3.3097 \(\int (a+b x)^m (c+d x)^{-4-m} (e+f x)^3 \, dx\)

Optimal. Leaf size=406 \[ \frac{2 b^2 (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (m+3) (b c-a d)^3}+\frac{3 f^2 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^3 (m+1) (b c-a d)}+\frac{(a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^3 (m+3) (b c-a d)}+\frac{3 f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^3 (m+2) (b c-a d)}+\frac{2 b (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^3 (m+2) (m+3) (b c-a d)^2}+\frac{3 b f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (b c-a d)^2}-\frac{f^3 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^4 m} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.279139, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{2 b^2 (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (m+3) (b c-a d)^3}+\frac{3 f^2 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^3 (m+1) (b c-a d)}+\frac{(a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^3 (m+3) (b c-a d)}+\frac{3 f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^3 (m+2) (b c-a d)}+\frac{2 b (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^3 (m+2) (m+3) (b c-a d)^2}+\frac{3 b f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (b c-a d)^2}-\frac{f^3 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^4 m} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x)^3,x]

[Out]

((d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(d^3*(b*c - a*d)*(3 + m)) + (3*f*(d*e - c*f)^2*(a + b*x)^
(1 + m)*(c + d*x)^(-2 - m))/(d^3*(b*c - a*d)*(2 + m)) + (2*b*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m
))/(d^3*(b*c - a*d)^2*(2 + m)*(3 + m)) + (3*f^2*(d*e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^3*(b*c -
a*d)*(1 + m)) + (3*b*f*(d*e - c*f)^2*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^3*(b*c - a*d)^2*(1 + m)*(2 + m))
+ (2*b^2*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^3*(b*c - a*d)^3*(1 + m)*(2 + m)*(3 + m)) - (f
^3*(a + b*x)^m*Hypergeometric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^4*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)^{-4-m} (e+f x)^3 \, dx &=\int \left (\frac{(d e-c f)^3 (a+b x)^m (c+d x)^{-4-m}}{d^3}+\frac{3 f (d e-c f)^2 (a+b x)^m (c+d x)^{-3-m}}{d^3}+\frac{3 f^2 (d e-c f) (a+b x)^m (c+d x)^{-2-m}}{d^3}+\frac{f^3 (a+b x)^m (c+d x)^{-1-m}}{d^3}\right ) \, dx\\ &=\frac{f^3 \int (a+b x)^m (c+d x)^{-1-m} \, dx}{d^3}+\frac{\left (3 f^2 (d e-c f)\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^3}+\frac{\left (3 f (d e-c f)^2\right ) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{d^3}+\frac{(d e-c f)^3 \int (a+b x)^m (c+d x)^{-4-m} \, dx}{d^3}\\ &=\frac{(d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^3 (b c-a d) (3+m)}+\frac{3 f (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^3 (b c-a d) (2+m)}+\frac{3 f^2 (d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^3 (b c-a d) (1+m)}+\frac{\left (3 b f (d e-c f)^2\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^3 (b c-a d) (2+m)}+\frac{\left (2 b (d e-c f)^3\right ) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{d^3 (b c-a d) (3+m)}+\frac{\left (f^3 (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^3}\\ &=\frac{(d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^3 (b c-a d) (3+m)}+\frac{3 f (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^3 (b c-a d) (2+m)}+\frac{2 b (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^3 (b c-a d)^2 (2+m) (3+m)}+\frac{3 f^2 (d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^3 (b c-a d) (1+m)}+\frac{3 b f (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-1-m}}{d^3 (b c-a d)^2 (1+m) (2+m)}-\frac{f^3 (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^4 m}+\frac{\left (2 b^2 (d e-c f)^3\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^3 (b c-a d)^2 (2+m) (3+m)}\\ &=\frac{(d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^3 (b c-a d) (3+m)}+\frac{3 f (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^3 (b c-a d) (2+m)}+\frac{2 b (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^3 (b c-a d)^2 (2+m) (3+m)}+\frac{3 f^2 (d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^3 (b c-a d) (1+m)}+\frac{3 b f (d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-1-m}}{d^3 (b c-a d)^2 (1+m) (2+m)}+\frac{2 b^2 (d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-1-m}}{d^3 (b c-a d)^3 (1+m) (2+m) (3+m)}-\frac{f^3 (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^4 m}\\ \end{align*}

