3.3080 \(\int (a+b x)^m (c+d x)^{-4-m} (e+f x)^3 \, dx\)

Optimal. Leaf size=406 \[ \frac{2 b^2 (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (m+3) (b c-a d)^3}-\frac{f^3 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^4 m}+\frac{3 f^2 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^3 (m+1) (b c-a d)}+\frac{(a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^3 (m+3) (b c-a d)}+\frac{3 f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^3 (m+2) (b c-a d)}+\frac{2 b (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^3 (m+2) (m+3) (b c-a d)^2}+\frac{3 b f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.741184, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2 b^2 (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (m+3) (b c-a d)^3}-\frac{f^3 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^4 m}+\frac{3 f^2 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^3 (m+1) (b c-a d)}+\frac{(a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^3 (m+3) (b c-a d)}+\frac{3 f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^3 (m+2) (b c-a d)}+\frac{2 b (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^3 (m+2) (m+3) (b c-a d)^2}+\frac{3 b f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 141.596, size = 342, normalized size = 0.84 \[ \frac{2 b^{2} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (c f - d e\right )^{3}}{d^{3} \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (a d - b c\right )^{3}} + \frac{3 b f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (c f - d e\right )^{2}}{d^{3} \left (m + 1\right ) \left (m + 2\right ) \left (a d - b c\right )^{2}} - \frac{2 b \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2} \left (c f - d e\right )^{3}}{d^{3} \left (m + 2\right ) \left (m + 3\right ) \left (a d - b c\right )^{2}} + \frac{3 f^{2} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (c f - d e\right )}{d^{3} \left (m + 1\right ) \left (a d - b c\right )} - \frac{3 f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2} \left (c f - d e\right )^{2}}{d^{3} \left (m + 2\right ) \left (a d - b c\right )} + \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 3} \left (c f - d e\right )^{3}}{d^{3} \left (m + 3\right ) \left (a d - b c\right )} - \frac{f^{3} \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d^{4} m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**(-4-m)*(f*x+e)**3,x)

[Out]

2*b**2*(a + b*x)**(m + 1)*(c + d*x)**(-m - 1)*(c*f - d*e)**3/(d**3*(m + 1)*(m +
2)*(m + 3)*(a*d - b*c)**3) + 3*b*f*(a + b*x)**(m + 1)*(c + d*x)**(-m - 1)*(c*f -
d*e)**2/(d**3*(m + 1)*(m + 2)*(a*d - b*c)**2) - 2*b*(a + b*x)**(m + 1)*(c + d*x
)**(-m - 2)*(c*f - d*e)**3/(d**3*(m + 2)*(m + 3)*(a*d - b*c)**2) + 3*f**2*(a + b
*x)**(m + 1)*(c + d*x)**(-m - 1)*(c*f - d*e)/(d**3*(m + 1)*(a*d - b*c)) - 3*f*(a
+ b*x)**(m + 1)*(c + d*x)**(-m - 2)*(c*f - d*e)**2/(d**3*(m + 2)*(a*d - b*c)) +
(a + b*x)**(m + 1)*(c + d*x)**(-m - 3)*(c*f - d*e)**3/(d**3*(m + 3)*(a*d - b*c)
) - f**3*(d*(a + b*x)/(a*d - b*c))**(-m)*(a + b*x)**m*(c + d*x)**(-m)*hyper((-m,
-m), (-m + 1,), b*(-c - d*x)/(a*d - b*c))/(d**4*m)

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Mathematica [C]  time = 57.4284, size = 1833, normalized size = 4.51 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x)^3,x]

[Out]

