3.3132 \(\int \frac{(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx\)

Optimal. Leaf size=311 \[ -\frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (5 d e-c f (3-m))+b^2 \left (-\left (c^2 f^2 \left (m^2-7 m+12\right )-10 c d e f (3-m)+20 d^2 e^2\right )\right )\right ) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{20 (m+1) (b e-a f)^6 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} (b (6 d e-c f (4-m))-a d f (m+2))}{20 (e+f x)^4 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)} \]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(5*(b*e - a*f)*(d*e - c*f)*(e + f*x)^5) - (f*(b*(6*d*e - c*f*(4 - m))
- a*d*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(20*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^4) - ((b*c -
a*d)^3*(2*a*b*d*f*(5*d*e - c*f*(3 - m))*(1 + m) - a^2*d^2*f^2*(2 + 3*m + m^2) - b^2*(20*d^2*e^2 - 10*c*d*e*f*(
3 - m) + c^2*f^2*(12 - 7*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m, 2 + m, ((
d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/(20*(b*e - a*f)^6*(d*e - c*f)^2*(1 + m))

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Rubi [A]  time = 0.323771, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {129, 151, 12, 131} \[ -\frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (5 d e-c f (3-m))+b^2 \left (-\left (c^2 f^2 \left (m^2-7 m+12\right )-10 c d e f (3-m)+20 d^2 e^2\right )\right )\right ) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{20 (m+1) (b e-a f)^6 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} (-a d f (m+2)-b c f (4-m)+6 b d e)}{20 (e+f x)^4 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^6,x]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(5*(b*e - a*f)*(d*e - c*f)*(e + f*x)^5) - (f*(6*b*d*e - b*c*f*(4 - m)
- a*d*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(20*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^4) - ((b*c -
a*d)^3*(2*a*b*d*f*(5*d*e - c*f*(3 - m))*(1 + m) - a^2*d^2*f^2*(2 + 3*m + m^2) - b^2*(20*d^2*e^2 - 10*c*d*e*f*(
3 - m) + c^2*f^2*(12 - 7*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m, 2 + m, ((
d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/(20*(b*e - a*f)^6*(d*e - c*f)^2*(1 + m))

Rule 129

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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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}}{(e+f x)^6} \, dx &=-\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac{\int \frac{(a+b x)^m (c+d x)^{2-m} (-b (5 d e-c f (4-m))+a d f (2+m)+b d f x)}{(e+f x)^5} \, dx}{5 (b e-a f) (d e-c f)}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac{f (6 b d e-b c f (4-m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{3-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}+\frac{\int \frac{\left (-2 a b d f (5 d e-c f (3-m)) (1+m)+a^2 d^2 f^2 \left (2+3 m+m^2\right )+b^2 \left (20 d^2 e^2-10 c d e f (3-m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (a+b x)^m (c+d x)^{2-m}}{(e+f x)^4} \, dx}{20 (b e-a f)^2 (d e-c f)^2}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac{f (6 b d e-b c f (4-m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{3-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}-\frac{\left (2 a b d f (5 d e-c f (3-m)) (1+m)-a^2 d^2 f^2 \left (2+3 m+m^2\right )-b^2 \left (20 d^2 e^2-10 c d e f (3-m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) \int \frac{(a+b x)^m (c+d x)^{2-m}}{(e+f x)^4} \, dx}{20 (b e-a f)^2 (d e-c f)^2}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{5 (b e-a f) (d e-c f) (e+f x)^5}-\frac{f (6 b d e-b c f (4-m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{3-m}}{20 (b e-a f)^2 (d e-c f)^2 (e+f x)^4}-\frac{(b c-a d)^3 \left (2 a b d f (5 d e-c f (3-m)) (1+m)-a^2 d^2 f^2 \left (2+3 m+m^2\right )-b^2 \left (20 d^2 e^2-10 c d e f (3-m)+c^2 f^2 \left (12-7 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (4,1+m;2+m;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{20 (b e-a f)^6 (d e-c f)^2 (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.659521, size = 271, normalized size = 0.87 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (\frac{(b c-a d)^3 \left (a^2 d^2 f^2 \left (m^2+3 m+2\right )-2 a b d f (m+1) (c f (m-3)+5 d e)+b^2 \left (c^2 f^2 \left (m^2-7 m+12\right )+10 c d e f (m-3)+20 d^2 e^2\right )\right ) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^5 (d e-c f)}-\frac{f (c+d x)^4 (-a d f (m+2)+b c f (m-4)+6 b d e)}{(e+f x)^4 (b e-a f) (d e-c f)}-\frac{4 f (c+d x)^4}{(e+f x)^5}\right )}{20 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^6,x]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*((-4*f*(c + d*x)^4)/(e + f*x)^5 - (f*(6*b*d*e + b*c*f*(-4 + m) - a*d*f*(
2 + m))*(c + d*x)^4)/((b*e - a*f)*(d*e - c*f)*(e + f*x)^4) + ((b*c - a*d)^3*(-2*a*b*d*f*(5*d*e + c*f*(-3 + m))
*(1 + m) + a^2*d^2*f^2*(2 + 3*m + m^2) + b^2*(20*d^2*e^2 + 10*c*d*e*f*(-3 + m) + c^2*f^2*(12 - 7*m + m^2)))*Hy
pergeometric2F1[4, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^5*(d*e - c*f)*
(1 + m))))/(20*(b*e - a*f)*(d*e - c*f))

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Maple [F]  time = 0.341, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}}{ \left ( fx+e \right ) ^{6}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^6,x)

[Out]

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^6,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 (f x + e\right )}^{6}}\,{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)/(f*x+e)^6,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^6, 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}}{f^{6} x^{6} + 6 \, e f^{5} x^{5} + 15 \, e^{2} f^{4} x^{4} + 20 \, e^{3} f^{3} x^{3} + 15 \, e^{4} f^{2} x^{2} + 6 \, e^{5} f x + e^{6}}, 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)/(f*x+e)^6,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 2)/(f^6*x^6 + 6*e*f^5*x^5 + 15*e^2*f^4*x^4 + 20*e^3*f^3*x^3 + 15*e^4*f^2*
x^2 + 6*e^5*f*x + e^6), 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)/(f*x+e)**6,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 (f x + e\right )}^{6}}\,{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)/(f*x+e)^6,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^6, x)