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

Optimal. Leaf size=117 \[ \frac{(a+b x)^{n+1} (c+d x)^{p+1} (a d (p+1)+b c (n+1)) \, _2F_1\left (1,n+p+2;p+2;\frac{b (c+d x)}{b c-a d}\right )}{b d (p+1) (n+p+2) (b c-a d)}+\frac{(a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+2)} \]

[Out]

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

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Rubi [A]  time = 0.14896, antiderivative size = 129, normalized size of antiderivative = 1.1, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{(a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+2)}-\frac{(a+b x)^{n+1} (c+d x)^p (a d (p+1)+b c (n+1)) \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^2 d (n+1) (n+p+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.7126, size = 104, normalized size = 0.89 \[ \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{p + 1}}{b d \left (n + p + 2\right )} - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- p} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{p} \left (a d \left (p + 1\right ) + b c \left (n + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b^{2} d \left (n + 1\right ) \left (n + p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.367063, size = 136, normalized size = 1.16 \[ \frac{3 a c x^2 (a+b x)^n (c+d x)^p F_1\left (2;-n,-p;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-n,-p;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 b c n x F_1\left (3;1-n,-p;4;-\frac{b x}{a},-\frac{d x}{c}\right )+2 a d p x F_1\left (3;-n,1-p;4;-\frac{b x}{a},-\frac{d x}{c}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*a*c*x^2*(a + b*x)^n*(c + d*x)^p*AppellF1[2, -n, -p, 3, -((b*x)/a), -((d*x)/c)
])/(6*a*c*AppellF1[2, -n, -p, 3, -((b*x)/a), -((d*x)/c)] + 2*b*c*n*x*AppellF1[3,
1 - n, -p, 4, -((b*x)/a), -((d*x)/c)] + 2*a*d*p*x*AppellF1[3, -n, 1 - p, 4, -((
b*x)/a), -((d*x)/c)])

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int x \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^n*(d*x+c)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^n*(d*x + c)^p*x, 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*(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. \[ \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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