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

Optimal. Leaf size=188 \[ \frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3}}{d (m+3) (b c-a d)}-\frac {(a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+3)-b (c f (m+1)+2 d e))}{d (m+2) (m+3) (b c-a d)^2}-\frac {b (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+3)-b (c f (m+1)+2 d e))}{d (m+1) (m+2) (m+3) (b c-a d)^3} \]

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Rubi [A]  time = 0.09, antiderivative size = 184, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 45, 37} \begin {gather*} \frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3}}{d (m+3) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-2} (-a d f (m+3)+b c f (m+1)+2 b d e)}{d (m+2) (m+3) (b c-a d)^2}+\frac {b (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)+b c f (m+1)+2 b d e)}{d (m+1) (m+2) (m+3) (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(d*(b*c - a*d)*(3 + m)) + ((2*b*d*e + b*c*f*(1 + m) - a*d*f
*(3 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d*(b*c - a*d)^2*(2 + m)*(3 + m)) + (b*(2*b*d*e + b*c*f*(1 + m
) - a*d*f*(3 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d*(b*c - a*d)^3*(1 + m)*(2 + m)*(3 + 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 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 79

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

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Mathematica [A]  time = 0.14, size = 179, normalized size = 0.95 \begin {gather*} \frac {(a+b x)^{m+1} (c+d x)^{-m-3} \left (a^2 d (m+1) (c f+d e (m+2)+d f (m+3) x)-a b \left (c^2 f (m+3)+2 c d \left (e \left (m^2+4 m+3\right )+f \left (m^2+4 m+5\right ) x\right )+d^2 x (2 e (m+1)+f (m+3) x)\right )+b^2 \left (c^2 (m+3) (e (m+2)+f (m+1) x)+c d x (2 e (m+3)+f (m+1) x)+2 d^2 e x^2\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-3 - m)*(a^2*d*(1 + m)*(c*f + d*e*(2 + m) + d*f*(3 + m)*x) + b^2*(2*d^2*e*x^2 +
c^2*(3 + m)*(e*(2 + m) + f*(1 + m)*x) + c*d*x*(2*e*(3 + m) + f*(1 + m)*x)) - a*b*(c^2*f*(3 + m) + d^2*x*(2*e*(
1 + m) + f*(3 + m)*x) + 2*c*d*(e*(3 + 4*m + m^2) + f*(5 + 4*m + m^2)*x))))/((b*c - a*d)^3*(1 + m)*(2 + m)*(3 +
m))

