\(\int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx\) [112]

Optimal result

Integrand size = 25, antiderivative size = 319 \[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {e^2 (e+f x)^{1+n}}{b d f^3 (1+n)}+\frac {(b c+a d) e (e+f x)^{1+n}}{b^2 d^2 f^2 (1+n)}+\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) (e+f x)^{1+n}}{b^3 d^3 f (1+n)}-\frac {2 e (e+f x)^{2+n}}{b d f^3 (2+n)}-\frac {(b c+a d) (e+f x)^{2+n}}{b^2 d^2 f^2 (2+n)}+\frac {(e+f x)^{3+n}}{b d f^3 (3+n)}-\frac {a^4 (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )}{b^3 (b c-a d) (b e-a f) (1+n)}+\frac {c^4 (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )}{d^3 (b c-a d) (d e-c f) (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {186, 45, 70} \[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=-\frac {a^4 (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b (e+f x)}{b e-a f}\right )}{b^3 (n+1) (b c-a d) (b e-a f)}+\frac {\left (a^2 d^2+a b c d+b^2 c^2\right ) (e+f x)^{n+1}}{b^3 d^3 f (n+1)}+\frac {e (a d+b c) (e+f x)^{n+1}}{b^2 d^2 f^2 (n+1)}-\frac {(a d+b c) (e+f x)^{n+2}}{b^2 d^2 f^2 (n+2)}+\frac {c^4 (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d (e+f x)}{d e-c f}\right )}{d^3 (n+1) (b c-a d) (d e-c f)}+\frac {e^2 (e+f x)^{n+1}}{b d f^3 (n+1)}-\frac {2 e (e+f x)^{n+2}}{b d f^3 (n+2)}+\frac {(e+f x)^{n+3}}{b d f^3 (n+3)} \]

[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))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 186

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) (e+f x)^n}{b^3 d^3}-\frac {(b c+a d) x (e+f x)^n}{b^2 d^2}+\frac {x^2 (e+f x)^n}{b d}+\frac {a^4 (e+f x)^n}{b^3 (b c-a d) (a+b x)}+\frac {c^4 (e+f x)^n}{d^3 (-b c+a d) (c+d x)}\right ) \, dx \\ & = \frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) (e+f x)^{1+n}}{b^3 d^3 f (1+n)}+\frac {\int x^2 (e+f x)^n \, dx}{b d}+\frac {a^4 \int \frac {(e+f x)^n}{a+b x} \, dx}{b^3 (b c-a d)}-\frac {c^4 \int \frac {(e+f x)^n}{c+d x} \, dx}{d^3 (b c-a d)}-\frac {(b c+a d) \int x (e+f x)^n \, dx}{b^2 d^2} \\ & = \frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) (e+f x)^{1+n}}{b^3 d^3 f (1+n)}-\frac {a^4 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{b^3 (b c-a d) (b e-a f) (1+n)}+\frac {c^4 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{d^3 (b c-a d) (d e-c f) (1+n)}+\frac {\int \left (\frac {e^2 (e+f x)^n}{f^2}-\frac {2 e (e+f x)^{1+n}}{f^2}+\frac {(e+f x)^{2+n}}{f^2}\right ) \, dx}{b d}-\frac {(b c+a d) \int \left (-\frac {e (e+f x)^n}{f}+\frac {(e+f x)^{1+n}}{f}\right ) \, dx}{b^2 d^2} \\ & = \frac {e^2 (e+f x)^{1+n}}{b d f^3 (1+n)}+\frac {(b c+a d) e (e+f x)^{1+n}}{b^2 d^2 f^2 (1+n)}+\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) (e+f x)^{1+n}}{b^3 d^3 f (1+n)}-\frac {2 e (e+f x)^{2+n}}{b d f^3 (2+n)}-\frac {(b c+a d) (e+f x)^{2+n}}{b^2 d^2 f^2 (2+n)}+\frac {(e+f x)^{3+n}}{b d f^3 (3+n)}-\frac {a^4 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{b^3 (b c-a d) (b e-a f) (1+n)}+\frac {c^4 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{d^3 (b c-a d) (d e-c f) (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {(e+f x)^{1+n} \left (-\frac {a^4 d^3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )}{(b c-a d) (b e-a f)}+\frac {-\left ((b c-a d) (-d e+c f) \left (a^2 d^2 f^2 \left (6+5 n+n^2\right )+a b d f (3+n) (c f (2+n)+d (e-f (1+n) x))+b^2 \left (c^2 f^2 \left (6+5 n+n^2\right )+c d f (3+n) (e-f (1+n) x)+d^2 \left (2 e^2-2 e f (1+n) x+f^2 \left (2+3 n+n^2\right ) x^2\right )\right )\right )\right )+b^3 c^4 f^3 \left (6+5 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )}{(-b c+a d) f^3 (-d e+c f) (2+n) (3+n)}\right )}{b^3 d^3 (1+n)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x^{4} \left (f x +e \right )^{n}}{\left (b x +a \right ) \left (d x +c \right )}d x\]

[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)

Fricas [F]

\[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{4}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]

[In]

integrate(x^4*(f*x+e)^n/(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)

Sympy [F(-2)]

Exception generated. \[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

\[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{4}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{4}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int \frac {x^4\,{\left (e+f\,x\right )}^n}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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