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

Optimal. Leaf size=199 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n} \left (a^2 d^2 \left (n^2-3 n+2\right )+2 a b c d \left (1-n^2\right )+b^2 c^2 \left (n^2+3 n+2\right )\right ) \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (n+1)}-\frac{(a+b x)^{n+1} (c+d x)^{1-n} (a d (2-n)+b c (n+2))}{6 b^2 d^2}+\frac{x (a+b x)^{n+1} (c+d x)^{1-n}}{3 b d} \]

[Out]

-((a*d*(2 - n) + b*c*(2 + n))*(a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(6*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)

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Rubi [A]  time = 0.367531, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n} \left (a^2 d^2 \left (n^2-3 n+2\right )+2 a b c d \left (1-n^2\right )+b^2 c^2 \left (n^2+3 n+2\right )\right ) \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (n+1)}-\frac{(a+b x)^{n+1} (c+d x)^{1-n} (a d (2-n)+b c (n+2))}{6 b^2 d^2}+\frac{x (a+b x)^{n+1} (c+d x)^{1-n}}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

-((a*d*(2 - n) + b*c*(2 + n))*(a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(6*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)

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Rubi in Sympy [A]  time = 34.1977, size = 160, normalized size = 0.8 \[ \frac{x \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n + 1}}{3 b d} - \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n + 1} \left (a d \left (- n + 2\right ) + b c \left (n + 2\right )\right )}{6 b^{2} d^{2}} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{n} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n} \left (- 2 a b c d + \left (a d \left (- n + 1\right ) + b c \left (n + 1\right )\right ) \left (a d \left (- n + 2\right ) + b c \left (n + 2\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} n, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{6 b^{3} d^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)**n/((d*x+c)**n),x)

[Out]

x*(a + b*x)**(n + 1)*(c + d*x)**(-n + 1)/(3*b*d) - (a + b*x)**(n + 1)*(c + d*x)*
*(-n + 1)*(a*d*(-n + 2) + b*c*(n + 2))/(6*b**2*d**2) + (b*(-c - d*x)/(a*d - b*c)
)**n*(a + b*x)**(n + 1)*(c + d*x)**(-n)*(-2*a*b*c*d + (a*d*(-n + 1) + b*c*(n + 1
))*(a*d*(-n + 2) + b*c*(n + 2)))*hyper((n, n + 1), (n + 2,), d*(a + b*x)/(a*d -
b*c))/(6*b**3*d**2*(n + 1))

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Mathematica [C]  time = 0.108486, size = 130, normalized size = 0.65 \[ \frac{4 a c x^3 (a+b x)^n (c+d x)^{-n} F_1\left (3;-n,n;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{12 a c F_1\left (3;-n,n;4;-\frac{b x}{a},-\frac{d x}{c}\right )+3 b c n x F_1\left (4;1-n,n;5;-\frac{b x}{a},-\frac{d x}{c}\right )-3 a d n x F_1\left (4;-n,n+1;5;-\frac{b x}{a},-\frac{d x}{c}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(4*a*c*x^3*(a + b*x)^n*AppellF1[3, -n, n, 4, -((b*x)/a), -((d*x)/c)])/((c + d*x)
^n*(12*a*c*AppellF1[3, -n, n, 4, -((b*x)/a), -((d*x)/c)] + 3*b*c*n*x*AppellF1[4,
1 - n, n, 5, -((b*x)/a), -((d*x)/c)] - 3*a*d*n*x*AppellF1[4, -n, 1 + n, 5, -((b
*x)/a), -((d*x)/c)]))

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n} x^{2}}{{\left (d x + c\right )}^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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