3.113 \(\int \frac{x^3 (e+f x)^n}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=216 \[ \frac{a^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^2 (n+1) (b c-a d) (b e-a f)}-\frac{(a d+b c) (e+f x)^{n+1}}{b^2 d^2 f (n+1)}-\frac{c^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^2 (n+1) (b c-a d) (d e-c f)}-\frac{e (e+f x)^{n+1}}{b d f^2 (n+1)}+\frac{(e+f x)^{n+2}}{b d f^2 (n+2)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.440412, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{a^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^2 (n+1) (b c-a d) (b e-a f)}-\frac{(a d+b c) (e+f x)^{n+1}}{b^2 d^2 f (n+1)}-\frac{c^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^2 (n+1) (b c-a d) (d e-c f)}-\frac{e (e+f x)^{n+1}}{b d f^2 (n+1)}+\frac{(e+f x)^{n+2}}{b d f^2 (n+2)} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(e + f*x)^n)/((a + b*x)*(c + d*x)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 65.1924, size = 170, normalized size = 0.79 \[ \frac{a^{3} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- e - f x\right )}{a f - b e}} \right )}}{b^{2} \left (n + 1\right ) \left (a d - b c\right ) \left (a f - b e\right )} - \frac{c^{3} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (- e - f x\right )}{c f - d e}} \right )}}{d^{2} \left (n + 1\right ) \left (a d - b c\right ) \left (c f - d e\right )} - \frac{e \left (e + f x\right )^{n + 1}}{b d f^{2} \left (n + 1\right )} + \frac{\left (e + f x\right )^{n + 2}}{b d f^{2} \left (n + 2\right )} - \frac{\left (e + f x\right )^{n + 1} \left (a d + b c\right )}{b^{2} d^{2} f \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(f*x+e)**n/(b*x+a)/(d*x+c),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.85509, size = 262, normalized size = 1.21 \[ \frac{5}{4} e x^4 (e+f x)^n \left (\frac{a b F_1\left (4;-n,1;5;-\frac{f x}{e},-\frac{b x}{a}\right )}{(a+b x) (b c-a d) \left (5 a e F_1\left (4;-n,1;5;-\frac{f x}{e},-\frac{b x}{a}\right )+a f n x F_1\left (5;1-n,1;6;-\frac{f x}{e},-\frac{b x}{a}\right )-b e x F_1\left (5;-n,2;6;-\frac{f x}{e},-\frac{b x}{a}\right )\right )}+\frac{c d F_1\left (4;-n,1;5;-\frac{f x}{e},-\frac{d x}{c}\right )}{(c+d x) (a d-b c) \left (5 c e F_1\left (4;-n,1;5;-\frac{f x}{e},-\frac{d x}{c}\right )+c f n x F_1\left (5;1-n,1;6;-\frac{f x}{e},-\frac{d x}{c}\right )-d e x F_1\left (5;-n,2;6;-\frac{f x}{e},-\frac{d x}{c}\right )\right )}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^3*(e + f*x)^n)/((a + b*x)*(c + d*x)),x]

[Out]

(5*e*x^4*(e + f*x)^n*((a*b*AppellF1[4, -n, 1, 5, -((f*x)/e), -((b*x)/a)])/((b*c
- a*d)*(a + b*x)*(5*a*e*AppellF1[4, -n, 1, 5, -((f*x)/e), -((b*x)/a)] + a*f*n*x*
AppellF1[5, 1 - n, 1, 6, -((f*x)/e), -((b*x)/a)] - b*e*x*AppellF1[5, -n, 2, 6, -
((f*x)/e), -((b*x)/a)])) + (c*d*AppellF1[4, -n, 1, 5, -((f*x)/e), -((d*x)/c)])/(
(-(b*c) + a*d)*(c + d*x)*(5*c*e*AppellF1[4, -n, 1, 5, -((f*x)/e), -((d*x)/c)] +
c*f*n*x*AppellF1[5, 1 - n, 1, 6, -((f*x)/e), -((d*x)/c)] - d*e*x*AppellF1[5, -n,
2, 6, -((f*x)/e), -((d*x)/c)]))))/4

_______________________________________________________________________________________

Maple [F]  time = 0.101, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( fx+e \right ) ^{n}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(f*x+e)^n/(b*x+a)/(d*x+c),x)

[Out]

int(x^3*(f*x+e)^n/(b*x+a)/(d*x+c),x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)^n*x^3/((b*x + a)*(d*x + c)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((f*x + e)^n*x^3/(b*d*x^2 + a*c + (b*c + a*d)*x), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(f*x+e)**n/(b*x+a)/(d*x+c),x)

[Out]

Integral(x**3*(e + f*x)**n/((a + b*x)*(c + d*x)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)^n*x^3/((b*x + a)*(d*x + c)), x)