3.32.8 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx\) [3108]

Optimal. Leaf size=268 \[ \frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-4-m}}{d (b c-a d) (4+m)}-\frac {(a d f (4+m)-b (3 d e+c f (1+m))) (a+b x)^{1+m} (c+d x)^{-3-m}}{d (b c-a d)^2 (3+m) (4+m)}-\frac {2 b (a d f (4+m)-b (3 d e+c f (1+m))) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d)^3 (2+m) (3+m) (4+m)}-\frac {2 b^2 (a d f (4+m)-b (3 d e+c f (1+m))) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d)^4 (1+m) (2+m) (3+m) (4+m)} \]

[Out]

(-c*f+d*e)*(b*x+a)^(1+m)*(d*x+c)^(-4-m)/d/(-a*d+b*c)/(4+m)-(a*d*f*(4+m)-b*(3*d*e+c*f*(1+m)))*(b*x+a)^(1+m)*(d*
x+c)^(-3-m)/d/(-a*d+b*c)^2/(3+m)/(4+m)-2*b*(a*d*f*(4+m)-b*(3*d*e+c*f*(1+m)))*(b*x+a)^(1+m)*(d*x+c)^(-2-m)/d/(-
a*d+b*c)^3/(2+m)/(3+m)/(4+m)-2*b^2*(a*d*f*(4+m)-b*(3*d*e+c*f*(1+m)))*(b*x+a)^(1+m)*(d*x+c)^(-1-m)/d/(-a*d+b*c)
^4/(1+m)/(2+m)/(3+m)/(4+m)

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Rubi [A]
time = 0.10, antiderivative size = 264, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 47, 37} \begin {gather*} \frac {2 b^2 (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+4)+b c f (m+1)+3 b d e)}{d (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4}}{d (m+4) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-3} (-a d f (m+4)+b c f (m+1)+3 b d e)}{d (m+3) (m+4) (b c-a d)^2}+\frac {2 b (a+b x)^{m+1} (c+d x)^{-m-2} (-a d f (m+4)+b c f (m+1)+3 b d e)}{d (m+2) (m+3) (m+4) (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/(d*(b*c - a*d)*(4 + m)) + ((3*b*d*e + b*c*f*(1 + m) - a*d*f
*(4 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(d*(b*c - a*d)^2*(3 + m)*(4 + m)) + (2*b*(3*b*d*e + b*c*f*(1 +
m) - a*d*f*(4 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d*(b*c - a*d)^3*(2 + m)*(3 + m)*(4 + m)) + (2*b^2*
(3*b*d*e + b*c*f*(1 + m) - a*d*f*(4 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d*(b*c - a*d)^4*(1 + m)*(2 +
m)*(3 + m)*(4 + 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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c
, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 1]

Rubi steps

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

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Mathematica [A]
time = 0.34, size = 144, normalized size = 0.54 \begin {gather*} -\frac {(a+b x)^{1+m} (c+d x)^{-4-m} \left (d e-c f+\frac {(3 b d e+b c f (1+m)-a d f (4+m)) (c+d x) \left ((b c-a d)^2 (1+m) (2+m)+2 b (c+d x) (-a d (1+m)+b c (2+m)+b d x)\right )}{(b c-a d)^3 (1+m) (2+m) (3+m)}\right )}{d (-b c+a d) (4+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(d*e - c*f + ((3*b*d*e + b*c*f*(1 + m) - a*d*f*(4 + m))*(c + d*x)*((b*
c - a*d)^2*(1 + m)*(2 + m) + 2*b*(c + d*x)*(-(a*d*(1 + m)) + b*c*(2 + m) + b*d*x)))/((b*c - a*d)^3*(1 + m)*(2
+ m)*(3 + m))))/(d*(-(b*c) + a*d)*(4 + m)))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1183\) vs. \(2(268)=536\).
time = 0.10, size = 1184, normalized size = 4.42

