3.3140 \(\int \frac{(a+b x)^{1-n} (c+d x)^{1+n}}{(b c+a d+2 b d x)^4} \, dx\)

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

[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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Rubi [A]  time = 0.0288944, antiderivative size = 71, normalized size of antiderivative = 1., 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 = {131} \[ \frac{(a+b x)^{2-n} (c+d x)^{n-2} \, _2F_1\left (4,2-n;3-n;-\frac{d (a+b x)}{b (c+d x)}\right )}{b^4 (2-n) (b c-a d)} \]

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 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)^{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*}

Mathematica [A]  time = 0.0372336, size = 70, normalized size = 0.99 \[ -\frac{(a+b x)^{2-n} (c+d x)^{n-2} \, _2F_1\left (4,2-n;3-n;-\frac{d (a+b x)}{b (c+d x)}\right )}{b^4 (n-2) (b c-a d)} \]

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 = 0.098, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{1+n} \left ( bx+a \right ) ^{1-n}}{ \left ( 2\,bdx+ad+bc \right ) ^{4}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{-n + 1}{\left (d x + c\right )}^{n + 1}}{{\left (2 \, b d x + b c + a d\right )}^{4}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{-n + 1}{\left (d x + c\right )}^{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 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + 24 \,{\left (b^{4} c^{2} d^{2} + 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 8 \,{\left (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}\right )} x}, x\right ) \end{align*}

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)]  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)**(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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{-n + 1}{\left (d x + c\right )}^{n + 1}}{{\left (2 \, b d x + b c + a d\right )}^{4}}\,{d x} \end{align*}

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)