3.3061 \(\int \frac{(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^4} \, dx\)

Optimal. Leaf size=520 \[ \frac{f (a+b x)^{m+1} (c+d x)^{1-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f (c f m (2 m+3)+d e (7 m+12))+b^2 \left (-c^2 f^2 (2-m) m+7 c d e f m+6 d^2 e^2\right )\right )}{6 m (e+f x)^2 (b c-a d) (b e-a f)^2 (d e-c f)^3}+\frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )-3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (c f m+3 d e)+3 a b^2 d f (m+1) \left (-c^2 f^2 (1-m) m+6 c d e f m+6 d^2 e^2\right )+b^3 \left (-\left (c^3 f^3 m \left (m^2-3 m+2\right )-9 c^2 d e f^2 (1-m) m+18 c d^2 e^2 f m+6 d^3 e^3\right )\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{6 m (m+1) (b e-a f)^4 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m} (a d f (m+3)-b (c f m+3 d e))}{3 m (e+f x)^3 (b c-a d) (b e-a f) (d e-c f)^2}+\frac{d (a+b x)^{m+1} (c+d x)^{-m}}{m (e+f x)^3 (b c-a d) (d e-c f)} \]

[Out]

-(f*(a*d*f*(3 + m) - b*(3*d*e + c*f*m))*(a + b*x)^(1 + m)*(c + d*x)^(1 - m))/(3*
(b*c - a*d)*(b*e - a*f)*(d*e - c*f)^2*m*(e + f*x)^3) + (d*(a + b*x)^(1 + m))/((b
*c - a*d)*(d*e - c*f)*m*(c + d*x)^m*(e + f*x)^3) + (f*(b^2*(6*d^2*e^2 + 7*c*d*e*
f*m - c^2*f^2*(2 - m)*m) + a^2*d^2*f^2*(6 + 5*m + m^2) - a*b*d*f*(c*f*m*(3 + 2*m
) + d*e*(12 + 7*m)))*(a + b*x)^(1 + m)*(c + d*x)^(1 - m))/(6*(b*c - a*d)*(b*e -
a*f)^2*(d*e - c*f)^3*m*(e + f*x)^2) + ((3*a*b^2*d*f*(1 + m)*(6*d^2*e^2 + 6*c*d*e
*f*m - c^2*f^2*(1 - m)*m) - 3*a^2*b*d^2*f^2*(3*d*e + c*f*m)*(2 + 3*m + m^2) + a^
3*d^3*f^3*(6 + 11*m + 6*m^2 + m^3) - b^3*(6*d^3*e^3 + 18*c*d^2*e^2*f*m - 9*c^2*d
*e*f^2*(1 - m)*m + c^3*f^3*m*(2 - 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 -
m)*Hypergeometric2F1[2, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c +
d*x))])/(6*(b*e - a*f)^4*(d*e - c*f)^3*m*(1 + m))

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Rubi [A]  time = 2.47722, antiderivative size = 520, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{f (a+b x)^{m+1} (c+d x)^{1-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f (c f m (2 m+3)+d e (7 m+12))+b^2 \left (-c^2 f^2 (2-m) m+7 c d e f m+6 d^2 e^2\right )\right )}{6 m (e+f x)^2 (b c-a d) (b e-a f)^2 (d e-c f)^3}+\frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )-3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (c f m+3 d e)+3 a b^2 d f (m+1) \left (-c^2 f^2 (1-m) m+6 c d e f m+6 d^2 e^2\right )+b^3 \left (-\left (c^3 f^3 m \left (m^2-3 m+2\right )-9 c^2 d e f^2 (1-m) m+18 c d^2 e^2 f m+6 d^3 e^3\right )\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{6 m (m+1) (b e-a f)^4 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m} (a d f (m+3)-b (c f m+3 d e))}{3 m (e+f x)^3 (b c-a d) (b e-a f) (d e-c f)^2}+\frac{d (a+b x)^{m+1} (c+d x)^{-m}}{m (e+f x)^3 (b c-a d) (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

-(f*(a*d*f*(3 + m) - b*(3*d*e + c*f*m))*(a + b*x)^(1 + m)*(c + d*x)^(1 - m))/(3*
(b*c - a*d)*(b*e - a*f)*(d*e - c*f)^2*m*(e + f*x)^3) + (d*(a + b*x)^(1 + m))/((b
*c - a*d)*(d*e - c*f)*m*(c + d*x)^m*(e + f*x)^3) + (f*(b^2*(6*d^2*e^2 + 7*c*d*e*
f*m - c^2*f^2*(2 - m)*m) + a^2*d^2*f^2*(6 + 5*m + m^2) - a*b*d*f*(c*f*m*(3 + 2*m
) + d*e*(12 + 7*m)))*(a + b*x)^(1 + m)*(c + d*x)^(1 - m))/(6*(b*c - a*d)*(b*e -
a*f)^2*(d*e - c*f)^3*m*(e + f*x)^2) + ((3*a*b^2*d*f*(1 + m)*(6*d^2*e^2 + 6*c*d*e
*f*m - c^2*f^2*(1 - m)*m) - 3*a^2*b*d^2*f^2*(3*d*e + c*f*m)*(2 + 3*m + m^2) + a^
3*d^3*f^3*(6 + 11*m + 6*m^2 + m^3) - b^3*(6*d^3*e^3 + 18*c*d^2*e^2*f*m - 9*c^2*d
*e*f^2*(1 - m)*m + c^3*f^3*m*(2 - 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 -
m)*Hypergeometric2F1[2, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c +
d*x))])/(6*(b*e - a*f)^4*(d*e - c*f)^3*m*(1 + m))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**4,x)

[Out]

Timed out

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Mathematica [C]  time = 19.7115, size = 7153, normalized size = 13.76 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [F]  time = 0.187, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-1-m}}{ \left ( fx+e \right ) ^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x+a)^m*(d*x+c)^(-1-m)/(f*x+e)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 1}}{{\left (f x + e\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^4,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 1}}{f^{4} x^{4} + 4 \, e f^{3} x^{3} + 6 \, e^{2} f^{2} x^{2} + 4 \, e^{3} f x + e^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^4,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m - 1)/(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)**(-1-m)/(f*x+e)**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 1}}{{\left (f x + e\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^4,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^4, x)