method	result	size
gosper	\(-\frac {\left (d x +c \right )^{-4-m} \left (b x +a \right )^{1+m} \left (a^{3} d^{3} f \,m^{3} x -3 a^{2} b c \,d^{2} f \,m^{3} x -2 a^{2} b \,d^{3} f \,m^{2} x^{2}+3 a \,b^{2} c^{2} d f \,m^{3} x +4 a \,b^{2} c \,d^{2} f \,m^{2} x^{2}+2 a \,b^{2} d^{3} f m \,x^{3}-b^{3} c^{3} f \,m^{3} x -2 b^{3} c^{2} d f \,m^{2} x^{2}-2 b^{3} c \,d^{2} f m \,x^{3}+a^{3} d^{3} e \,m^{3}+7 a^{3} d^{3} f \,m^{2} x -3 a^{2} b c \,d^{2} e \,m^{3}-22 a^{2} b c \,d^{2} f \,m^{2} x -3 a^{2} b \,d^{3} e \,m^{2} x -10 a^{2} b \,d^{3} f m \,x^{2}+3 a \,b^{2} c^{2} d e \,m^{3}+23 a \,b^{2} c^{2} d f \,m^{2} x +6 a \,b^{2} c \,d^{2} e \,m^{2} x +20 a \,b^{2} c \,d^{2} f m \,x^{2}+6 a \,b^{2} d^{3} e m \,x^{2}+8 a \,b^{2} d^{3} f \,x^{3}-b^{3} c^{3} e \,m^{3}-8 b^{3} c^{3} f \,m^{2} x -3 b^{3} c^{2} d e \,m^{2} x -10 b^{3} c^{2} d f m \,x^{2}-6 b^{3} c \,d^{2} e m \,x^{2}-2 b^{3} c \,d^{2} f \,x^{3}-6 b^{3} d^{3} e \,x^{3}+a^{3} c \,d^{2} f \,m^{2}+6 a^{3} d^{3} e \,m^{2}+14 a^{3} d^{3} f m x -2 a^{2} b \,c^{2} d f \,m^{2}-21 a^{2} b c \,d^{2} e \,m^{2}-53 a^{2} b c \,d^{2} f m x -9 a^{2} b \,d^{3} e m x -8 a^{2} b \,d^{3} f \,x^{2}+a \,b^{2} c^{3} f \,m^{2}+24 a \,b^{2} c^{2} d e \,m^{2}+58 a \,b^{2} c^{2} d f m x +30 a \,b^{2} c \,d^{2} e m x +34 a \,b^{2} c \,d^{2} f \,x^{2}+6 a \,b^{2} d^{3} e \,x^{2}-9 b^{3} c^{3} e \,m^{2}-19 b^{3} c^{3} f m x -21 b^{3} c^{2} d e m x -8 b^{3} c^{2} d f \,x^{2}-24 b^{3} c \,d^{2} e \,x^{2}+3 a^{3} c \,d^{2} f m +11 a^{3} d^{3} e m +8 a^{3} d^{3} f x -10 a^{2} b \,c^{2} d f m -42 a^{2} b c \,d^{2} e m -34 a^{2} b c \,d^{2} f x -6 a^{2} b \,d^{3} e x +7 a \,b^{2} c^{3} f m +57 a \,b^{2} c^{2} d e m +56 a \,b^{2} c^{2} d f x +24 a \,b^{2} c \,d^{2} e x -26 b^{3} c^{3} e m -12 b^{3} c^{3} f x -36 b^{3} c^{2} d e x +2 a^{3} c \,d^{2} f +6 a^{3} d^{3} e -8 a^{2} b \,c^{2} d f -24 a^{2} b c \,d^{2} e +12 a \,b^{2} c^{3} f +36 a \,b^{2} c^{2} d e -24 b^{3} c^{3} e \right )}{a^{4} d^{4} m^{4}-4 a^{3} b c \,d^{3} m^{4}+6 a^{2} b^{2} c^{2} d^{2} m^{4}-4 a \,b^{3} c^{3} d \,m^{4}+b^{4} c^{4} m^{4}+10 a^{4} d^{4} m^{3}-40 a^{3} b c \,d^{3} m^{3}+60 a^{2} b^{2} c^{2} d^{2} m^{3}-40 a \,b^{3} c^{3} d \,m^{3}+10 b^{4} c^{4} m^{3}+35 a^{4} d^{4} m^{2}-140 a^{3} b c \,d^{3} m^{2}+210 a^{2} b^{2} c^{2} d^{2} m^{2}-140 a \,b^{3} c^{3} d \,m^{2}+35 b^{4} c^{4} m^{2}+50 a^{4} d^{4} m -200 a^{3} b c \,d^{3} m +300 a^{2} b^{2} c^{2} d^{2} m -200 a \,b^{3} c^{3} d m +50 b^{4} c^{4} m +24 a^{4} d^{4}-96 a^{3} b c \,d^{3}+144 a^{2} b^{2} c^{2} d^{2}-96 a \,b^{3} c^{3} d +24 b^{4} c^{4}}\)	\(1184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*x+c)^(-4-m)*(b*x+a)^(1+m)*(a^3*d^3*f*m^3*x-3*a^2*b*c*d^2*f*m^3*x-2*a^2*b*d^3*f*m^2*x^2+3*a*b^2*c^2*d*f*m^3
*x+4*a*b^2*c*d^2*f*m^2*x^2+2*a*b^2*d^3*f*m*x^3-b^3*c^3*f*m^3*x-2*b^3*c^2*d*f*m^2*x^2-2*b^3*c*d^2*f*m*x^3+a^3*d
^3*e*m^3+7*a^3*d^3*f*m^2*x-3*a^2*b*c*d^2*e*m^3-22*a^2*b*c*d^2*f*m^2*x-3*a^2*b*d^3*e*m^2*x-10*a^2*b*d^3*f*m*x^2
+3*a*b^2*c^2*d*e*m^3+23*a*b^2*c^2*d*f*m^2*x+6*a*b^2*c*d^2*e*m^2*x+20*a*b^2*c*d^2*f*m*x^2+6*a*b^2*d^3*e*m*x^2+8
*a*b^2*d^3*f*x^3-b^3*c^3*e*m^3-8*b^3*c^3*f*m^2*x-3*b^3*c^2*d*e*m^2*x-10*b^3*c^2*d*f*m*x^2-6*b^3*c*d^2*e*m*x^2-
2*b^3*c*d^2*f*x^3-6*b^3*d^3*e*x^3+a^3*c*d^2*f*m^2+6*a^3*d^3*e*m^2+14*a^3*d^3*f*m*x-2*a^2*b*c^2*d*f*m^2-21*a^2*
b*c*d^2*e*m^2-53*a^2*b*c*d^2*f*m*x-9*a^2*b*d^3*e*m*x-8*a^2*b*d^3*f*x^2+a*b^2*c^3*f*m^2+24*a*b^2*c^2*d*e*m^2+58
*a*b^2*c^2*d*f*m*x+30*a*b^2*c*d^2*e*m*x+34*a*b^2*c*d^2*f*x^2+6*a*b^2*d^3*e*x^2-9*b^3*c^3*e*m^2-19*b^3*c^3*f*m*
x-21*b^3*c^2*d*e*m*x-8*b^3*c^2*d*f*x^2-24*b^3*c*d^2*e*x^2+3*a^3*c*d^2*f*m+11*a^3*d^3*e*m+8*a^3*d^3*f*x-10*a^2*
b*c^2*d*f*m-42*a^2*b*c*d^2*e*m-34*a^2*b*c*d^2*f*x-6*a^2*b*d^3*e*x+7*a*b^2*c^3*f*m+57*a*b^2*c^2*d*e*m+56*a*b^2*
c^2*d*f*x+24*a*b^2*c*d^2*e*x-26*b^3*c^3*e*m-12*b^3*c^3*f*x-36*b^3*c^2*d*e*x+2*a^3*c*d^2*f+6*a^3*d^3*e-8*a^2*b*
c^2*d*f-24*a^2*b*c*d^2*e+12*a*b^2*c^3*f+36*a*b^2*c^2*d*e-24*b^3*c^3*e)/(a^4*d^4*m^4-4*a^3*b*c*d^3*m^4+6*a^2*b^
2*c^2*d^2*m^4-4*a*b^3*c^3*d*m^4+b^4*c^4*m^4+10*a^4*d^4*m^3-40*a^3*b*c*d^3*m^3+60*a^2*b^2*c^2*d^2*m^3-40*a*b^3*
c^3*d*m^3+10*b^4*c^4*m^3+35*a^4*d^4*m^2-140*a^3*b*c*d^3*m^2+210*a^2*b^2*c^2*d^2*m^2-140*a*b^3*c^3*d*m^2+35*b^4
*c^4*m^2+50*a^4*d^4*m-200*a^3*b*c*d^3*m+300*a^2*b^2*c^2*d^2*m-200*a*b^3*c^3*d*m+50*b^4*c^4*m+24*a^4*d^4-96*a^3
*b*c*d^3+144*a^2*b^2*c^2*d^2-96*a*b^3*c^3*d+24*b^4*c^4)

