3.953 \(\int (b x)^m (c+d x)^n (e+f x) \, dx\)

Optimal. Leaf size=108 \[ \frac{f (b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}-\frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (c f (m+1)-d e (m+n+2)) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{b d (m+1) (m+n+2)} \]

[Out]

(f*(b*x)^(1 + m)*(c + d*x)^(1 + n))/(b*d*(2 + m + n)) - ((c*f*(1 + m) - d*e*(2 + m + n))*(b*x)^(1 + m)*(c + d*
x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*x)/c)])/(b*d*(1 + m)*(2 + m + n)*(1 + (d*x)/c)^n)

________________________________________________________________________________________

Rubi [A]  time = 0.0559674, antiderivative size = 99, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 66, 64} \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (\frac{e}{m+1}-\frac{c f}{d (m+n+2)}\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{b}+\frac{f (b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)} \]

Antiderivative was successfully verified.

[In]

Int[(b*x)^m*(c + d*x)^n*(e + f*x),x]

[Out]

(f*(b*x)^(1 + m)*(c + d*x)^(1 + n))/(b*d*(2 + m + n)) + ((e/(1 + m) - (c*f)/(d*(2 + m + n)))*(b*x)^(1 + m)*(c
+ d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*x)/c)])/(b*(1 + (d*x)/c)^n)

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), 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, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0]))
||  !RationalQ[n])

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

\begin{align*} \int (b x)^m (c+d x)^n (e+f x) \, dx &=\frac{f (b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}+\left (e-\frac{c f (1+m)}{d (2+m+n)}\right ) \int (b x)^m (c+d x)^n \, dx\\ &=\frac{f (b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}+\left (\left (e-\frac{c f (1+m)}{d (2+m+n)}\right ) (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 \, dx\\ &=\frac{f (b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}+\frac{\left (e-\frac{c f (1+m)}{d (2+m+n)}\right ) (b x)^{1+m} (c+d x)^n \left (1+\frac{d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac{d x}{c}\right )}{b (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0380015, size = 81, normalized size = 0.75 \[ \frac{x (b x)^m (c+d x)^n \left (\frac{\left (\frac{d x}{c}+1\right )^{-n} (d e (m+n+2)-c f (m+1)) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{m+1}+f (c+d x)\right )}{d (m+n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*x)^m*(c + d*x)^n*(f*(c + d*x) + ((-(c*f*(1 + m)) + d*e*(2 + m + n))*Hypergeometric2F1[1 + m, -n, 2 + m,
-((d*x)/c)])/((1 + m)*(1 + (d*x)/c)^n)))/(d*(2 + m + n))

________________________________________________________________________________________

Maple [F]  time = 0.045, size = 0, normalized size = 0. \begin{align*} \int \left ( bx \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 9.10348, size = 82, normalized size = 0.76 \begin{align*} \frac{b^{m} c^{n} e x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac{b^{m} c^{n} f x^{2} x^{m} \Gamma \left (m + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 2 \\ m + 3 \end{matrix}\middle |{\frac{d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**m*c**n*e*x*x**m*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), d*x*exp_polar(I*pi)/c)/gamma(m + 2) + b**m*c**n*f
*x**2*x**m*gamma(m + 2)*hyper((-n, m + 2), (m + 3,), d*x*exp_polar(I*pi)/c)/gamma(m + 3)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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