3.3144 \(\int \frac{(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^4} \, dx\)

Optimal. Leaf size=65 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;-\frac{d (a+b x)}{b (c+d x)}\right )}{b^4 (m+1) (b c-a d)} \]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m, 2 + m, -((d*(a + b*x))/(b*(c + d*x)))])/(b^4
*(b*c - a*d)*(1 + m))

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Rubi [A]  time = 0.0254304, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {131} \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;-\frac{d (a+b x)}{b (c+d x)}\right )}{b^4 (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m, 2 + m, -((d*(a + b*x))/(b*(c + d*x)))])/(b^4
*(b*c - a*d)*(1 + m))

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

Mathematica [A]  time = 0.0350698, size = 65, normalized size = 1. \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;-\frac{d (a+b x)}{b (c+d x)}\right )}{b^4 (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m, 2 + m, -((d*(a + b*x))/(b*(c + d*x)))])/(b^4
*(b*c - a*d)*(1 + m))

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Maple [F]  time = 0.104, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}}{ \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)^m*(d*x+c)^(2-m)/(2*b*d*x+a*d+b*c)^4,x)

[Out]

int((b*x+a)^m*(d*x+c)^(2-m)/(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 )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\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)^m*(d*x+c)^(2-m)/(2*b*d*x+a*d+b*c)^4,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(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 )}^{m}{\left (d x + c\right )}^{-m + 2}}{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)^m*(d*x+c)^(2-m)/(2*b*d*x+a*d+b*c)^4,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 2)/(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)**m*(d*x+c)**(2-m)/(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 )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\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)^m*(d*x+c)^(2-m)/(2*b*d*x+a*d+b*c)^4,x, algorithm="giac")

[Out]

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