3.32.40 \(\int \frac {(a+b x)^{1-n} (c+d x)^{1+n}}{(b c+a d+2 b d x)^4} \, dx\) [3140]

Optimal. Leaf size=71 \[ \frac {(a+b x)^{2-n} (c+d x)^{-2+n} \, _2F_1\left (4,2-n;3-n;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (b c-a d) (2-n)} \]

[Out]

(b*x+a)^(2-n)*(d*x+c)^(-2+n)*hypergeom([4, 2-n],[3-n],-d*(b*x+a)/b/(d*x+c))/b^4/(-a*d+b*c)/(2-n)

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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {133} \begin {gather*} \frac {(a+b x)^{2-n} (c+d x)^{n-2} \, _2F_1\left (4,2-n;3-n;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (2-n) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(1 - n)*(c + d*x)^(1 + n))/(b*c + a*d + 2*b*d*x)^4,x]

[Out]

((a + b*x)^(2 - n)*(c + d*x)^(-2 + n)*Hypergeometric2F1[4, 2 - n, 3 - n, -((d*(a + b*x))/(b*(c + d*x)))])/(b^4
*(b*c - a*d)*(2 - n))

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]

Rubi steps

\begin {align*} \int \frac {(a+b x)^{1-n} (c+d x)^{1+n}}{(b c+a d+2 b d x)^4} \, dx &=\frac {(a+b x)^{2-n} (c+d x)^{-2+n} \, _2F_1\left (4,2-n;3-n;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (b c-a d) (2-n)}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 70, normalized size = 0.99 \begin {gather*} -\frac {(a+b x)^{2-n} (c+d x)^{-2+n} \, _2F_1\left (4,2-n;3-n;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (b c-a d) (-2+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(1 - n)*(c + d*x)^(1 + n))/(b*c + a*d + 2*b*d*x)^4,x]

[Out]

-(((a + b*x)^(2 - n)*(c + d*x)^(-2 + n)*Hypergeometric2F1[4, 2 - n, 3 - n, -((d*(a + b*x))/(b*(c + d*x)))])/(b
^4*(b*c - a*d)*(-2 + n)))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{1-n} \left (d x +c \right )^{1+n}}{\left (2 b d x +a d +b c \right )^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1-n)*(d*x+c)^(1+n)/(2*b*d*x+a*d+b*c)^4,x)

[Out]

int((b*x+a)^(1-n)*(d*x+c)^(1+n)/(2*b*d*x+a*d+b*c)^4,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)^(1-n)*(d*x+c)^(1+n)/(2*b*d*x+a*d+b*c)^4,x, algorithm="maxima")

[Out]

integrate((b*x + a)^(-n + 1)*(d*x + c)^(n + 1)/(2*b*d*x + b*c + a*d)^4, 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)^(1-n)*(d*x+c)^(1+n)/(2*b*d*x+a*d+b*c)^4,x, algorithm="fricas")

[Out]

integral((b*x + a)^(-n + 1)*(d*x + c)^(n + 1)/(16*b^4*d^4*x^4 + b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 +
4*a^3*b*c*d^3 + a^4*d^4 + 32*(b^4*c*d^3 + a*b^3*d^4)*x^3 + 24*(b^4*c^2*d^2 + 2*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2
+ 8*(b^4*c^3*d + 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 + a^3*b*d^4)*x), 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)**(1-n)*(d*x+c)**(1+n)/(2*b*d*x+a*d+b*c)**4,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)^(1-n)*(d*x+c)^(1+n)/(2*b*d*x+a*d+b*c)^4,x, algorithm="giac")

[Out]

integrate((b*x + a)^(-n + 1)*(d*x + c)^(n + 1)/(2*b*d*x + b*c + a*d)^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{1-n}\,{\left (c+d\,x\right )}^{n+1}}{{\left (a\,d+b\,c+2\,b\,d\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(1 - n)*(c + d*x)^(n + 1))/(a*d + b*c + 2*b*d*x)^4,x)

[Out]

int(((a + b*x)^(1 - n)*(c + d*x)^(n + 1))/(a*d + b*c + 2*b*d*x)^4, x)

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