3.3063 \(\int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx\)

Optimal. Leaf size=135 \[ \frac{f (a+b x)^{m+1} (c+d x)^{1-m}}{2 b d}-\frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f (1-m)-b (2 d e-c f (m+1))) \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^2 d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0599319, antiderivative size = 134, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {80, 70, 69} \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (-a d f (1-m)-b c f (m+1)+2 b d e) \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^2 d (m+1)}+\frac{f (a+b x)^{m+1} (c+d x)^{1-m}}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0891403, size = 109, normalized size = 0.81 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (b f (c+d x)-\frac{\left (\frac{b (c+d x)}{b c-a d}\right )^m (-a d f (m-1)+b c f (m+1)-2 b d e) \, _2F_1\left (m,m+1;m+2;\frac{d (a+b x)}{a d-b c}\right )}{m+1}\right )}{2 b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{m}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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