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IntegrateAlgebraic [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^m (c+d x)^{-4-m} (e+f x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.35, size = 902, normalized size = 4.80 \begin {gather*} \frac {{\left ({\left (2 \, b^{3} d^{3} e + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} f m + {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f\right )} x^{4} + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} e m^{2} + {\left (8 \, b^{3} c d^{2} e + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f m^{2} + 4 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} f + {\left (2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e + {\left (5 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f\right )} m\right )} x^{3} + {\left (12 \, b^{3} c^{2} d e + {\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} f\right )} m^{2} + 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} f + {\left ({\left (7 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e + 4 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} f\right )} m\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} e - {\left (3 \, a^{2} b c^{3} - a^{3} c^{2} d\right )} f + {\left ({\left (5 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} e - {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f\right )} m + {\left ({\left ({\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} e + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} f\right )} m^{2} + 2 \, {\left (3 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} e - 4 \, {\left (3 \, a^{2} b c^{2} d - a^{3} c d^{2}\right )} f + {\left ({\left (5 \, b^{3} c^{3} - a b^{2} c^{2} d - 7 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} e + {\left (3 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} f\right )} m\right )} x\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4}}{6 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 6 \, a^{3} d^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} m^{3} + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} m^{2} + 11 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} m} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*b^3*d^3*e + (b^3*c*d^2 - a*b^2*d^3)*f*m + (b^3*c*d^2 - 3*a*b^2*d^3)*f)*x^4 + (a*b^2*c^3 - 2*a^2*b*c^2*d +
a^3*c*d^2)*e*m^2 + (8*b^3*c*d^2*e + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*f*m^2 + 4*(b^3*c^2*d - 3*a*b^2*c*d
^2)*f + (2*(b^3*c*d^2 - a*b^2*d^3)*e + (5*b^3*c^2*d - 8*a*b^2*c*d^2 + 3*a^2*b*d^3)*f)*m)*x^3 + (12*b^3*c^2*d*e
+ ((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*f)*m^2 + 3*(b^
3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*f + ((7*b^3*c^2*d - 8*a*b^2*c*d^2 + a^2*b*d^3)*e + 4*(b^3*c^3
- a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*f)*m)*x^2 + 2*(3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c*d^2)*e - (3*a^2*b*c
^3 - a^3*c^2*d)*f + ((5*a*b^2*c^3 - 8*a^2*b*c^2*d + 3*a^3*c*d^2)*e - (a^2*b*c^3 - a^3*c^2*d)*f)*m + (((b^3*c^3
- a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*e + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f)*m^2 + 2*(3*b^3*c^3 + 3*
a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*e - 4*(3*a^2*b*c^2*d - a^3*c*d^2)*f + ((5*b^3*c^3 - a*b^2*c^2*d - 7*a^2
*b*c*d^2 + 3*a^3*d^3)*e + (3*a*b^2*c^3 - 8*a^2*b*c^2*d + 5*a^3*c*d^2)*f)*m)*x)*(b*x + a)^m*(d*x + c)^(-m - 4)/
(6*b^3*c^3 - 18*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 6*a^3*d^3 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)
*m^3 + 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*m^2 + 11*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2