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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),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1778 vs. \(2 (273) = 546\).
time = 1.11, size = 1778, normalized size = 6.63 \begin {gather*} \frac {{\left (2 \, {\left ({\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} f m + {\left (b^{4} c d^{3} - 4 \, a b^{3} d^{4}\right )} f\right )} x^{5} + 2 \, {\left ({\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} f m^{2} + 2 \, {\left (3 \, b^{4} c^{2} d^{2} - 5 \, a b^{3} c d^{3} + 2 \, a^{2} b^{2} d^{4}\right )} f m + 5 \, {\left (b^{4} c^{2} d^{2} - 4 \, a b^{3} c d^{3}\right )} f\right )} x^{4} - {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} f m^{2} + {\left ({\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 )} f m^{3} + 5 \, {\left (2 \, b^{4} c^{3} d - 5 \, a b^{3} c^{2} d^{2} + 4 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} f m^{2} + {\left (29 \, b^{4} c^{3} d - 66 \, a b^{3} c^{2} d^{2} + 41 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} f m + 20 \, {\left (b^{4} c^{3} d - 4 \, a b^{3} c^{2} d^{2}\right )} f\right )} x^{3} - {\left (7 \, a^{2} b^{2} c^{4} - 10 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} f m + {\left ({\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} f m^{3} + {\left (8 \, b^{4} c^{4} - 14 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2} + 16 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} f m^{2} + {\left (19 \, b^{4} c^{4} - 36 \, a b^{3} c^{3} d - 15 \, a^{2} b^{2} c^{2} d^{2} + 46 \, a^{3} b c d^{3} - 14 \, a^{4} d^{4}\right )} f m + 4 \, {\left (3 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 8 \, a^{3} b c d^{3} - 2 \, a^{4} d^{4}\right )} f\right )} x^{2} - 2 \, {\left (6 \, a^{2} b^{2} c^{4} - 4 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} f + {\left ({\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} f m^{3} + {\left (7 \, a b^{3} c^{4} - 22 \, a^{2} b^{2} c^{3} d + 23 \, a^{3} b c^{2} d^{2} - 8 \, a^{4} c d^{3}\right )} f m^{2} + {\left (12 \, a b^{3} c^{4} - 55 \, a^{2} b^{2} c^{3} d + 60 \, a^{3} b c^{2} d^{2} - 17 \, a^{4} c d^{3}\right )} f m - 10 \, {\left (6 \, a^{2} b^{2} c^{3} d - 4 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} f\right )} x + {\left (6 \, b^{4} d^{4} x^{5} + 24 \, a b^{3} c^{4} - 36 \, a^{2} b^{2} c^{3} d + 24 \, a^{3} b c^{2} d^{2} - 6 \, a^{4} c d^{3} + 6 \, {\left (5 \, b^{4} c d^{3} + {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} m\right )} x^{4} + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} m^{3} + 3 \, {\left (20 \, b^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} m^{2} + {\left (9 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} m\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c^{4} - 8 \, a^{2} b^{2} c^{3} d + 7 \, a^{3} b c^{2} d^{2} - 2 \, a^{4} c d^{3}\right )} m^{2} + {\left (60 \, b^{4} c^{3} d + {\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 )} m^{3} + 3 \, {\left (4 \, b^{4} c^{3} d - 9 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} m^{2} + {\left (47 \, b^{4} c^{3} d - 60 \, a b^{3} c^{2} d^{2} + 15 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} m\right )} x^{2} + {\left (26 \, a b^{3} c^{4} - 57 \, a^{2} b^{2} c^{3} d + 42 \, a^{3} b c^{2} d^{2} - 11 \, a^{4} c d^{3}\right )} m + {\left (24 \, b^{4} c^{4} + 24 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} + 24 \, a^{3} b c d^{3} - 6 \, a^{4} d^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} m^{3} + 3 \, {\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2} + 6 \, a^{3} b c d^{3} - 2 \, a^{4} d^{4}\right )} m^{2} + {\left (26 \, b^{4} c^{4} - 10 \, a b^{3} c^{3} d - 45 \, a^{2} b^{2} c^{2} d^{2} + 40 \, a^{3} b c d^{3} - 11 \, a^{4} d^{4}\right )} m\right )} x\right )} e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}}{24 \, b^{4} c^{4} - 96 \, a b^{3} c^{3} d + 144 \, a^{2} b^{2} c^{2} d^{2} - 96 \, a^{3} b c d^{3} + 24 \, a^{4} d^{4} + {\left (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}\right )} m^{4} + 10 \, {\left (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}\right )} m^{3} + 35 \, {\left (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}\right )} m^{2} + 50 \, {\left (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}\right )} m} \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),x, algorithm="fricas")

