3.10.95 \(\int x^m (a+b x)^{1+n} (c+d x)^n \, dx\) [995]

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

[Out]

a*x^(1+m)*(b*x+a)^n*(d*x+c)^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 = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {140, 138} \begin {gather*} \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} \end {gather*}

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 138

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

Rule 140

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]

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*}

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Mathematica [A]
time = 0.18, size = 120, normalized size = 1.52 \begin {gather*} \frac {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} \left (a (2+m) F_1\left (1+m;-n,-n;2+m;-\frac {b x}{a},-\frac {d x}{c}\right )+b (1+m) x F_1\left (2+m;-n,-n;3+m;-\frac {b x}{a},-\frac {d x}{c}\right )\right )}{(1+m) (2+m)} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.03, size = 0, normalized size = 0.00 \[\int x^{m} \left (b x +a \right )^{1+n} \left (d x +c \right )^{n}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

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]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^m\,{\left (a+b\,x\right )}^{n+1}\,{\left (c+d\,x\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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