3.32.5 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx\) [3105]

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

[Out]

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

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Rubi [A]
time = 0.35, antiderivative size = 650, normalized size of antiderivative = 1.00, 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, 47, 37, 72, 71} \begin {gather*} \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 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 {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}+\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 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 {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} \end {gather*}

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 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 47

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 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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]))

Rule 72

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 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]))

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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 24.69, size = 3080, normalized size = 4.74 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + b*x)^m*(c + d*x)^(-4 - m)*((60*c^4*d*e*f^3*m*x^2*(1 + (d*x)/c)^m*(b^4*c^4*(6 + 11*m + 6*m^2 + m^3)*x^4*(
(c*(a + b*x))/(a*(c + d*x)))^m - a*b^3*c^3*(2 + 3*m + m^2)*x^3*((c*(a + b*x))/(a*(c + d*x)))^m*(-(c*m) + 3*d*(
4 + m)*x) + 3*a^2*b^2*c^2*(1 + m)*x^2*((c*(a + b*x))/(a*(c + d*x)))^m*(-(c^2*m) - c*d*m*(4 + m)*x + d^2*(12 +
7*m + m^2)*x^2) - a^3*b*c*x*((c*(a + b*x))/(a*(c + d*x)))^m*(-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 + 6*c^4*(-1 + ((c*(a + b*x))/(a*(c + d*x
)))^m) + 6*c^3*d*x*(-4 + 4*((c*(a + b*x))/(a*(c + d*x)))^m + m*((c*(a + b*x))/(a*(c + d*x)))^m) + 3*c^2*d^2*x^
2*(7*m*((c*(a + b*x))/(a*(c + d*x)))^m + m^2*((c*(a + b*x))/(a*(c + d*x)))^m + 12*(-1 + ((c*(a + b*x))/(a*(c +
d*x)))^m)) + c*d^3*x^3*(26*m*((c*(a + b*x))/(a*(c + d*x)))^m + 9*m^2*((c*(a + b*x))/(a*(c + d*x)))^m + m^3*((
c*(a + b*x))/(a*(c + d*x)))^m + 24*(-1 + ((c*(a + b*x))/(a*(c + d*x)))^m))))*Gamma[-m])/(1 + (b*x)/a)^m + (3*d
*(b*c - a*d)^4*f^4*m*(1 + m)*(2 + m)*(3 + m)*(4 + m)*x^7*(c + d*x)^4*(1 + (d*x)/c)^m*AppellF1[5, -m, 5 + m, 6,
-((b*x)/a), -((d*x)/c)]*Gamma[-m])/(1 + (b*x)/a)^m + 30*c^3*d*e^2*f^2*x*((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]) + 10*c^2*d*e^3*f
*(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))))*G
amma[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) + ...

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (d x +c \right )^{-5-m} \left (f x +e \right )^{4}\, dx\]

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 = 0.00, 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)^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 = 0.00, 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)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3879 deep

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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)^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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^4\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+5}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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