[Out]

(2*((b^4*c*d^3 - a*b^3*d^4)*f*m + (b^4*c*d^3 - 4*a*b^3*d^4)*f)*x^5 + 2*((b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2
*d^4)*f*m^2 + 2*(3*b^4*c^2*d^2 - 5*a*b^3*c*d^3 + 2*a^2*b^2*d^4)*f*m + 5*(b^4*c^2*d^2 - 4*a*b^3*c*d^3)*f)*x^4 -
(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*f*m^2 + ((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)*f*m^3 + 5*(2*b^4*c^3*d - 5*a*b^3*c^2*d^2 + 4*a^2*b^2*c*d^3 - a^3*b*d^4)*f*m^2 + (29*b^4*c^3*d - 66*a*b^3*c
^2*d^2 + 41*a^2*b^2*c*d^3 - 4*a^3*b*d^4)*f*m + 20*(b^4*c^3*d - 4*a*b^3*c^2*d^2)*f)*x^3 - (7*a^2*b^2*c^4 - 10*a
^3*b*c^3*d + 3*a^4*c^2*d^2)*f*m + ((b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*f*m^3 + (8*b^4*c^4 - 14
*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 - 7*a^4*d^4)*f*m^2 + (19*b^4*c^4 - 36*a*b^3*c^3*d - 15*a^2*b
^2*c^2*d^2 + 46*a^3*b*c*d^3 - 14*a^4*d^4)*f*m + 4*(3*b^4*c^4 - 12*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 8*a^3*b*c
*d^3 - 2*a^4*d^4)*f)*x^2 - 2*(6*a^2*b^2*c^4 - 4*a^3*b*c^3*d + a^4*c^2*d^2)*f + ((a*b^3*c^4 - 3*a^2*b^2*c^3*d +
3*a^3*b*c^2*d^2 - a^4*c*d^3)*f*m^3 + (7*a*b^3*c^4 - 22*a^2*b^2*c^3*d + 23*a^3*b*c^2*d^2 - 8*a^4*c*d^3)*f*m^2
+ (12*a*b^3*c^4 - 55*a^2*b^2*c^3*d + 60*a^3*b*c^2*d^2 - 17*a^4*c*d^3)*f*m - 10*(6*a^2*b^2*c^3*d - 4*a^3*b*c^2*
d^2 + a^4*c*d^3)*f)*x + (6*b^4*d^4*x^5 + 24*a*b^3*c^4 - 36*a^2*b^2*c^3*d + 24*a^3*b*c^2*d^2 - 6*a^4*c*d^3 + 6*
(5*b^4*c*d^3 + (b^4*c*d^3 - a*b^3*d^4)*m)*x^4 + (a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*m^
3 + 3*(20*b^4*c^2*d^2 + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*m^2 + (9*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + a^
2*b^2*d^4)*m)*x^3 + 3*(3*a*b^3*c^4 - 8*a^2*b^2*c^3*d + 7*a^3*b*c^2*d^2 - 2*a^4*c*d^3)*m^2 + (60*b^4*c^3*d + (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)*m^3 + 3*(4*b^4*c^3*d - 9*a*b^3*c^2*d^2 + 6*a^2*b^2*c
*d^3 - a^3*b*d^4)*m^2 + (47*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 15*a^2*b^2*c*d^3 - 2*a^3*b*d^4)*m)*x^2 + (26*a*b^3*
c^4 - 57*a^2*b^2*c^3*d + 42*a^3*b*c^2*d^2 - 11*a^4*c*d^3)*m + (24*b^4*c^4 + 24*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^
2 + 24*a^3*b*c*d^3 - 6*a^4*d^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*m^3 + 3*(3*b^4*c^4 - 4*a*
b^3*c^3*d - 3*a^2*b^2*c^2*d^2 + 6*a^3*b*c*d^3 - 2*a^4*d^4)*m^2 + (26*b^4*c^4 - 10*a*b^3*c^3*d - 45*a^2*b^2*c^2
*d^2 + 40*a^3*b*c*d^3 - 11*a^4*d^4)*m)*x)*e)*(b*x + a)^m*(d*x + c)^(-m - 5)/(24*b^4*c^4 - 96*a*b^3*c^3*d + 144
*a^2*b^2*c^2*d^2 - 96*a^3*b*c*d^3 + 24*a^4*d^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)*m^4 + 10*(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)*m^3 + 35*(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)*m^2 + 50*(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)*m)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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),x)

