3.32.49 \(\int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx\) [3149]

Optimal. Leaf size=124 \[ \frac {(a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^{-1+n} \, _2F_1\left (1+m,m+n;2+m;-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {134} \begin {gather*} \frac {(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \, _2F_1\left (m+1,m+n;m+2;-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.11, size = 123, normalized size = 0.99 \begin {gather*} \frac {(a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^{-1+n} \, _2F_1\left (1+m,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)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (d x +c \right )^{-m -n} \left (f x +e \right )^{-2+n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^{n-2}\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+n}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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