3.3105 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx\)

Optimal. Leaf size=650 \[ \frac{6 b^2 (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-2}}{d^4 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{8 b^2 f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (m+3) (b c-a d)^3}+\frac{6 b^3 (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{6 f^2 (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^4 (m+2) (b c-a d)}+\frac{4 f^3 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^4 (m+1) (b c-a d)}+\frac{6 b f^2 (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (b c-a d)^2}+\frac{(a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-4}}{d^4 (m+4) (b c-a d)}+\frac{4 f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^4 (m+3) (b c-a d)}+\frac{3 b (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-3}}{d^4 (m+3) (m+4) (b c-a d)^2}+\frac{8 b f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^4 (m+2) (m+3) (b c-a d)^2}-\frac{f^4 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^5 m} \]

[Out]

((d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/(d^4*(b*c - a*d)*(4 + m)) + (4*f*(d*e - c*f)^3*(a + b*x)^
(1 + m)*(c + d*x)^(-3 - m))/(d^4*(b*c - a*d)*(3 + m)) + (3*b*(d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m
))/(d^4*(b*c - a*d)^2*(3 + m)*(4 + m)) + (6*f^2*(d*e - c*f)^2*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^4*(b*c
- a*d)*(2 + m)) + (8*b*f*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^4*(b*c - a*d)^2*(2 + m)*(3 + m
)) + (6*b^2*(d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^4*(b*c - a*d)^3*(2 + m)*(3 + m)*(4 + m)) +
(4*f^3*(d*e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)*(1 + m)) + (6*b*f^2*(d*e - c*f)^2*(a
+ b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^2*(1 + m)*(2 + m)) + (8*b^2*f*(d*e - c*f)^3*(a + b*x)^(1
+ m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^3*(1 + m)*(2 + m)*(3 + m)) + (6*b^3*(d*e - c*f)^4*(a + b*x)^(1 + m)*
(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) - (f^4*(a + b*x)^m*Hypergeometric2F1[-
m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^5*m*(-((d*(a + b*x))/(b*c - a*d)))^m*(c + d*x)^m)

________________________________________________________________________________________

Rubi [A]  time = 0.499441, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {128, 45, 37, 70, 69} \[ \frac{6 b^2 (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-2}}{d^4 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{8 b^2 f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (m+3) (b c-a d)^3}+\frac{6 b^3 (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{6 f^2 (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^4 (m+2) (b c-a d)}+\frac{4 f^3 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^4 (m+1) (b c-a d)}+\frac{6 b f^2 (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (b c-a d)^2}+\frac{(a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-4}}{d^4 (m+4) (b c-a d)}+\frac{4 f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^4 (m+3) (b c-a d)}+\frac{3 b (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-3}}{d^4 (m+3) (m+4) (b c-a d)^2}+\frac{8 b f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^4 (m+2) (m+3) (b c-a d)^2}-\frac{f^4 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^5 m} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^4,x]

[Out]

((d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/(d^4*(b*c - a*d)*(4 + m)) + (4*f*(d*e - c*f)^3*(a + b*x)^
(1 + m)*(c + d*x)^(-3 - m))/(d^4*(b*c - a*d)*(3 + m)) + (3*b*(d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m
))/(d^4*(b*c - a*d)^2*(3 + m)*(4 + m)) + (6*f^2*(d*e - c*f)^2*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^4*(b*c
- a*d)*(2 + m)) + (8*b*f*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^4*(b*c - a*d)^2*(2 + m)*(3 + m
)) + (6*b^2*(d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^4*(b*c - a*d)^3*(2 + m)*(3 + m)*(4 + m)) +
(4*f^3*(d*e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)*(1 + m)) + (6*b*f^2*(d*e - c*f)^2*(a
+ b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^2*(1 + m)*(2 + m)) + (8*b^2*f*(d*e - c*f)^3*(a + b*x)^(1
+ m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^3*(1 + m)*(2 + m)*(3 + m)) + (6*b^3*(d*e - c*f)^4*(a + b*x)^(1 + m)*
(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) - (f^4*(a + b*x)^m*Hypergeometric2F1[-
m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^5*m*(-((d*(a + b*x))/(b*c - a*d)))^m*(c + d*x)^m)