[Out]

Timed out

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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),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.54, size = 1658, normalized size = 6.19 \begin {gather*} \frac {2\,b^3\,d^3\,x^5\,{\left (a+b\,x\right )}^m\,\left (b\,c\,f-4\,a\,d\,f+3\,b\,d\,e-a\,d\,f\,m+b\,c\,f\,m\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}-\frac {x^2\,{\left (a+b\,x\right )}^m\,\left (f\,a^4\,d^4\,m^3+7\,f\,a^4\,d^4\,m^2+14\,f\,a^4\,d^4\,m+8\,f\,a^4\,d^4-2\,f\,a^3\,b\,c\,d^3\,m^3-16\,f\,a^3\,b\,c\,d^3\,m^2-46\,f\,a^3\,b\,c\,d^3\,m-32\,f\,a^3\,b\,c\,d^3+e\,a^3\,b\,d^4\,m^3+3\,e\,a^3\,b\,d^4\,m^2+2\,e\,a^3\,b\,d^4\,m+3\,f\,a^2\,b^2\,c^2\,d^2\,m^2+15\,f\,a^2\,b^2\,c^2\,d^2\,m+48\,f\,a^2\,b^2\,c^2\,d^2-3\,e\,a^2\,b^2\,c\,d^3\,m^3-18\,e\,a^2\,b^2\,c\,d^3\,m^2-15\,e\,a^2\,b^2\,c\,d^3\,m+2\,f\,a\,b^3\,c^3\,d\,m^3+14\,f\,a\,b^3\,c^3\,d\,m^2+36\,f\,a\,b^3\,c^3\,d\,m+48\,f\,a\,b^3\,c^3\,d+3\,e\,a\,b^3\,c^2\,d^2\,m^3+27\,e\,a\,b^3\,c^2\,d^2\,m^2+60\,e\,a\,b^3\,c^2\,d^2\,m-f\,b^4\,c^4\,m^3-8\,f\,b^4\,c^4\,m^2-19\,f\,b^4\,c^4\,m-12\,f\,b^4\,c^4-e\,b^4\,c^3\,d\,m^3-12\,e\,b^4\,c^3\,d\,m^2-47\,e\,b^4\,c^3\,d\,m-60\,e\,b^4\,c^3\,d\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}-\frac {x\,{\left (a+b\,x\right )}^m\,\left (f\,a^4\,c\,d^3\,m^3+8\,f\,a^4\,c\,d^3\,m^2+17\,f\,a^4\,c\,d^3\,m+10\,f\,a^4\,c\,d^3+e\,a^4\,d^4\,m^3+6\,e\,a^4\,d^4\,m^2+11\,e\,a^4\,d^4\,m+6\,e\,a^4\,d^4-3\,f\,a^3\,b\,c^2\,d^2\,m^3-23\,f\,a^3\,b\,c^2\,d^2\,m^2-60\,f\,a^3\,b\,c^2\,d^2\,m-40\,f\,a^3\,b\,c^2\,d^2-2\,e\,a^3\,b\,c\,d^3\,m^3-18\,e\,a^3\,b\,c\,d^3\,m^2-40\,e\,a^3\,b\,c\,d^3\,m-24\,e\,a^3\,b\,c\,d^3+3\,f\,a^2\,b^2\,c^3\,d\,m^3+22\,f\,a^2\,b^2\,c^3\,d\,m^2+55\,f\,a^2\,b^2\,c^3\,d\,m+60\,f\,a^2\,b^2\,c^3\,d+9\,e\,a^2\,b^2\,c^2\,d^2\,m^2+45\,e\,a^2\,b^2\,c^2\,d^2\,m+36\,e\,a^2\,b^2\,c^2\,d^2-f\,a\,b^3\,c^4\,m^3-7\,f\,a\,b^3\,c^4\,m^2-12\,f\,a\,b^3\,c^4\,m+2\,e\,a\,b^3\,c^3\,d\,m^3+12\,e\,a\,b^3\,c^3\,d\,m^2+10\,e\,a\,b^3\,c^3\,d\,m-24\,e\,a\,b^3\,c^3\,d-e\,b^4\,c^4\,m^3-9\,e\,b^4\,c^4\,m^2-26\,e\,b^4\,c^4\,m-24\,e\,b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}-\frac {{\left (a+b\,x\right )}^m\,\left (f\,a^4\,c^2\,d^2\,m^2+3\,f\,a^4\,c^2\,d^2\,m+2\,f\,a^4\,c^2\,d^2+e\,a^4\,c\,d^3\,m^3+6\,e\,a^4\,c\,d^3\,m^2+11\,e\,a^4\,c\,d^3\,m+6\,e\,a^4\,c\,d^3-2\,f\,a^3\,b\,c^3\,d\,m^2-10\,f\,a^3\,b\,c^3\,d\,m-8\,f\,a^3\,b\,c^3\,d-3\,e\,a^3\,b\,c^2\,d^2\,m^3-21\,e\,a^3\,b\,c^2\,d^2\,m^2-42\,e\,a^3\,b\,c^2\,d^2\,m-24\,e\,a^3\,b\,c^2\,d^2+f\,a^2\,b^2\,c^4\,m^2+7\,f\,a^2\,b^2\,c^4\,m+12\,f\,a^2\,b^2\,c^4+3\,e\,a^2\,b^2\,c^3\,d\,m^3+24\,e\,a^2\,b^2\,c^3\,d\,m^2+57\,e\,a^2\,b^2\,c^3\,d\,m+36\,e\,a^2\,b^2\,c^3\,d-e\,a\,b^3\,c^4\,m^3-9\,e\,a\,b^3\,c^4\,m^2-26\,e\,a\,b^3\,c^4\,m-24\,e\,a\,b^3\,c^4\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {b\,d\,x^3\,{\left (a+b\,x\right )}^m\,\left (b\,c\,f-4\,a\,d\,f+3\,b\,d\,e-a\,d\,f\,m+b\,c\,f\,m\right )\,\left (a^2\,d^2\,m^2+a^2\,d^2\,m-2\,a\,b\,c\,d\,m^2-10\,a\,b\,c\,d\,m+b^2\,c^2\,m^2+9\,b^2\,c^2\,m+20\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {2\,b^2\,d^2\,x^4\,{\left (a+b\,x\right )}^m\,\left (5\,b\,c-a\,d\,m+b\,c\,m\right )\,\left (b\,c\,f-4\,a\,d\,f+3\,b\,d\,e-a\,d\,f\,m+b\,c\,f\,m\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{m+5}\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^3*d^3*x^5*(a + b*x)^m*(b*c*f - 4*a*d*f + 3*b*d*e - a*d*f*m + b*c*f*m))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(
50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - (x^2*(a + b*x)^m*(8*a^4*d^4*f - 12*b^4*c^4*f + 7*a^4*d^4*f*m^2 - 8*b^4*c
^4*f*m^2 + a^4*d^4*f*m^3 - b^4*c^4*f*m^3 - 60*b^4*c^3*d*e + 14*a^4*d^4*f*m - 19*b^4*c^4*f*m + 48*a*b^3*c^3*d*f
- 32*a^3*b*c*d^3*f + 2*a^3*b*d^4*e*m - 47*b^4*c^3*d*e*m + 3*a^3*b*d^4*e*m^2 + a^3*b*d^4*e*m^3 - 12*b^4*c^3*d*
e*m^2 - b^4*c^3*d*e*m^3 + 48*a^2*b^2*c^2*d^2*f + 27*a*b^3*c^2*d^2*e*m^2 - 18*a^2*b^2*c*d^3*e*m^2 + 3*a*b^3*c^2
*d^2*e*m^3 - 3*a^2*b^2*c*d^3*e*m^3 + 15*a^2*b^2*c^2*d^2*f*m + 36*a*b^3*c^3*d*f*m - 46*a^3*b*c*d^3*f*m + 3*a^2*
