3.112 \(\int \frac{x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=319 \[ -\frac{a^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^3 (n+1) (b c-a d) (b e-a f)}+\frac{\left (a^2 d^2+a b c d+b^2 c^2\right ) (e+f x)^{n+1}}{b^3 d^3 f (n+1)}+\frac{e (a d+b c) (e+f x)^{n+1}}{b^2 d^2 f^2 (n+1)}-\frac{(a d+b c) (e+f x)^{n+2}}{b^2 d^2 f^2 (n+2)}+\frac{c^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^3 (n+1) (b c-a d) (d e-c f)}+\frac{e^2 (e+f x)^{n+1}}{b d f^3 (n+1)}-\frac{2 e (e+f x)^{n+2}}{b d f^3 (n+2)}+\frac{(e+f x)^{n+3}}{b d f^3 (n+3)} \]

[Out]

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

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Rubi [A]  time = 0.719059, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{a^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^3 (n+1) (b c-a d) (b e-a f)}+\frac{\left (a^2 d^2+a b c d+b^2 c^2\right ) (e+f x)^{n+1}}{b^3 d^3 f (n+1)}+\frac{e (a d+b c) (e+f x)^{n+1}}{b^2 d^2 f^2 (n+1)}-\frac{(a d+b c) (e+f x)^{n+2}}{b^2 d^2 f^2 (n+2)}+\frac{c^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^3 (n+1) (b c-a d) (d e-c f)}+\frac{e^2 (e+f x)^{n+1}}{b d f^3 (n+1)}-\frac{2 e (e+f x)^{n+2}}{b d f^3 (n+2)}+\frac{(e+f x)^{n+3}}{b d f^3 (n+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 97.9068, size = 267, normalized size = 0.84 \[ - \frac{a^{4} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- e - f x\right )}{a f - b e}} \right )}}{b^{3} \left (n + 1\right ) \left (a d - b c\right ) \left (a f - b e\right )} + \frac{c^{4} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (- e - f x\right )}{c f - d e}} \right )}}{d^{3} \left (n + 1\right ) \left (a d - b c\right ) \left (c f - d e\right )} + \frac{e^{2} \left (e + f x\right )^{n + 1}}{b d f^{3} \left (n + 1\right )} - \frac{2 e \left (e + f x\right )^{n + 2}}{b d f^{3} \left (n + 2\right )} + \frac{\left (e + f x\right )^{n + 3}}{b d f^{3} \left (n + 3\right )} + \frac{e \left (e + f x\right )^{n + 1} \left (a d + b c\right )}{b^{2} d^{2} f^{2} \left (n + 1\right )} - \frac{\left (e + f x\right )^{n + 2} \left (a d + b c\right )}{b^{2} d^{2} f^{2} \left (n + 2\right )} + \frac{\left (e + f x\right )^{n + 1} \left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right )}{b^{3} d^{3} f \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(f*x+e)**n/(b*x+a)/(d*x+c),x)

[Out]

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

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Mathematica [C]  time = 0.884739, size = 262, normalized size = 0.82 \[ \frac{6}{5} e x^5 (e+f x)^n \left (\frac{a b F_1\left (5;-n,1;6;-\frac{f x}{e},-\frac{b x}{a}\right )}{(a+b x) (b c-a d) \left (6 a e F_1\left (5;-n,1;6;-\frac{f x}{e},-\frac{b x}{a}\right )+a f n x F_1\left (6;1-n,1;7;-\frac{f x}{e},-\frac{b x}{a}\right )-b e x F_1\left (6;-n,2;7;-\frac{f x}{e},-\frac{b x}{a}\right )\right )}+\frac{c d F_1\left (5;-n,1;6;-\frac{f x}{e},-\frac{d x}{c}\right )}{(c+d x) (a d-b c) \left (6 c e F_1\left (5;-n,1;6;-\frac{f x}{e},-\frac{d x}{c}\right )+c f n x F_1\left (6;1-n,1;7;-\frac{f x}{e},-\frac{d x}{c}\right )-d e x F_1\left (6;-n,2;7;-\frac{f x}{e},-\frac{d x}{c}\right )\right )}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(6*e*x^5*(e + f*x)^n*((a*b*AppellF1[5, -n, 1, 6, -((f*x)/e), -((b*x)/a)])/((b*c
- a*d)*(a + b*x)*(6*a*e*AppellF1[5, -n, 1, 6, -((f*x)/e), -((b*x)/a)] + a*f*n*x*
AppellF1[6, 1 - n, 1, 7, -((f*x)/e), -((b*x)/a)] - b*e*x*AppellF1[6, -n, 2, 7, -
((f*x)/e), -((b*x)/a)])) + (c*d*AppellF1[5, -n, 1, 6, -((f*x)/e), -((d*x)/c)])/(
(-(b*c) + a*d)*(c + d*x)*(6*c*e*AppellF1[5, -n, 1, 6, -((f*x)/e), -((d*x)/c)] +
c*f*n*x*AppellF1[6, 1 - n, 1, 7, -((f*x)/e), -((d*x)/c)] - d*e*x*AppellF1[6, -n,
2, 7, -((f*x)/e), -((d*x)/c)]))))/5

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Maple [F]  time = 0.171, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4} \left ( fx+e \right ) ^{n}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((f*x + e)^n*x^4/(b*d*x^2 + a*c + (b*c + a*d)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(f*x+e)**n/(b*x+a)/(d*x+c),x)

[Out]

Integral(x**4*(e + f*x)**n/((a + b*x)*(c + d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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