3.32.57 \(\int (a+b x)^m (c+d x)^n (e+f x)^{-3-m-n} \, dx\) [3157]

Optimal. Leaf size=227 \[ -\frac {f (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f) (d e-c f) (2+m+n)}-\frac {(a d f (1+m)+b (c f (1+n)-d e (2+m+n))) (a+b x)^{1+m} (c+d x)^n \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x)^{-1-m-n} \, _2F_1\left (1+m,-n;2+m;-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^2 (d e-c f) (1+m) (2+m+n)} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {98, 134} \begin {gather*} -\frac {(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} (a d f (m+1)+b c f (n+1)-b d e (m+n+2)) \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (b e-a f)^2 (d e-c f)}-\frac {f (a+b x)^{m+1} (c+d x)^{n+1} (e+f x)^{-m-n-2}}{(m+n+2) (b e-a f) (d e-c f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^(-3 - m - n),x]

[Out]

-((f*(a + b*x)^(1 + m)*(c + d*x)^(1 + n)*(e + f*x)^(-2 - m - n))/((b*e - a*f)*(d*e - c*f)*(2 + m + n))) - ((a*
d*f*(1 + m) + b*c*f*(1 + n) - b*d*e*(2 + m + n))*(a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-1 - m - n)*Hypergeo
metric2F1[1 + m, -n, 2 + m, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)^2*(d*e - c*f)*(1
+ m)*(2 + m + n)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^n)

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 134

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

Rubi steps

\begin {align*} \int (a+b x)^m (c+d x)^n (e+f x)^{-3-m-n} \, dx &=-\frac {f (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f) (d e-c f) (2+m+n)}-\frac {(a d f (1+m)+b c f (1+n)-b d e (2+m+n)) \int (a+b x)^m (c+d x)^n (e+f x)^{-2-m-n} \, dx}{(b e-a f) (d e-c f) (2+m+n)}\\ &=-\frac {f (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f) (d e-c f) (2+m+n)}-\frac {(a d f (1+m)+b c f (1+n)-b d e (2+m+n)) (a+b x)^{1+m} (c+d x)^n \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x)^{-1-m-n} \, _2F_1\left (1+m,-n;2+m;-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^2 (d e-c f) (1+m) (2+m+n)}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 189, normalized size = 0.83 \begin {gather*} -\frac {(a+b x)^{1+m} (c+d x)^n (e+f x)^{-2-m-n} \left (f (c+d x)-\frac {(-a d f (1+m)-b c f (1+n)+b d e (2+m+n)) \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x) \, _2F_1\left (1+m,-n;2+m;\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)}\right )}{(b e-a f) (d e-c f) (2+m+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^(-3 - m - n),x]

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{-3-m -n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-3-m-n),x)

[Out]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-3-m-n),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)^n*(f*x+e)^(-3-m-n),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 3), 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)^n*(f*x+e)^(-3-m-n),x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 3), 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)**n*(f*x+e)**(-3-m-n),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)^n*(f*x+e)^(-3-m-n),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 3), 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 )}^n}{{\left (e+f\,x\right )}^{m+n+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^m*(c + d*x)^n)/(e + f*x)^(m + n + 3),x)

[Out]

int(((a + b*x)^m*(c + d*x)^n)/(e + f*x)^(m + n + 3), x)

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