b^2*c^2*d^2*f*m^2 + 60*a*b^3*c^2*d^2*e*m - 15*a^2*b^2*c*d^3*e*m + 14*a*b^3*c^3*d*f*m^2 - 16*a^3*b*c*d^3*f*m^2
+ 2*a*b^3*c^3*d*f*m^3 - 2*a^3*b*c*d^3*f*m^3))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 +
24)) - (x*(a + b*x)^m*(6*a^4*d^4*e - 24*b^4*c^4*e + 6*a^4*d^4*e*m^2 - 9*b^4*c^4*e*m^2 + a^4*d^4*e*m^3 - b^4*c
^4*e*m^3 + 10*a^4*c*d^3*f + 11*a^4*d^4*e*m - 26*b^4*c^4*e*m - 24*a*b^3*c^3*d*e - 24*a^3*b*c*d^3*e - 12*a*b^3*c
^4*f*m + 17*a^4*c*d^3*f*m + 60*a^2*b^2*c^3*d*f - 40*a^3*b*c^2*d^2*f - 7*a*b^3*c^4*f*m^2 - a*b^3*c^4*f*m^3 + 8*
a^4*c*d^3*f*m^2 + a^4*c*d^3*f*m^3 + 36*a^2*b^2*c^2*d^2*e + 45*a^2*b^2*c^2*d^2*e*m + 22*a^2*b^2*c^3*d*f*m^2 - 2
3*a^3*b*c^2*d^2*f*m^2 + 3*a^2*b^2*c^3*d*f*m^3 - 3*a^3*b*c^2*d^2*f*m^3 + 10*a*b^3*c^3*d*e*m - 40*a^3*b*c*d^3*e*
m + 9*a^2*b^2*c^2*d^2*e*m^2 + 12*a*b^3*c^3*d*e*m^2 - 18*a^3*b*c*d^3*e*m^2 + 2*a*b^3*c^3*d*e*m^3 - 2*a^3*b*c*d^
3*e*m^3 + 55*a^2*b^2*c^3*d*f*m - 60*a^3*b*c^2*d^2*f*m))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m
^3 + m^4 + 24)) - ((a + b*x)^m*(12*a^2*b^2*c^4*f + 2*a^4*c^2*d^2*f - 24*a*b^3*c^4*e + 6*a^4*c*d^3*e - 8*a^3*b*
c^3*d*f - 26*a*b^3*c^4*e*m + 11*a^4*c*d^3*e*m + 36*a^2*b^2*c^3*d*e - 24*a^3*b*c^2*d^2*e - 9*a*b^3*c^4*e*m^2 -
a*b^3*c^4*e*m^3 + 7*a^2*b^2*c^4*f*m + 6*a^4*c*d^3*e*m^2 + a^4*c*d^3*e*m^3 + 3*a^4*c^2*d^2*f*m + a^2*b^2*c^4*f*
m^2 + a^4*c^2*d^2*f*m^2 + 24*a^2*b^2*c^3*d*e*m^2 - 21*a^3*b*c^2*d^2*e*m^2 + 3*a^2*b^2*c^3*d*e*m^3 - 3*a^3*b*c^
2*d^2*e*m^3 - 10*a^3*b*c^3*d*f*m + 57*a^2*b^2*c^3*d*e*m - 42*a^3*b*c^2*d^2*e*m - 2*a^3*b*c^3*d*f*m^2))/((a*d -
b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (b*d*x^3*(a + b*x)^m*(b*c*f - 4*a*d*f + 3*b*d
*e - a*d*f*m + b*c*f*m)*(20*b^2*c^2 + a^2*d^2*m + 9*b^2*c^2*m + a^2*d^2*m^2 + b^2*c^2*m^2 - 10*a*b*c*d*m - 2*a
*b*c*d*m^2))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (2*b^2*d^2*x^4*(a + b*x)^
m*(5*b*c - a*d*m + b*c*m)*(b*c*f - 4*a*d*f + 3*b*d*e - a*d*f*m + b*c*f*m))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(5
0*m + 35*m^2 + 10*m^3 + m^4 + 24))

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