Rule 128

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (IGtQ[m, 0] || (
ILtQ[m, 0] && ILtQ[n, 0]))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx &=\int \left (\frac{(d e-c f)^4 (a+b x)^m (c+d x)^{-5-m}}{d^4}+\frac{4 f (d e-c f)^3 (a+b x)^m (c+d x)^{-4-m}}{d^4}+\frac{6 f^2 (d e-c f)^2 (a+b x)^m (c+d x)^{-3-m}}{d^4}+\frac{4 f^3 (d e-c f) (a+b x)^m (c+d x)^{-2-m}}{d^4}+\frac{f^4 (a+b x)^m (c+d x)^{-1-m}}{d^4}\right ) \, dx\\ &=\frac{f^4 \int (a+b x)^m (c+d x)^{-1-m} \, dx}{d^4}+\frac{\left (4 f^3 (d e-c f)\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^4}+\frac{\left (6 f^2 (d e-c f)^2\right ) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{d^4}+\frac{\left (4 f (d e-c f)^3\right ) \int (a+b x)^m (c+d x)^{-4-m} \, dx}{d^4}+\frac{(d e-c f)^4 \int (a+b x)^m (c+d x)^{-5-m} \, dx}{d^4}\\ &=\frac{(d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-4-m}}{d^4 (b c-a d) (4+m)}+\frac{4 f (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^4 (b c-a d) (3+m)}+\frac{6 f^2 (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^4 (b c-a d) (2+m)}+\frac{4 f^3 (d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d) (1+m)}+\frac{\left (6 b f^2 (d e-c f)^2\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^4 (b c-a d) (2+m)}+\frac{\left (8 b f (d e-c f)^3\right ) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{d^4 (b c-a d) (3+m)}+\frac{\left (3 b (d e-c f)^4\right ) \int (a+b x)^m (c+d x)^{-4-m} \, dx}{d^4 (b c-a d) (4+m)}+\frac{\left (f^4 (a+b x)^m \left (\frac{d (a+b x)}{-b c+a d}\right )^{-m}\right ) \int (c+d x)^{-1-m} \left (-\frac{a d}{b c-a d}-\frac{b d x}{b c-a d}\right )^m \, dx}{d^4}\\ &=\frac{(d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-4-m}}{d^4 (b c-a d) (4+m)}+\frac{4 f (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^4 (b c-a d) (3+m)}+\frac{3 b (d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^4 (b c-a d)^2 (3+m) (4+m)}+\frac{6 f^2 (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^4 (b c-a d) (2+m)}+\frac{8 b f (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^4 (b c-a d)^2 (2+m) (3+m)}+\frac{4 f^3 (d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d) (1+m)}+\frac{6 b f^2 (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d)^2 (1+m) (2+m)}-\frac{f^4 (a+b x)^m \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} (c+d x)^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^5 m}+\frac{\left (8 b^2 f (d e-c f)^3\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^4 (b c-a d)^2 (2+m) (3+m)}+\frac{\left (6 b^2 (d e-c f)^4\right ) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{d^4 (b c-a d)^2 (3+m) (4+m)}\\ &=\frac{(d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-4-m}}{d^4 (b c-a d) (4+m)}+\frac{4 f (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^4 (b c-a d) (3+m)}+\frac{3 b (d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^4 (b c-a d)^2 (3+m) (4+m)}+\frac{6 f^2 (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^4 (b c-a d) (2+m)}+\frac{8 b f (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^4 (b c-a d)^2 (2+m) (3+m)}+\frac{6 b^2 (d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^4 (b c-a d)^3 (2+m) (3+m) (4+m)}+\frac{4 f^3 (d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d) (1+m)}+\frac{6 b f^2 (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d)^2 (1+m) (2+m)}+\frac{8 b^2 f (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d)^3 (1+m) (2+m) (3+m)}-\frac{f^4 (a+b x)^m \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} (c+d x)^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^5 m}+\frac{\left (6 b^3 (d e-c f)^4\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^4 (b c-a d)^3 (2+m) (3+m) (4+m)}\\ &=\frac{(d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-4-m}}{d^4 (b c-a d) (4+m)}+\frac{4 f (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^4 (b c-a d) (3+m)}+\frac{3 b (d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^4 (b c-a d)^2 (3+m) (4+m)}+\frac{6 f^2 (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^4 (b c-a d) (2+m)}+\frac{8 b f (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^4 (b c-a d)^2 (2+m) (3+m)}+\frac{6 b^2 (d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^4 (b c-a d)^3 (2+m) (3+m) (4+m)}+\frac{4 f^3 (d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d) (1+m)}+\frac{6 b f^2 (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d)^2 (1+m) (2+m)}+\frac{8 b^2 f (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d)^3 (1+m) (2+m) (3+m)}+\frac{6 b^3 (d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d)^4 (1+m) (2+m) (3+m) (4+m)}-\frac{f^4 (a+b x)^m \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} (c+d x)^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^5 m}\\ \end{align*}

