3.3093 \(\int \frac{(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx\)

Optimal. Leaf size=570 \[ \frac{d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (d e-c f (m+4))+b^2 \left (c^2 f^2 \left (m^2+9 m+26\right )-2 c d e f (m+10)+6 d^2 e^2\right )\right )}{(m+2) (m+3) (m+4) (b c-a d)^3 (d e-c f)^3}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (d e-c f (3 m+7))+a b^2 d f (m+4) \left (c^2 f^2 \left (3 m^2+17 m+26\right )-2 c d e f (m+5)+2 d^2 e^2\right )+b^3 \left (-c^3 f^3 \left (m^3+10 m^2+35 m+50\right )+c^2 d e f^2 \left (m^2+11 m+46\right )-2 c d^2 e^2 f (m+13)+6 d^3 e^3\right )\right )}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4 (d e-c f)^4}+\frac{f^4 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f) (d e-c f)^4}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)+b (3 d e-c f (m+7)))}{(m+3) (m+4) (b c-a d)^2 (d e-c f)^2} \]

[Out]

(d*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/((b*c - a*d)*(d*e - c*f)*(4 + m)) + (d*
(a*d*f*(4 + m) + b*(3*d*e - c*f*(7 + m)))*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/
((b*c - a*d)^2*(d*e - c*f)^2*(3 + m)*(4 + m)) + (d*(a^2*d^2*f^2*(12 + 7*m + m^2)
+ 2*a*b*d*f*(4 + m)*(d*e - c*f*(4 + m)) + b^2*(6*d^2*e^2 - 2*c*d*e*f*(10 + m) +
c^2*f^2*(26 + 9*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)^3
*(d*e - c*f)^3*(2 + m)*(3 + m)*(4 + m)) + (d*(a^3*d^3*f^3*(24 + 26*m + 9*m^2 + m
^3) + a^2*b*d^2*f^2*(12 + 7*m + m^2)*(d*e - c*f*(7 + 3*m)) + a*b^2*d*f*(4 + m)*(
2*d^2*e^2 - 2*c*d*e*f*(5 + m) + c^2*f^2*(26 + 17*m + 3*m^2)) + b^3*(6*d^3*e^3 -
2*c*d^2*e^2*f*(13 + m) + c^2*d*e*f^2*(46 + 11*m + m^2) - c^3*f^3*(50 + 35*m + 10
*m^2 + m^3)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/((b*c - a*d)^4*(d*e - c*f)^4
*(1 + m)*(2 + m)*(3 + m)*(4 + m)) + (f^4*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hy
pergeometric2F1[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)^4*(1 + m))

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Rubi [A]  time = 3.51892, antiderivative size = 569, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (d e-c f (m+4))+b^2 \left (c^2 f^2 \left (m^2+9 m+26\right )-2 c d e f (m+10)+6 d^2 e^2\right )\right )}{(m+2) (m+3) (m+4) (b c-a d)^3 (d e-c f)^3}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (d e-c f (3 m+7))+a b^2 d f (m+4) \left (c^2 f^2 \left (3 m^2+17 m+26\right )-2 c d e f (m+5)+2 d^2 e^2\right )+b^3 \left (-c^3 f^3 \left (m^3+10 m^2+35 m+50\right )+c^2 d e f^2 \left (m^2+11 m+46\right )-2 c d^2 e^2 f (m+13)+6 d^3 e^3\right )\right )}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4 (d e-c f)^4}+\frac{f^4 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f) (d e-c f)^4}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b c f (m+7)+3 b d e)}{(m+3) (m+4) (b c-a d)^2 (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(-5 - m))/(e + f*x),x]

[Out]

(d*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/((b*c - a*d)*(d*e - c*f)*(4 + m)) + (d*
(3*b*d*e + a*d*f*(4 + m) - b*c*f*(7 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/
((b*c - a*d)^2*(d*e - c*f)^2*(3 + m)*(4 + m)) + (d*(a^2*d^2*f^2*(12 + 7*m + m^2)
+ 2*a*b*d*f*(4 + m)*(d*e - c*f*(4 + m)) + b^2*(6*d^2*e^2 - 2*c*d*e*f*(10 + m) +
c^2*f^2*(26 + 9*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)^3
*(d*e - c*f)^3*(2 + m)*(3 + m)*(4 + m)) + (d*(a^3*d^3*f^3*(24 + 26*m + 9*m^2 + m
^3) + a^2*b*d^2*f^2*(12 + 7*m + m^2)*(d*e - c*f*(7 + 3*m)) + a*b^2*d*f*(4 + m)*(
2*d^2*e^2 - 2*c*d*e*f*(5 + m) + c^2*f^2*(26 + 17*m + 3*m^2)) + b^3*(6*d^3*e^3 -
2*c*d^2*e^2*f*(13 + m) + c^2*d*e*f^2*(46 + 11*m + m^2) - c^3*f^3*(50 + 35*m + 10
*m^2 + m^3)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/((b*c - a*d)^4*(d*e - c*f)^4
*(1 + m)*(2 + m)*(3 + m)*(4 + m)) + (f^4*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hy
pergeometric2F1[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)^4*(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)**(-5-m)/(f*x+e),x)

[Out]

Timed out

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Mathematica [C]  time = 43.6279, size = 50481, normalized size = 88.56 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(-5 - m))/(e + f*x),x]

[Out]

Result too large to show

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Maple [F]  time = 0.103, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-5-m}}{fx+e}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x)

[Out]

int((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}}{f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}}{f x + e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e),x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m - 5)/(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)**(-5-m)/(f*x+e),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}}{f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e), x)