3.3161 \(\int (a+b x)^m (c+d x)^n (e+f x) \, dx\)

Optimal. Leaf size=131 \[ \frac{f (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}-\frac{(a+b x)^{m+1} (c+d x)^{n+1} (b d e (m+n+2)-f (a d (n+1)+b c (m+1))) \, _2F_1\left (1,m+n+2;n+2;\frac{b (c+d x)}{b c-a d}\right )}{b d (n+1) (m+n+2) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0776548, antiderivative size = 141, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {80, 70, 69} \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (b d e (m+n+2)-f (a d (n+1)+b c (m+1))) \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^2 d (m+1) (m+n+2)}+\frac{f (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(1 + n))/(b*d*(2 + m + n)) + ((b*d*e*(2 + m + n) - f*(b*c*(1 + m) + a*d*(1 + n)
))*(a + b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b^2*d*(1
+ m)*(2 + m + n)*((b*(c + d*x))/(b*c - a*d))^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 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

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

Rubi steps

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

Mathematica [A]  time = 0.115729, size = 117, normalized size = 0.89 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{\left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (-a d f (n+1)-b c f (m+1)+b d e (m+n+2)) \, _2F_1\left (m+1,-n;m+2;\frac{d (a+b x)}{a d-b c}\right )}{m+1}+b f (c+d x)\right )}{b^2 d (m+n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + m)*(c + d*x)^n*(b*f*(c + d*x) + ((-(b*c*f*(1 + m)) - a*d*f*(1 + n) + b*d*e*(2 + m + n))*Hyperg
eometric2F1[1 + m, -n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)])/((1 + m)*((b*(c + d*x))/(b*c - a*d))^n)))/(b^2*d*
(2 + m + n))

________________________________________________________________________________________

Maple [F]  time = 0.058, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \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+a)^m*(d*x+c)^n*(f*x+e),x)

[Out]

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

[Out]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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