- a^3*d^3)*m)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 503, normalized size = 2.68 \begin {gather*} -\frac {\left (a^{2} d^{2} f \,m^{2} x -2 a b c d f \,m^{2} x -a b \,d^{2} f m \,x^{2}+b^{2} c^{2} f \,m^{2} x +b^{2} c d f m \,x^{2}+a^{2} d^{2} e \,m^{2}+4 a^{2} d^{2} f m x -2 a b c d e \,m^{2}-8 a b c d f m x -2 a b \,d^{2} e m x -3 a b \,d^{2} f \,x^{2}+b^{2} c^{2} e \,m^{2}+4 b^{2} c^{2} f m x +2 b^{2} c d e m x +b^{2} c d f \,x^{2}+2 b^{2} d^{2} e \,x^{2}+a^{2} c d f m +3 a^{2} d^{2} e m +3 a^{2} d^{2} f x -a b \,c^{2} f m -8 a b c d e m -10 a b c d f x -2 a b \,d^{2} e x +5 b^{2} c^{2} e m +3 b^{2} c^{2} f x +6 b^{2} c d e x +a^{2} c d f +2 a^{2} d^{2} e -3 a b \,c^{2} f -6 a b c d e +6 b^{2} c^{2} e \right ) \left (b x +a \right )^{m +1} \left (d x +c \right )^{-m -3}}{a^{3} d^{3} m^{3}-3 a^{2} b c \,d^{2} m^{3}+3 a \,b^{2} c^{2} d \,m^{3}-b^{3} c^{3} m^{3}+6 a^{3} d^{3} m^{2}-18 a^{2} b c \,d^{2} m^{2}+18 a \,b^{2} c^{2} d \,m^{2}-6 b^{3} c^{3} m^{2}+11 a^{3} d^{3} m -33 a^{2} b c \,d^{2} m +33 a \,b^{2} c^{2} d m -11 b^{3} c^{3} m +6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x+a)^(m+1)*(d*x+c)^(-m-3)*(a^2*d^2*f*m^2*x-2*a*b*c*d*f*m^2*x-a*b*d^2*f*m*x^2+b^2*c^2*f*m^2*x+b^2*c*d*f*m*x
^2+a^2*d^2*e*m^2+4*a^2*d^2*f*m*x-2*a*b*c*d*e*m^2-8*a*b*c*d*f*m*x-2*a*b*d^2*e*m*x-3*a*b*d^2*f*x^2+b^2*c^2*e*m^2
+4*b^2*c^2*f*m*x+2*b^2*c*d*e*m*x+b^2*c*d*f*x^2+2*b^2*d^2*e*x^2+a^2*c*d*f*m+3*a^2*d^2*e*m+3*a^2*d^2*f*x-a*b*c^2
*f*m-8*a*b*c*d*e*m-10*a*b*c*d*f*x-2*a*b*d^2*e*x+5*b^2*c^2*e*m+3*b^2*c^2*f*x+6*b^2*c*d*e*x+a^2*c*d*f+2*a^2*d^2*
e-3*a*b*c^2*f-6*a*b*c*d*e+6*b^2*c^2*e)/(a^3*d^3*m^3-3*a^2*b*c*d^2*m^3+3*a*b^2*c^2*d*m^3-b^3*c^3*m^3+6*a^3*d^3*
m^2-18*a^2*b*c*d^2*m^2+18*a*b^2*c^2*d*m^2-6*b^3*c^3*m^2+11*a^3*d^3*m-33*a^2*b*c*d^2*m+33*a*b^2*c^2*d*m-11*b^3*
c^3*m+6*a^3*d^3-18*a^2*b*c*d^2+18*a*b^2*c^2*d-6*b^3*c^3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.44, size = 874, normalized size = 4.65 \begin {gather*} -\frac {x\,{\left (a+b\,x\right )}^m\,\left (f\,a^3\,c\,d^2\,m^2+5\,f\,a^3\,c\,d^2\,m+4\,f\,a^3\,c\,d^2+e\,a^3\,d^3\,m^2+3\,e\,a^3\,d^3\,m+2\,e\,a^3\,d^3-2\,f\,a^2\,b\,c^2\,d\,m^2-8\,f\,a^2\,b\,c^2\,d\,m-12\,f\,a^2\,b\,c^2\,d-e\,a^2\,b\,c\,d^2\,m^2-7\,e\,a^2\,b\,c\,d^2\,m-6\,e\,a^2\,b\,c\,d^2+f\,a\,b^2\,c^3\,m^2+3\,f\,a\,b^2\,c^3\,m-e\,a\,b^2\,c^2\,d\,m^2-e\,a\,b^2\,c^2\,d\,m+6\,e\,a\,b^2\,c^2\,d+e\,b^3\,c^3\,m^2+5\,e\,b^3\,c^3\,m+6\,e\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {x^2\,{\left (a+b\,x\right )}^m\,\left (f\,a^3\,d^3\,m^2+4\,f\,a^3\,d^3\,m+3\,f\,a^3\,d^3-f\,a^2\,b\,c\,d^2\,m^2-4\,f\,a^2\,b\,c\,d^2\,m-9\,f\,a^2\,b\,c\,d^2+e\,a^2\,b\,d^3\,m^2+e\,a^2\,b\,d^3\,m-f\,a\,b^2\,c^2\,d\,m^2-4\,f\,a\,b^2\,c^2\,d\,m-9\,f\,a\,b^2\,c^2\,d-2\,e\,a\,b^2\,c\,d^2\,m^2-8\,e\,a\,b^2\,c\,d^2\,m+f\,b^3\,c^3\,m^2+4\,f\,b^3\,c^3\,m+3\,f\,b^3\,c^3+e\,b^3\,c^2\,d\,m^2+7\,e\,b^3\,c^2\,d\,m+12\,e\,b^3\,c^2\,d\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^m\,\left (f\,a^2\,c\,d\,m+f\,a^2\,c\,d+e\,a^2\,d^2\,m^2+3\,e\,a^2\,d^2\,m+2\,e\,a^2\,d^2-f\,a\,b\,c^2\,m-3\,f\,a\,b\,c^2-2\,e\,a\,b\,c\,d\,m^2-8\,e\,a\,b\,c\,d\,m-6\,e\,a\,b\,c\,d+e\,b^2\,c^2\,m^2+5\,e\,b^2\,c^2\,m+6\,e\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {b^2\,d^2\,x^4\,{\left (a+b\,x\right )}^m\,\left (b\,c\,f-3\,a\,d\,f+2\,b\,d\,e-a\,d\,f\,m+b\,c\,f\,m\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {b\,d\,x^3\,{\left (a+b\,x\right )}^m\,\left (4\,b\,c-a\,d\,m+b\,c\,m\right )\,\left (b\,c\,f-3\,a\,d\,f+2\,b\,d\,e-a\,d\,f\,m+b\,c\,f\,m\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\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 + 4),x)

