3.3088 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx\)

Optimal. Leaf size=650 \[ \frac{6 b^3 (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{6 b^2 (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-2}}{d^4 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{8 b^2 f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (m+3) (b c-a d)^3}-\frac{f^4 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^5 m}+\frac{4 f^3 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^4 (m+1) (b c-a d)}+\frac{6 f^2 (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^4 (m+2) (b c-a d)}+\frac{6 b f^2 (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (b c-a d)^2}+\frac{(a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-4}}{d^4 (m+4) (b c-a d)}+\frac{4 f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^4 (m+3) (b c-a d)}+\frac{3 b (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-3}}{d^4 (m+3) (m+4) (b c-a d)^2}+\frac{8 b f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^4 (m+2) (m+3) (b c-a d)^2} \]

[Out]

((d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/(d^4*(b*c - a*d)*(4 + m)) +
(4*f*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(d^4*(b*c - a*d)*(3 +
m)) + (3*b*(d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(d^4*(b*c - a*d)^
2*(3 + m)*(4 + m)) + (6*f^2*(d*e - c*f)^2*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/
(d^4*(b*c - a*d)*(2 + m)) + (8*b*f*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-2
- m))/(d^4*(b*c - a*d)^2*(2 + m)*(3 + m)) + (6*b^2*(d*e - c*f)^4*(a + b*x)^(1 +
m)*(c + d*x)^(-2 - m))/(d^4*(b*c - a*d)^3*(2 + m)*(3 + m)*(4 + m)) + (4*f^3*(d*
e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)*(1 + m)) + (6*b*
f^2*(d*e - c*f)^2*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^2*(1 +
m)*(2 + m)) + (8*b^2*f*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*
(b*c - a*d)^3*(1 + m)*(2 + m)*(3 + m)) + (6*b^3*(d*e - c*f)^4*(a + b*x)^(1 + m)*
(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) - (f^4*(
a + b*x)^m*Hypergeometric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^5*m*(
-((d*(a + b*x))/(b*c - a*d)))^m*(c + d*x)^m)

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Rubi [A]  time = 1.32347, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{6 b^3 (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{6 b^2 (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-2}}{d^4 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{8 b^2 f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (m+3) (b c-a d)^3}-\frac{f^4 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^5 m}+\frac{4 f^3 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^4 (m+1) (b c-a d)}+\frac{6 f^2 (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^4 (m+2) (b c-a d)}+\frac{6 b f^2 (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (b c-a d)^2}+\frac{(a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-4}}{d^4 (m+4) (b c-a d)}+\frac{4 f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^4 (m+3) (b c-a d)}+\frac{3 b (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-3}}{d^4 (m+3) (m+4) (b c-a d)^2}+\frac{8 b f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^4 (m+2) (m+3) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^4,x]

[Out]

((d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/(d^4*(b*c - a*d)*(4 + m)) +
(4*f*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(d^4*(b*c - a*d)*(3 +
m)) + (3*b*(d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(d^4*(b*c - a*d)^
2*(3 + m)*(4 + m)) + (6*f^2*(d*e - c*f)^2*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/
(d^4*(b*c - a*d)*(2 + m)) + (8*b*f*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-2
- m))/(d^4*(b*c - a*d)^2*(2 + m)*(3 + m)) + (6*b^2*(d*e - c*f)^4*(a + b*x)^(1 +
m)*(c + d*x)^(-2 - m))/(d^4*(b*c - a*d)^3*(2 + m)*(3 + m)*(4 + m)) + (4*f^3*(d*
e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)*(1 + m)) + (6*b*
f^2*(d*e - c*f)^2*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^2*(1 +
m)*(2 + m)) + (8*b^2*f*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*
(b*c - a*d)^3*(1 + m)*(2 + m)*(3 + m)) + (6*b^3*(d*e - c*f)^4*(a + b*x)^(1 + m)*
(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) - (f^4*(
a + b*x)^m*Hypergeometric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^5*m*(
-((d*(a + b*x))/(b*c - a*d)))^m*(c + d*x)^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)**4,x)

[Out]

Timed out

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Mathematica [C]  time = 168.581, size = 5118, normalized size = 7.87 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^4,x]

[Out]

Result too large to show

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-5-m} \left ( fx+e \right ) ^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x)

[Out]

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{4}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m - 5),x, algorithm="maxima")

[Out]

integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m - 5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f^{4} x^{4} + 4 \, e f^{3} x^{3} + 6 \, e^{2} f^{2} x^{2} + 4 \, e^{3} f x + e^{4}\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m - 5),x, algorithm="fricas")

[Out]

integral((f^4*x^4 + 4*e*f^3*x^3 + 6*e^2*f^2*x^2 + 4*e^3*f*x + e^4)*(b*x + a)^m*(
d*x + c)^(-m - 5), 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)**4,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m - 5),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError