3.137 \(\int \frac{(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx\)

Optimal. Leaf size=233 \[ -\frac{(a+b x)^m (B e-A f) (c+d x)^{-m} \, _2F_1\left (1,-m;1-m;\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f^2 m}-\frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right ) (a B d f m-b (-A d f+B c f m+B d e))}{b f^2 m (m+1) (b c-a d)}-\frac{d (a+b x)^{m+1} (B e-A f) (c+d x)^{-m}}{f^2 m (b c-a d)} \]

[Out]

-((d*(B*e - A*f)*(a + b*x)^(1 + m))/((b*c - a*d)*f^2*m*(c + d*x)^m)) - ((B*e - A*f)*(a + b*x)^m*Hypergeometric
2F1[1, -m, 1 - m, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/(f^2*m*(c + d*x)^m) - ((a*B*d*f*m - b*(B*d
*e - A*d*f + B*c*f*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*(b*c - a*d)*f^2*m*(1 + m)*(c + d*x)^m)

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Rubi [A]  time = 0.12704, antiderivative size = 220, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {157, 70, 69, 105, 131} \[ \frac{(a+b x)^m (B e-A f) (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^2 m}-\frac{(a+b x)^m (B e-A f) (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{f^2 m}+\frac{B (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b f (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((B*e - A*f)*(a + b*x)^m*Hypergeometric2F1[1, m, 1 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/(f^2
*m*(c + d*x)^m) - ((B*e - A*f)*(a + b*x)^m*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[m, m, 1 + m, -((d*(
a + b*x))/(b*c - a*d))])/(f^2*m*(c + d*x)^m) + (B*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometr
ic2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b*f*(1 + m)*(c + d*x)^m)

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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]))

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

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

Mathematica [A]  time = 0.136575, size = 174, normalized size = 0.75 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\left (\frac{b (c+d x)}{b c-a d}\right )^m \left (B f m (a+b x) \, _2F_1\left (m,m+1;m+2;\frac{d (a+b x)}{a d-b c}\right )-b (m+1) (B e-A f) \, _2F_1\left (m,m;m+1;\frac{d (a+b x)}{a d-b c}\right )\right )+b (m+1) (B e-A f) \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{b f^2 m (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( Bx+A \right ) \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{m} \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((B*x + A)*(b*x + a)^m/((f*x + e)*(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 (B x + A\right )}{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}{\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*(B*x+A)/((d*x+c)^m)/(f*x+e),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

[Out]

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