3.10.56 \(\int (b x)^m (c+d x)^n (e+f x)^p \, dx\) [956]

Optimal. Leaf size=81 \[ \frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p} F_1\left (1+m;-n,-p;2+m;-\frac {d x}{c},-\frac {f x}{e}\right )}{b (1+m)} \]

[Out]

(b*x)^(1+m)*(d*x+c)^n*(f*x+e)^p*AppellF1(1+m,-n,-p,2+m,-d*x/c,-f*x/e)/b/(1+m)/((1+d*x/c)^n)/((1+f*x/e)^p)

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Rubi [A]
time = 0.03, antiderivative size = 81, 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 {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac {f x}{e}+1\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac {d x}{c},-\frac {f x}{e}\right )}{b (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x)^m*(c + d*x)^n*(e + f*x)^p,x]

[Out]

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

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

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Mathematica [A]
time = 0.11, size = 79, normalized size = 0.98 \begin {gather*} \frac {x (b x)^m (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} (e+f x)^p \left (\frac {e+f x}{e}\right )^{-p} F_1\left (1+m;-n,-p;2+m;-\frac {d x}{c},-\frac {f x}{e}\right )}{1+m} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x)^m*(c + d*x)^n*(e + f*x)^p,x]

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^m*(d*x+c)^n*(f*x+e)^p,x)

[Out]

int((b*x)^m*(d*x+c)^n*(f*x+e)^p,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((b*x)^m*(d*x+c)^n*(f*x+e)^p,x, algorithm="maxima")

[Out]

integrate((b*x)^m*(d*x + c)^n*(f*x + e)^p, 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((b*x)^m*(d*x+c)^n*(f*x+e)^p,x, algorithm="fricas")

[Out]

integral((b*x)^m*(d*x + c)^n*(f*x + e)^p, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)**m*(d*x+c)**n*(f*x+e)**p,x)

[Out]

Timed out

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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((b*x)^m*(d*x+c)^n*(f*x+e)^p,x, algorithm="giac")

[Out]

integrate((b*x)^m*(d*x + c)^n*(f*x + e)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)^p*(b*x)^m*(c + d*x)^n,x)

[Out]

int((e + f*x)^p*(b*x)^m*(c + d*x)^n, x)

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