(3*e*f^2*(a + b*x)^m*(c + d*x)^(-3 - m)*(b^3*c^3*(2 + 3*m + m^2)*x^3*((c*(a + b*
x))/(a*(c + d*x)))^m - a*b^2*c^2*(1 + m)*x^2*((c*(a + b*x))/(a*(c + d*x)))^m*(-(
c*m) + 2*d*(3 + m)*x) + a^2*b*c*x*((c*(a + b*x))/(a*(c + d*x)))^m*(-2*c^2*m - 2*
c*d*m*(3 + m)*x + d^2*(6 + 5*m + m^2)*x^2) + a^3*(-2*d^3*x^3 + 2*c^3*(-1 + ((c*(
a + b*x))/(a*(c + d*x)))^m) + 2*c^2*d*x*(-3 + 3*((c*(a + b*x))/(a*(c + d*x)))^m
+ m*((c*(a + b*x))/(a*(c + d*x)))^m) + c*d^2*x^2*(-6 + 6*((c*(a + b*x))/(a*(c +
d*x)))^m + 5*m*((c*(a + b*x))/(a*(c + d*x)))^m + m^2*((c*(a + b*x))/(a*(c + d*x)
))^m))))/(c*(b*c - a*d)^3*(1 + m)*(2 + m)*(3 + m)*((c*(a + b*x))/(a*(c + d*x)))^
m) - (5*a*c*f^3*x^4*(a + b*x)^m*(c + d*x)^(-4 - m)*AppellF1[4, -m, 4 + m, 5, -((
b*x)/a), -((d*x)/c)])/(4*(-5*a*c*AppellF1[4, -m, 4 + m, 5, -((b*x)/a), -((d*x)/c
)] - b*c*m*x*AppellF1[5, 1 - m, 4 + m, 6, -((b*x)/a), -((d*x)/c)] + a*d*(4 + m)*
x*AppellF1[5, -m, 5 + m, 6, -((b*x)/a), -((d*x)/c)])) + (3*e^2*f*x^2*(a + b*x)^m
*(c + d*x)^(-4 - m)*(1 + (d*x)/c)*((c + d*x)*(b^3*c^3*m*(1 + m)*x^3 + a*b^2*c^2*
m*x^2*(c*(-3 + m) - 2*d*(3 + m)*x) - a^2*b*c*x*(d^2*(3 + m)*x^2*(-2 - m + 2*((a*
(c + d*x))/(c*(a + b*x)))^m) + 2*c*d*(3 + m)*x*(-2 + m + 2*((a*(c + d*x))/(c*(a
+ b*x)))^m) + 2*c^2*(-3 + 2*m + 3*((a*(c + d*x))/(c*(a + b*x)))^m + m*((a*(c + d
*x))/(c*(a + b*x)))^m)) + a^3*(2*d^3*m*x^3*((a*(c + d*x))/(c*(a + b*x)))^m - 6*c
^3*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m) + 2*c^2*d*x*(6 - 6*((a*(c + d*x))/(c*(
a + b*x)))^m + m*(2 + ((a*(c + d*x))/(c*(a + b*x)))^m)) + c*d^2*x^2*(6 + m^2 - 6
*((a*(c + d*x))/(c*(a + b*x)))^m + m*(5 + 4*((a*(c + d*x))/(c*(a + b*x)))^m))))*
Gamma[1 - m] + m*(3*c + d*x)*(b^3*c^3*(2 + 3*m + m^2)*x^3 + a*b^2*c^2*(1 + m)*x^
2*(c*m - 2*d*(3 + m)*x) + a^2*b*c*x*(-2*c^2*m - 2*c*d*m*(3 + m)*x + d^2*(6 + 5*m
+ m^2)*x^2) + a^3*(-2*d^3*x^3*((a*(c + d*x))/(c*(a + b*x)))^m - 2*c^3*(-1 + ((a
*(c + d*x))/(c*(a + b*x)))^m) - 2*c^2*d*x*(-3 - m + 3*((a*(c + d*x))/(c*(a + b*x
)))^m) - c*d^2*x^2*(-6 - 5*m - m^2 + 6*((a*(c + d*x))/(c*(a + b*x)))^m)))*Gamma[
-m]))/((c + d*x)*(b^3*c^3*m*(2 + 3*m + m^2)*x^3 - 3*a*b^2*c^2*m*(1 + m)*x^2*(c +
d*(3 + m)*x) + 3*a^2*b*c*m*x*(2*c^2 + 2*c*d*(3 + m)*x + d^2*(6 + 5*m + m^2)*x^2
) + a^3*(6*c^3*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m) + 6*c^2*d*x*(-3 - m + 3*((
a*(c + d*x))/(c*(a + b*x)))^m) + 3*c*d^2*x^2*(-6 - 5*m - m^2 + 6*((a*(c + d*x))/
(c*(a + b*x)))^m) + d^3*x^3*(-6 - 11*m - 6*m^2 - m^3 + 6*((a*(c + d*x))/(c*(a +
b*x)))^m)))*Gamma[1 - m] + m*(b^3*c^3*(2 + 3*m + m^2)*x^3*(3*c*(2 + m) + d*m*x)
- 3*a*b^2*c^2*(1 + m)*x^2*(c^2*m + c*d*(12 + 14*m + 3*m^2)*x + d^2*m*(3 + m)*x^2
) + 3*a^2*b*c*x*(2*c^3*m + 2*c^2*d*m*(4 + m)*x + c*d^2*(12 + 34*m + 19*m^2 + 3*m
^3)*x^2 + d^3*m*(6 + 5*m + m^2)*x^3) + a^3*(6*c^4*(-1 + ((a*(c + d*x))/(c*(a + b
*x)))^m) + 6*c^3*d*x*(-4 - m + 4*((a*(c + d*x))/(c*(a + b*x)))^m) + d^4*x^4*(-6
- 11*m - 6*m^2 - m^3 + 6*((a*(c + d*x))/(c*(a + b*x)))^m) + 3*c*d^3*x^3*(-12 - 1
6*m - 7*m^2 - m^3 + 8*((a*(c + d*x))/(c*(a + b*x)))^m) + 3*c^2*d^2*x^2*(-7*m - m
^2 + 12*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m))))*Gamma[-m]) - (e^3*(c + d*x)^(-
3 - m)*(a - (b*c)/d + (b*(c + d*x))/d)^m*Hypergeometric2F1[-3 - m, -m, -2 - m, -
((b*(c + d*x))/((a - (b*c)/d)*d))])/(d*(3 + m)*(1 + (b*(c + d*x))/((a - (b*c)/d)
*d))^m)

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-4-m} \left ( fx+e \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="giac")

[Out]

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