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

Optimal. Leaf size=108 \[ -\frac{(a d+b c) (a+b x)^{n+1}}{b^2 d^2 (n+1)}+\frac{(a+b x)^{n+2}}{b^2 d (n+2)}+\frac{c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d^2 (n+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.155036, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{(a d+b c) (a+b x)^{n+1}}{b^2 d^2 (n+1)}+\frac{(a+b x)^{n+2}}{b^2 d (n+2)}+\frac{c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d^2 (n+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.9726, size = 85, normalized size = 0.79 \[ - \frac{c^{2} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{d^{2} \left (n + 1\right ) \left (a d - b c\right )} + \frac{\left (a + b x\right )^{n + 2}}{b^{2} d \left (n + 2\right )} - \frac{\left (a + b x\right )^{n + 1} \left (a d + b c\right )}{b^{2} d^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.394099, size = 164, normalized size = 1.52 \[ \frac{(a+b x)^n \left (-\frac{a^2 d^2 \left (\left (\frac{b x}{a}+1\right )^n-1\right ) \left (\frac{b x}{a}+1\right )^{-n}}{b^2 (n+1) (n+2)}+\frac{c^2 \left (\frac{d (a+b x)}{b (c+d x)}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{b c-a d}{b c+b d x}\right )}{n}-\frac{c d (a+b x)}{b n+b}+\frac{a d^2 n x}{b \left (n^2+3 n+2\right )}+\frac{d^2 x^2}{n+2}\right )}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( bx+a \right ) ^{n}}{dx+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^n/(d*x+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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