∫ (a+bx)m(c+dx)2−m ________________________

3.32.28 \(\int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^2} \, dx\) [3128]

Optimal. Leaf size=316 \[ -\frac {2 d^2 (d e-c f) (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) f^3 m}+\frac {(d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-m}}{f^2 (b e-a f) (e+f x)}+\frac {(d e-c f) (a d f (2-m)-b (2 d e-c f m)) (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,-m;1-m;\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f^3 (b e-a f) m}+\frac {d^2 (b (2 d e-c f (2-m))-a d f m) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d) f^3 m (1+m)} \]

[Out]

-2*d^2*(-c*f+d*e)*(b*x+a)^(1+m)/(-a*d+b*c)/f^3/m/((d*x+c)^m)+(-c*f+d*e)^2*(b*x+a)^(1+m)/f^2/(-a*f+b*e)/((d*x+c

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

c*f+d*e)/(b*x+a))/f^3/(-a*f+b*e)/m/((d*x+c)^m)+d^2*(b*(2*d*e-c*f*(2-m))-a*d*f*m)*(b*x+a)^(1+m)*(b*(d*x+c)/(-a*

d+b*c))^m*hypergeom([m, 1+m],[2+m],-d*(b*x+a)/(-a*d+b*c))/b/(-a*d+b*c)/f^3/m/(1+m)/((d*x+c)^m)

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Rubi [A]
time = 0.20, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {131, 80, 72, 71, 156, 12, 133} \begin {gather*} \frac {d^2 (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 (2-m)+2 b d e) \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{b f^3 m (m+1) (b c-a d)}-\frac {2 d^2 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m}}{f^3 m (b c-a d)}+\frac {(a+b x)^m (d e-c f) (c+d x)^{-m} (a d f (2-m)-b (2 d e-c f 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^3 m (b e-a f)}+\frac {(a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m}}{f^2 (e+f x) (b e-a f)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*d^2*(d*e - c*f)*(a + b*x)^(1 + m))/((b*c - a*d)*f^3*m*(c + d*x)^m) + ((d*e - c*f)^2*(a + b*x)^(1 + m))/(f^

2*(b*e - a*f)*(c + d*x)^m*(e + f*x)) + ((d*e - c*f)*(a*d*f*(2 - m) - b*(2*d*e - c*f*m))*(a + b*x)^m*Hypergeome

tric2F1[1, -m, 1 - m, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/(f^3*(b*e - a*f)*m*(c + d*x)^m) + (d^2

*(2*b*d*e - b*c*f*(2 - m) - a*d*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^3*m*(1 + m)*(c + d*x)^m)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c

- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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 72

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 80

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 131

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[f^(p -

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

*x)^m*((e + f*x)^p/(c + d*x)^(m + 1))*ExpandToSum[f^(-p + 1)*(c + d*x)^(-p + 1) - (d*e*p - c*f*(p - 1) + d*f*x

)/(d^p*(e + f*x)^p), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p, 0] && ILtQ[p, 0] && (L

tQ[m, 0] || SumSimplerQ[m, 1] || !(LtQ[n, 0] || SumSimplerQ[n, 1]))

Rule 133

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

(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p

+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && !ILtQ[m, 0]

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb

ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f

))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*

g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p

+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {(a+b x)^m (c+d x)^{2-m}}{(e+f x)^2} \, dx &=\frac {\left ((b c-a d)^2 (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int \frac {(a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{2-m}}{(e+f x)^2} \, dx}{b^2}\\ &=\frac {(b c-a d)^2 (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m F_1\left (1+m;-2+m,2;2+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b (b e-a f)^2 (1+m)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 0.20, size = 103, normalized size = 0.33 \begin {gather*} \frac {b (a+b x)^{1+m} (c+d x)^{2-m} \left (\frac {b (c+d x)}{b c-a d}\right )^{-2+m} F_1\left (1+m;-2+m,2;2+m;\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )}{(b e-a f)^2 (1+m)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*((b*(c + d*x))/(b*c - a*d))^(-2 + m)*AppellF1[1 + m, -2 + m, 2, 2 + m,

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{2-m}}{\left (f x +e \right )^{2}}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^2,x)

[Out]

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^2, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^2,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 2)/(f^2*x^2 + 2*f*x*e + e^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(d*x+c)**(2-m)/(f*x+e)**2,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{2-m}}{{\left (e+f\,x\right )}^2} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^2,x)

[Out]

int(((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^2, x)

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