Mathematica [C]  time = 31.0636, size = 4261, normalized size = 6.56 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^4,x]

[Out]

(-4*e*f^3*(a + b*x)^m*(c + d*x)^(-5 - m)*((c + d*x)/c)^(5 + m)*(-6*a^4*c^4 - 24*a^4*c^3*d*x - 36*a^4*c^2*d^2*x
^2 - 24*a^4*c*d^3*x^3 - 6*a^4*d^4*x^4 + 6*a^4*c^4*((a*c + b*c*x)/(a*(c + d*x)))^m + 24*a^4*c^3*d*x*((a*c + b*c
*x)/(a*(c + d*x)))^m - 6*a^3*b*c^4*m*x*((a*c + b*c*x)/(a*(c + d*x)))^m + 6*a^4*c^3*d*m*x*((a*c + b*c*x)/(a*(c
+ d*x)))^m + 36*a^4*c^2*d^2*x^2*((a*c + b*c*x)/(a*(c + d*x)))^m + 3*a^2*b^2*c^4*m*x^2*((a*c + b*c*x)/(a*(c + d
*x)))^m - 24*a^3*b*c^3*d*m*x^2*((a*c + b*c*x)/(a*(c + d*x)))^m + 21*a^4*c^2*d^2*m*x^2*((a*c + b*c*x)/(a*(c + d
*x)))^m + 3*a^2*b^2*c^4*m^2*x^2*((a*c + b*c*x)/(a*(c + d*x)))^m - 6*a^3*b*c^3*d*m^2*x^2*((a*c + b*c*x)/(a*(c +
d*x)))^m + 3*a^4*c^2*d^2*m^2*x^2*((a*c + b*c*x)/(a*(c + d*x)))^m + 24*a^4*c*d^3*x^3*((a*c + b*c*x)/(a*(c + d*
x)))^m - 2*a*b^3*c^4*m*x^3*((a*c + b*c*x)/(a*(c + d*x)))^m + 12*a^2*b^2*c^3*d*m*x^3*((a*c + b*c*x)/(a*(c + d*x
)))^m - 36*a^3*b*c^2*d^2*m*x^3*((a*c + b*c*x)/(a*(c + d*x)))^m + 26*a^4*c*d^3*m*x^3*((a*c + b*c*x)/(a*(c + d*x
)))^m - 3*a*b^3*c^4*m^2*x^3*((a*c + b*c*x)/(a*(c + d*x)))^m + 15*a^2*b^2*c^3*d*m^2*x^3*((a*c + b*c*x)/(a*(c +
d*x)))^m - 21*a^3*b*c^2*d^2*m^2*x^3*((a*c + b*c*x)/(a*(c + d*x)))^m + 9*a^4*c*d^3*m^2*x^3*((a*c + b*c*x)/(a*(c
+ d*x)))^m - a*b^3*c^4*m^3*x^3*((a*c + b*c*x)/(a*(c + d*x)))^m + 3*a^2*b^2*c^3*d*m^3*x^3*((a*c + b*c*x)/(a*(c
+ d*x)))^m - 3*a^3*b*c^2*d^2*m^3*x^3*((a*c + b*c*x)/(a*(c + d*x)))^m + a^4*c*d^3*m^3*x^3*((a*c + b*c*x)/(a*(c
+ d*x)))^m - 6*b^4*c^4*x^4*((a*c + b*c*x)/(a*(c + d*x)))^m + 24*a*b^3*c^3*d*x^4*((a*c + b*c*x)/(a*(c + d*x)))
^m - 36*a^2*b^2*c^2*d^2*x^4*((a*c + b*c*x)/(a*(c + d*x)))^m + 24*a^3*b*c*d^3*x^4*((a*c + b*c*x)/(a*(c + d*x)))
^m - 11*b^4*c^4*m*x^4*((a*c + b*c*x)/(a*(c + d*x)))^m + 42*a*b^3*c^3*d*m*x^4*((a*c + b*c*x)/(a*(c + d*x)))^m -
57*a^2*b^2*c^2*d^2*m*x^4*((a*c + b*c*x)/(a*(c + d*x)))^m + 26*a^3*b*c*d^3*m*x^4*((a*c + b*c*x)/(a*(c + d*x)))
^m - 6*b^4*c^4*m^2*x^4*((a*c + b*c*x)/(a*(c + d*x)))^m + 21*a*b^3*c^3*d*m^2*x^4*((a*c + b*c*x)/(a*(c + d*x)))^
