d*x))/((b
*c - a*d)*(e + f*x)))^(1 + m)*Hypergeometric2F1[-2 + m, 1 + m, 2 + m, ((-(d*e) +
c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))])/((b*e - a*f)^4*(1 + m))

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Maple [F]  time = 0.185, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}}{ \left ( fx+e \right ) ^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^4,x)

[Out]

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^4,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{f^{4} x^{4} + 4 \, e f^{3} x^{3} + 6 \, e^{2} f^{2} x^{2} + 4 \, e^{3} f x + e^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^4,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 2)/(f^4*x^4 + 4*e*f^3*x^3 + 6*e^2*f^2*x^2 +
4*e^3*f*x + e^4), 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)**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^4,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^4, x)