[Out]

- (x*(a + b*x)^m*(2*a^3*d^3*e + 6*b^3*c^3*e + a^3*d^3*e*m^2 + b^3*c^3*e*m^2 + 4*a^3*c*d^2*f + 3*a^3*d^3*e*m +
5*b^3*c^3*e*m + 6*a*b^2*c^2*d*e - 6*a^2*b*c*d^2*e - 12*a^2*b*c^2*d*f + 3*a*b^2*c^3*f*m + 5*a^3*c*d^2*f*m + a*b
^2*c^3*f*m^2 + a^3*c*d^2*f*m^2 - a*b^2*c^2*d*e*m - 7*a^2*b*c*d^2*e*m - 8*a^2*b*c^2*d*f*m - a*b^2*c^2*d*e*m^2 -
a^2*b*c*d^2*e*m^2 - 2*a^2*b*c^2*d*f*m^2))/((a*d - b*c)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) - (x^2*(
a + b*x)^m*(3*a^3*d^3*f + 3*b^3*c^3*f + a^3*d^3*f*m^2 + b^3*c^3*f*m^2 + 12*b^3*c^2*d*e + 4*a^3*d^3*f*m + 4*b^3
*c^3*f*m - 9*a*b^2*c^2*d*f - 9*a^2*b*c*d^2*f + a^2*b*d^3*e*m + 7*b^3*c^2*d*e*m + a^2*b*d^3*e*m^2 + b^3*c^2*d*e
*m^2 - 8*a*b^2*c*d^2*e*m - 4*a*b^2*c^2*d*f*m - 4*a^2*b*c*d^2*f*m - 2*a*b^2*c*d^2*e*m^2 - a*b^2*c^2*d*f*m^2 - a
^2*b*c*d^2*f*m^2))/((a*d - b*c)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) - (a*c*(a + b*x)^m*(2*a^2*d^2*e
+ 6*b^2*c^2*e + a^2*d^2*e*m^2 + b^2*c^2*e*m^2 - 3*a*b*c^2*f + a^2*c*d*f + 3*a^2*d^2*e*m + 5*b^2*c^2*e*m - 6*a*
b*c*d*e - a*b*c^2*f*m + a^2*c*d*f*m - 2*a*b*c*d*e*m^2 - 8*a*b*c*d*e*m))/((a*d - b*c)^3*(c + d*x)^(m + 4)*(11*m
+ 6*m^2 + m^3 + 6)) - (b^2*d^2*x^4*(a + b*x)^m*(b*c*f - 3*a*d*f + 2*b*d*e - a*d*f*m + b*c*f*m))/((a*d - b*c)^
3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) - (b*d*x^3*(a + b*x)^m*(4*b*c - a*d*m + b*c*m)*(b*c*f - 3*a*d*f
+ 2*b*d*e - a*d*f*m + b*c*f*m))/((a*d - b*c)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6))

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sympy [F(-1)]  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)**(-4-m)*(f*x+e),x)

[Out]

Timed out

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