m - 24*a^2*b^2*c^2*d^2*m^2*x^4*((a*c + b*c*x)/(a*(c + d*x)))^m + 9*a^3*b*c*d^3*m^2*x^4*((a*c + b*c*x)/(a*(c +
d*x)))^m - b^4*c^4*m^3*x^4*((a*c + b*c*x)/(a*(c + d*x)))^m + 3*a*b^3*c^3*d*m^3*x^4*((a*c + b*c*x)/(a*(c + d*x)
))^m - 3*a^2*b^2*c^2*d^2*m^3*x^4*((a*c + b*c*x)/(a*(c + d*x)))^m + a^3*b*c*d^3*m^3*x^4*((a*c + b*c*x)/(a*(c +
d*x)))^m))/((b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)*((a + b*x)/a)^m*(1 + (d*x)/c)^4) + (f^4*x^5*(a + b*x
)^m*(c + d*x)^(-5 - m)*((c + d*x)/c)^(5 + m)*AppellF1[5, -m, 5 + m, 6, -((b*x)/a), -((d*x)/c)])/(5*((a + b*x)/
a)^m) + (2*e^2*f^2*(a + b*x)^m*(1 + (b*x)/a)^m*(c + d*x)^(-5 - m)*((c + d*x)/c)^(5 + m)*(1 + (d*x)/c)^(-4 - m)
*((c + d*x)*(b^4*c^4*m*(2 + 3*m + m^2)*x^4 + a*b^3*c^3*m*(1 + m)*x^3*(c*(-4 + m) - 3*d*(4 + m)*x) + 3*a^2*b^2*
c^2*m*x^2*(-2*c^2*(-2 + m) - c*d*(-12 + m + m^2)*x + d^2*(12 + 7*m + m^2)*x^2) + a^4*(-6*d^4*m*x^4*((a*(c + d*
x))/(c*(a + b*x)))^m + 24*c^4*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m) - 6*c^2*d^2*x^2*(12 + 7*m + m^2 - 12*((a*
(c + d*x))/(c*(a + b*x)))^m + 3*m*((a*(c + d*x))/(c*(a + b*x)))^m) - c*d^3*x^3*(24 + 26*m + 9*m^2 + m^3 - 24*(
(a*(c + d*x))/(c*(a + b*x)))^m + 18*m*((a*(c + d*x))/(c*(a + b*x)))^m) - 6*c^3*d*x*(-12*(-1 + ((a*(c + d*x))/(
c*(a + b*x)))^m) + m*(3 + ((a*(c + d*x))/(c*(a + b*x)))^m))) + a^3*b*c*x*(-(d^3*(4 + m)*x^3*(6 + 5*m + m^2 - 6
*((a*(c + d*x))/(c*(a + b*x)))^m)) + 6*c^2*d*(4 + m)*x*(-3 + 2*m + 3*((a*(c + d*x))/(c*(a + b*x)))^m) + 3*c*d^
2*(4 + m)*x^2*(-6 + m + m^2 + 6*((a*(c + d*x))/(c*(a + b*x)))^m) + 6*c^3*(4*(-1 + ((a*(c + d*x))/(c*(a + b*x))
)^m) + m*(3 + ((a*(c + d*x))/(c*(a + b*x)))^m))))*Gamma[1 - m] + m*(4*c + d*x)*(b^4*c^4*(6 + 11*m + 6*m^2 + m^
3)*x^4 + a*b^3*c^3*(2 + 3*m + m^2)*x^3*(c*m - 3*d*(4 + m)*x) - 3*a^2*b^2*c^2*(1 + m)*x^2*(c^2*m + c*d*m*(4 + m
)*x - d^2*(12 + 7*m + m^2)*x^2) + a^3*b*c*x*(6*c^3*m + 6*c^2*d*m*(4 + m)*x + 3*c*d^2*m*(12 + 7*m + m^2)*x^2 -
d^3*(24 + 26*m + 9*m^2 + m^3)*x^3) + a^4*(6*d^4*x^4*((a*(c + d*x))/(c*(a + b*x)))^m + 6*c^4*(-1 + ((a*(c + d*x
