3.946 \(\int x^2 (a+b x)^n (c+d x)^p \, dx\)

Optimal. Leaf size=206 \[ -\frac{(a+b x)^{n+1} (c+d x)^{p+1} \left (a^2 d^2 \left (p^2+3 p+2\right )+2 a b c d (n+1) (p+1)+b^2 c^2 \left (n^2+3 n+2\right )\right ) \, _2F_1\left (1,n+p+2;p+2;\frac{b (c+d x)}{b c-a d}\right )}{b^2 d^2 (p+1) (n+p+2) (n+p+3) (b c-a d)}-\frac{(a+b x)^{n+1} (c+d x)^{p+1} (a d (p+2)+b c (n+2))}{b^2 d^2 (n+p+2) (n+p+3)}+\frac{x (a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+3)} \]

[Out]

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

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Rubi [A]  time = 0.424608, antiderivative size = 216, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{(a+b x)^{n+1} (c+d x)^p \left (a^2 d^2 \left (p^2+3 p+2\right )+2 a b c d (n+1) (p+1)+b^2 c^2 \left (n^2+3 n+2\right )\right ) \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^3 d^2 (n+1) (n+p+2) (n+p+3)}-\frac{(a+b x)^{n+1} (c+d x)^{p+1} (a d (p+2)+b c (n+2))}{b^2 d^2 (n+p+2) (n+p+3)}+\frac{x (a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 60.6779, size = 185, normalized size = 0.9 \[ \frac{x \left (a + b x\right )^{n + 1} \left (c + d x\right )^{p + 1}}{b d \left (n + p + 3\right )} - \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{p + 1} \left (a d \left (p + 2\right ) + b c \left (n + 2\right )\right )}{b^{2} d^{2} \left (n + p + 2\right ) \left (n + p + 3\right )} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- p} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{p} \left (- a b c d \left (n + p + 2\right ) + \left (a d \left (p + 1\right ) + b c \left (n + 1\right )\right ) \left (a d \left (p + 2\right ) + b c \left (n + 2\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b^{3} d^{2} \left (n + 1\right ) \left (n + p + 2\right ) \left (n + p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.390902, size = 136, normalized size = 0.66 \[ \frac{4 a c x^3 (a+b x)^n (c+d x)^p F_1\left (3;-n,-p;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 \left (4 a c F_1\left (3;-n,-p;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c n x F_1\left (4;1-n,-p;5;-\frac{b x}{a},-\frac{d x}{c}\right )+a d p x F_1\left (4;-n,1-p;5;-\frac{b x}{a},-\frac{d x}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^n*(d*x+c)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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