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

Optimal. Leaf size=122 \[ \frac{c^2 (a+b x)^{n+1}}{d^2 (c+d x) (b c-a d)}+\frac{c (a+b x)^{n+1} (2 a d-b c (n+2)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d^2 (n+1) (b c-a d)^2}+\frac{(a+b x)^{n+1}}{b d^2 (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.22471, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{c^2 (a+b x)^{n+1}}{d^2 (c+d x) (b c-a d)}+\frac{c (a+b x)^{n+1} (2 a d-b c (n+2)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d^2 (n+1) (b c-a d)^2}+\frac{(a+b x)^{n+1}}{b d^2 (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.6717, size = 102, normalized size = 0.84 \[ - \frac{c^{2} \left (a + b x\right )^{n + 1}}{d^{2} \left (c + d x\right ) \left (a d - b c\right )} + \frac{c \left (a + b x\right )^{n + 1} \left (2 a d - b c n - 2 b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{d^{2} \left (n + 1\right ) \left (a d - b c\right )^{2}} + \frac{\left (a + b x\right )^{n + 1}}{b d^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.335153, size = 126, normalized size = 1.03 \[ \frac{4 a c x^3 (a+b x)^n F_1\left (3;-n,2;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 (c+d x)^2 \left (4 a c F_1\left (3;-n,2;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c n x F_1\left (4;1-n,2;5;-\frac{b x}{a},-\frac{d x}{c}\right )-2 a d x F_1\left (4;-n,3;5;-\frac{b x}{a},-\frac{d x}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(4*a*c*x^3*(a + b*x)^n*AppellF1[3, -n, 2, 4, -((b*x)/a), -((d*x)/c)])/(3*(c + d*
x)^2*(4*a*c*AppellF1[3, -n, 2, 4, -((b*x)/a), -((d*x)/c)] + b*c*n*x*AppellF1[4,
1 - n, 2, 5, -((b*x)/a), -((d*x)/c)] - 2*a*d*x*AppellF1[4, -n, 3, 5, -((b*x)/a),
-((d*x)/c)]))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^n/(d*x+c)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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