))/(c*(a + b*x)))^m) + 6*c^3*d*x*(-4 - m + 4*((a*(c + d*x))/(c*(a + b*x)))^m) + 3*c^2*d^2*x^2*(-7*m - m^2 + 12
*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m)) + c*d^3*x^3*(-26*m - 9*m^2 - m^3 + 24*(-1 + ((a*(c + d*x))/(c*(a + b*
x)))^m))))*Gamma[-m]))/(c*(b*c - a*d)^4*m*(1 + m)*(2 + m)*(3 + m)*(4 + m)*x*((a + b*x)/a)^m*Gamma[-m]) + (2*e^
3*f*(a + b*x)^m*(1 + (b*x)/a)^m*(c + d*x)^(-5 - m)*((c + d*x)/c)^(5 + m)*(1 + (d*x)/c)^(-4 - m)*(2*(c + d*x)*(
b^4*c^4*m*(1 + m)*x^4*(3*c*(3 + m) + d*(5 + m)*x) - a*b^3*c^3*m*x^3*(-3*c^2*(-8 - m + m^2) + 4*c*d*(19 + 12*m
+ 2*m^2)*x + 3*d^2*(12 + 7*m + m^2)*x^2) + a^3*b*c*x*(36*c^4*m + 6*c*d^3*(4 + m)*x^3*(-3 - m + 3*((a*(c + d*x)
)/(c*(a + b*x)))^m + 3*m*((a*(c + d*x))/(c*(a + b*x)))^m) + 3*c^2*d^2*(4 + m)*x^2*(-6 + 7*m + 3*m^2 + 6*((a*(c
+ d*x))/(c*(a + b*x)))^m + 6*m*((a*(c + d*x))/(c*(a + b*x)))^m) - d^4*(4 + m)*x^4*(6 + m^2 - 6*((a*(c + d*x))
/(c*(a + b*x)))^m + m*(5 - 6*((a*(c + d*x))/(c*(a + b*x)))^m)) + 6*c^3*d*x*(4*(-1 + ((a*(c + d*x))/(c*(a + b*x
)))^m) + m^2*(5 + ((a*(c + d*x))/(c*(a + b*x)))^m) + m*(17 + 5*((a*(c + d*x))/(c*(a + b*x)))^m))) - 3*a^2*b^2*
c^2*x^2*(d^3*(12 + 7*m + m^2)*x^3*(-1 - m + ((a*(c + d*x))/(c*(a + b*x)))^m) + c*d^2*(12 + 7*m + m^2)*x^2*(-3
- 2*m + 3*((a*(c + d*x))/(c*(a + b*x)))^m) + c^3*(12*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m) + m^2*(5 + ((a*(c
+ d*x))/(c*(a + b*x)))^m) + m*(-1 + 7*((a*(c + d*x))/(c*(a + b*x)))^m)) + c^2*d*x*(3*m^3 + 36*(-1 + ((a*(c + d
*x))/(c*(a + b*x)))^m) + m^2*(8 + 3*((a*(c + d*x))/(c*(a + b*x)))^m) + m*(-19 + 21*((a*(c + d*x))/(c*(a + b*x)
))^m))) + a^4*(-3*d^5*m*(3 + m)*x^5*((a*(c + d*x))/(c*(a + b*x)))^m + 36*c^5*(-1 + ((a*(c + d*x))/(c*(a + b*x)
))^m) + 12*c^4*d*x*(-3*m + 11*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m)) - 3*c^3*d^2*x^2*(-60*(-1 + ((a*(c + d*x)
)/(c*(a + b*x)))^m) + m^2*(5 + ((a*(c + d*x))/(c*(a + b*x)))^m) + m*(35 + 3*((a*(c + d*x))/(c*(a + b*x)))^m))
- 3*c^2*d^3*x^3*(m^3 - 36*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m) + m^2*(10 + 3*((a*(c + d*x))/(c*(a + b*x)))^m
