3.3073 \(\int (a+b x)^m (c+d x)^{-1-m} (e+f x) \, dx\)

Optimal. Leaf size=152 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m}}{d m (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f m+b (d e-c f (m+1))) \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b d m (m+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0799128, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {79, 70, 69} \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m}}{d m (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f m-b c f (m+1)+b d e) \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b d m (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 79

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

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

Mathematica [A]  time = 0.0831963, size = 114, normalized size = 0.75 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{\left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f m-b c f (m+1)+b d e) \, _2F_1\left (m,m+1;m+2;\frac{d (a+b x)}{a d-b c}\right )}{b (m+1)}+c f-d e\right )}{d m (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + m)*(-(d*e) + c*f + ((b*d*e + a*d*f*m - b*c*f*(1 + m))*((b*(c + d*x))/(b*c - a*d))^m*Hypergeome
tric2F1[m, 1 + m, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(1 + m))))/(d*(-(b*c) + a*d)*m*(c + d*x)^m)

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-1-m} \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)^(-1-m)*(f*x+e),x)

[Out]

int((b*x+a)^m*(d*x+c)^(-1-m)*(f*x+e),x)

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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 )}^{-m - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-m)*(f*x+e),x, algorithm="maxima")

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 1), x)

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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 )}^{-m - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-m)*(f*x+e),x, algorithm="fricas")

[Out]

integral((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 1), 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*(d*x+c)**(-1-m)*(f*x+e),x)

[Out]

Timed out

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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 )}^{-m - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-m)*(f*x+e),x, algorithm="giac")

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 1), x)