\(\int x^2 (a+b x)^n (c+d x)^{-n} \, dx\) [971]

Optimal result

Integrand size = 20, antiderivative size = 199 \[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=-\frac {(a d (2-n)+b c (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 b^2 d^2}+\frac {x (a+b x)^{1+n} (c+d x)^{1-n}}{3 b d}+\frac {\left (2 a b c d \left (1-n^2\right )+a^2 d^2 \left (2-3 n+n^2\right )+b^2 c^2 \left (2+3 n+n^2\right )\right ) (a+b x)^{1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (1+n)} \]

[Out]

-1/6*(a*d*(2-n)+b*c*(2+n))*(b*x+a)^(1+n)*(d*x+c)^(1-n)/b^2/d^2+1/3*x*(b*x+a)^(1+n)*(d*x+c)^(1-n)/b/d+1/6*(2*a*
b*c*d*(-n^2+1)+a^2*d^2*(n^2-3*n+2)+b^2*c^2*(n^2+3*n+2))*(b*x+a)^(1+n)*(b*(d*x+c)/(-a*d+b*c))^n*hypergeom([n, 1
+n],[2+n],-d*(b*x+a)/(-a*d+b*c))/b^3/d^2/(1+n)/((d*x+c)^n)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {92, 81, 72, 71} \[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=\frac {(a+b x)^{n+1} (c+d x)^{-n} \left (a^2 d^2 \left (n^2-3 n+2\right )+2 a b c d \left (1-n^2\right )+b^2 c^2 \left (n^2+3 n+2\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (n+1)}-\frac {(a+b x)^{n+1} (c+d x)^{1-n} (a d (2-n)+b c (n+2))}{6 b^2 d^2}+\frac {x (a+b x)^{n+1} (c+d x)^{1-n}}{3 b d} \]

[In]

Int[(x^2*(a + b*x)^n)/(c + d*x)^n,x]

[Out]

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

Rule 71

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

Rule 72

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

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b x)^{1+n} (c+d x)^{1-n}}{3 b d}+\frac {\int (a+b x)^n (c+d x)^{-n} (-a c-(a d (2-n)+b c (2+n)) x) \, dx}{3 b d} \\ & = -\frac {(a d (2-n)+b c (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 b^2 d^2}+\frac {x (a+b x)^{1+n} (c+d x)^{1-n}}{3 b d}+\frac {\left (2 a b c d \left (1-n^2\right )+a^2 d^2 \left (2-3 n+n^2\right )+b^2 c^2 \left (2+3 n+n^2\right )\right ) \int (a+b x)^n (c+d x)^{-n} \, dx}{6 b^2 d^2} \\ & = -\frac {(a d (2-n)+b c (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 b^2 d^2}+\frac {x (a+b x)^{1+n} (c+d x)^{1-n}}{3 b d}+\frac {\left (\left (2 a b c d \left (1-n^2\right )+a^2 d^2 \left (2-3 n+n^2\right )+b^2 c^2 \left (2+3 n+n^2\right )\right ) (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx}{6 b^2 d^2} \\ & = -\frac {(a d (2-n)+b c (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 b^2 d^2}+\frac {x (a+b x)^{1+n} (c+d x)^{1-n}}{3 b d}+\frac {\left (2 a b c d \left (1-n^2\right )+a^2 d^2 \left (2-3 n+n^2\right )+b^2 c^2 \left (2+3 n+n^2\right )\right ) (a+b x)^{1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.77 \[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-n} \left (b (a d (-2+n)-b c (2+n)) (c+d x)+2 b^2 d x (c+d x)+\frac {\left (-2 a b c d \left (-1+n^2\right )+a^2 d^2 \left (2-3 n+n^2\right )+b^2 c^2 \left (2+3 n+n^2\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )}{1+n}\right )}{6 b^3 d^2} \]

[In]

Integrate[(x^2*(a + b*x)^n)/(c + d*x)^n,x]

[Out]

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

Maple [F]

\[\int x^{2} \left (b x +a \right )^{n} \left (d x +c \right )^{-n}d x\]

[In]

int(x^2*(b*x+a)^n/((d*x+c)^n),x)

[Out]

int(x^2*(b*x+a)^n/((d*x+c)^n),x)

Fricas [F]

\[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{{\left (d x + c\right )}^{n}} \,d x } \]

[In]

integrate(x^2*(b*x+a)^n/((d*x+c)^n),x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate(x**2*(b*x+a)**n/((d*x+c)**n),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

\[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{{\left (d x + c\right )}^{n}} \,d x } \]

[In]

integrate(x^2*(b*x+a)^n/((d*x+c)^n),x, algorithm="maxima")

[Out]

integrate((b*x + a)^n*x^2/(d*x + c)^n, x)

Giac [F]

\[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{{\left (d x + c\right )}^{n}} \,d x } \]

[In]

integrate(x^2*(b*x+a)^n/((d*x+c)^n),x, algorithm="giac")

[Out]

integrate((b*x + a)^n*x^2/(d*x + c)^n, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=\int \frac {x^2\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n} \,d x \]

[In]

int((x^2*(a + b*x)^n)/(c + d*x)^n,x)

[Out]

int((x^2*(a + b*x)^n)/(c + d*x)^n, x)