) + m*(33 + 9*((a*(c + d*x))/(c*(a + b*x)))^m)) - c*d^4*x^4*(m^3 - 24*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m) +
9*m^2*(1 + ((a*(c + d*x))/(c*(a + b*x)))^m) + m*(26 + 27*((a*(c + d*x))/(c*(a + b*x)))^m))))*Gamma[1 - m] + 2
*m*(6*c^2 + 4*c*d*x + d^2*x^2)*(b^4*c^4*(6 + 11*m + 6*m^2 + m^3)*x^4 + a*b^3*c^3*(2 + 3*m + m^2)*x^3*(c*m - 3*
d*(4 + m)*x) - 3*a^2*b^2*c^2*(1 + m)*x^2*(c^2*m + c*d*m*(4 + m)*x - d^2*(12 + 7*m + m^2)*x^2) + a^3*b*c*x*(6*c
^3*m + 6*c^2*d*m*(4 + m)*x + 3*c*d^2*m*(12 + 7*m + m^2)*x^2 - d^3*(24 + 26*m + 9*m^2 + m^3)*x^3) + a^4*(6*d^4*
x^4*((a*(c + d*x))/(c*(a + b*x)))^m + 6*c^4*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m) + 6*c^3*d*x*(-4 - m + 4*((a
*(c + d*x))/(c*(a + b*x)))^m) + 3*c^2*d^2*x^2*(-7*m - m^2 + 12*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m)) + c*d^3
*x^3*(-26*m - 9*m^2 - m^3 + 24*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m))))*Gamma[-m]))/(3*c^2*(b*c - a*d)^4*m*(1
+ m)*(2 + m)*(3 + m)*(4 + m)*x^2*((a + b*x)/a)^m*Gamma[-m]) - (e^4*(c + d*x)^(-4 - m)*(a - (b*c)/d + (b*(c +
d*x))/d)^m*Hypergeometric2F1[-4 - m, -m, -3 - m, -((b*(c + d*x))/((a - (b*c)/d)*d))])/(d*(4 + m)*(1 + (b*(c +
d*x))/((a - (b*c)/d)*d))^m)

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-5-m} \left ( fx+e \right ) ^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x)

[Out]

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{4}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x, algorithm="maxima")

[Out]

integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m - 5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (f^{4} x^{4} + 4 \, e f^{3} x^{3} + 6 \, e^{2} f^{2} x^{2} + 4 \, e^{3} f x + e^{4}\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x, algorithm="fricas")

[Out]

integral((f^4*x^4 + 4*e*f^3*x^3 + 6*e^2*f^2*x^2 + 4*e^3*f*x + e^4)*(b*x + a)^m*(d*x + c)^(-m - 5), 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)**(-5-m)*(f*x+e)**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{4}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x, algorithm="giac")

[Out]

integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m - 5), x)