3.995 \(\int x^m (a+b x)^{1+n} (c+d x)^n \, dx\)

Optimal. Leaf size=79 \[ \frac{a x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n-1,-n;m+2;-\frac{b x}{a},-\frac{d x}{c}\right )}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.0452354, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {135, 133} \[ \frac{a x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n-1,-n;m+2;-\frac{b x}{a},-\frac{d x}{c}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x)^(1 + n)*(c + d*x)^n,x]

[Out]

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

Rule 135

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c^IntPart[n]*(c +
d*x)^FracPart[n])/(1 + (d*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d
, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

\begin{align*} \int x^m (a+b x)^{1+n} (c+d x)^n \, dx &=\left (a (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n}\right ) \int x^m \left (1+\frac{b x}{a}\right )^{1+n} (c+d x)^n \, dx\\ &=\left (a (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n} (c+d x)^n \left (1+\frac{d x}{c}\right )^{-n}\right ) \int x^m \left (1+\frac{b x}{a}\right )^{1+n} \left (1+\frac{d x}{c}\right )^n \, dx\\ &=\frac{a x^{1+m} (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n} (c+d x)^n \left (1+\frac{d x}{c}\right )^{-n} F_1\left (1+m;-1-n,-n;2+m;-\frac{b x}{a},-\frac{d x}{c}\right )}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.177015, size = 120, normalized size = 1.52 \[ \frac{x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (a (m+2) F_1\left (m+1;-n,-n;m+2;-\frac{b x}{a},-\frac{d x}{c}\right )+b (m+1) x F_1\left (m+2;-n,-n;m+3;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}{(m+1) (m+2)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^m*(a + b*x)^(1 + n)*(c + d*x)^n,x]

[Out]

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

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Maple [F]  time = 0.139, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( bx+a \right ) ^{1+n} \left ( dx+c \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x)

[Out]

int(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n + 1}{\left (d x + c\right )}^{n} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x, algorithm="maxima")

[Out]

integrate((b*x + a)^(n + 1)*(d*x + c)^n*x^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x + a\right )}^{n + 1}{\left (d x + c\right )}^{n} x^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x, algorithm="fricas")

[Out]

integral((b*x + a)^(n + 1)*(d*x + c)^n*x^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x+a)**(1+n)*(d*x+c)**n,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n + 1}{\left (d x + c\right )}^{n} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x, algorithm="giac")

[Out]

integrate((b*x + a)^(n + 1)*(d*x + c)^n*x^m, x)