Mathematica [C]  time = 32.386, size = 1150, normalized size = 2.83 \[ \frac{1}{4} (a+b x)^m (c+d x)^{-m} \left (-\frac{4 e^3 \, _2F_1\left (-m-3,-m;-m-2;\frac{b (c+d x)}{b c-a d}\right ) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m}}{d (m+3) (c+d x)^3}+\frac{12 e f^2 \left (\frac{b x}{a}+1\right )^{-m} \left (\frac{d x}{c}+1\right )^m \left (b^3 c^3 \left (m^2+3 m+2\right ) x^3 \left (\frac{c (a+b x)}{a (c+d x)}\right )^m-a b^2 c^2 (m+1) x^2 (2 d (m+3) x-c m) \left (\frac{c (a+b x)}{a (c+d x)}\right )^m+a^2 b c x \left (-2 m c^2-2 d m (m+3) x c+d^2 \left (m^2+5 m+6\right ) x^2\right ) \left (\frac{c (a+b x)}{a (c+d x)}\right )^m+a^3 \left (2 \left (\left (\frac{c (a+b x)}{a (c+d x)}\right )^m-1\right ) c^3+2 d x \left (m \left (\frac{c (a+b x)}{a (c+d x)}\right )^m+3 \left (\frac{c (a+b x)}{a (c+d x)}\right )^m-3\right ) c^2+d^2 x^2 \left (m^2 \left (\frac{c (a+b x)}{a (c+d x)}\right )^m+5 m \left (\frac{c (a+b x)}{a (c+d x)}\right )^m+6 \left (\frac{c (a+b x)}{a (c+d x)}\right )^m-6\right ) c-2 d^3 x^3\right )\right )}{c (b c-a d)^3 (m+1) (m+2) (m+3) (c+d x)^3}+\frac{f^3 x^4 \left (\frac{b x}{a}+1\right )^{-m} \left (\frac{d x}{c}+1\right )^m F_1\left (4;-m,m+4;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{c^4}+\frac{6 e^2 f \left ((c+d x) \left (\left (2 d^3 m x^3 \left (\frac{a (c+d x)}{c (a+b x)}\right )^m-6 c^3 \left (\left (\frac{a (c+d x)}{c (a+b x)}\right )^m-1\right )+2 c^2 d x \left (-6 \left (\frac{a (c+d x)}{c (a+b x)}\right )^m+m \left (\left (\frac{a (c+d x)}{c (a+b x)}\right )^m+2\right )+6\right )+c d^2 x^2 \left (-6 \left (\frac{a (c+d x)}{c (a+b x)}\right )^m+m^2+m \left (4 \left (\frac{a (c+d x)}{c (a+b x)}\right )^m+5\right )+6\right )\right ) a^3-b c x \left (2 \left (m \left (\frac{a (c+d x)}{c (a+b x)}\right )^m+3 \left (\frac{a (c+d x)}{c (a+b x)}\right )^m+2 m-3\right ) c^2+2 d (m+3) x \left (2 \left (\frac{a (c+d x)}{c (a+b x)}\right )^m+m-2\right ) c+d^2 (m+3) x^2 \left (2 \left (\frac{a (c+d x)}{c (a+b x)}\right )^m-m-2\right )\right ) a^2+b^2 c^2 m x^2 (c (m-3)-2 d (m+3) x) a+b^3 c^3 m (m+1) x^3\right ) \text{Gamma}(1-m)+m (3 c+d x) \left (\left (-2 d^3 x^3 \left (\frac{a (c+d x)}{c (a+b x)}\right )^m-2 c^3 \left (\left (\frac{a (c+d x)}{c (a+b x)}\right )^m-1\right )-2 c^2 d x \left (3 \left (\frac{a (c+d x)}{c (a+b x)}\right )^m-m-3\right )-c d^2 x^2 \left (6 \left (\frac{a (c+d x)}{c (a+b x)}\right )^m-m^2-5 m-6\right )\right ) a^3+b c x \left (-2 m c^2-2 d m (m+3) x c+d^2 \left (m^2+5 m+6\right ) x^2\right ) a^2+b^2 c^2 (m+1) x^2 (c m-2 d (m+3) x) a+b^3 c^3 \left (m^2+3 m+2\right ) x^3\right ) \text{Gamma}(-m)\right )}{c^2 (b c-a d)^3 m (m+1) (m+2) (m+3) x (c+d x)^3 \text{Gamma}(-m)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x)^3,x]

[Out]

((a + b*x)^m*((12*e*f^2*(1 + (d*x)/c)^m*(b^3*c^3*(2 + 3*m + m^2)*x^3*((c*(a + b*x))/(a*(c + d*x)))^m - a*b^2*c
^2*(1 + m)*x^2*((c*(a + b*x))/(a*(c + d*x)))^m*(-(c*m) + 2*d*(3 + m)*x) + a^2*b*c*x*((c*(a + b*x))/(a*(c + d*x
)))^m*(-2*c^2*m - 2*c*d*m*(3 + m)*x + d^2*(6 + 5*m + m^2)*x^2) + a^3*(-2*d^3*x^3 + 2*c^3*(-1 + ((c*(a + b*x))/
(a*(c + d*x)))^m) + 2*c^2*d*x*(-3 + 3*((c*(a + b*x))/(a*(c + d*x)))^m + m*((c*(a + b*x))/(a*(c + d*x)))^m) + c
*d^2*x^2*(-6 + 6*((c*(a + b*x))/(a*(c + d*x)))^m + 5*m*((c*(a + b*x))/(a*(c + d*x)))^m + m^2*((c*(a + b*x))/(a
*(c + d*x)))^m))))/(c*(b*c - a*d)^3*(1 + m)*(2 + m)*(3 + m)*(1 + (b*x)/a)^m*(c + d*x)^3) + (f^3*x^4*(1 + (d*x)
/c)^m*AppellF1[4, -m, 4 + m, 5, -((b*x)/a), -((d*x)/c)])/(c^4*(1 + (b*x)/a)^m) + (6*e^2*f*((c + d*x)*(b^3*c^3*
m*(1 + m)*x^3 + a*b^2*c^2*m*x^2*(c*(-3 + m) - 2*d*(3 + m)*x) - a^2*b*c*x*(d^2*(3 + m)*x^2*(-2 - m + 2*((a*(c +
d*x))/(c*(a + b*x)))^m) + 2*c*d*(3 + m)*x*(-2 + m + 2*((a*(c + d*x))/(c*(a + b*x)))^m) + 2*c^2*(-3 + 2*m + 3*
((a*(c + d*x))/(c*(a + b*x)))^m + m*((a*(c + d*x))/(c*(a + b*x)))^m)) + a^3*(2*d^3*m*x^3*((a*(c + d*x))/(c*(a
+ b*x)))^m - 6*c^3*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m) + 2*c^2*d*x*(6 - 6*((a*(c + d*x))/(c*(a + b*x)))^m +
m*(2 + ((a*(c + d*x))/(c*(a + b*x)))^m)) + c*d^2*x^2*(6 + m^2 - 6*((a*(c + d*x))/(c*(a + b*x)))^m + m*(5 + 4*
((a*(c + d*x))/(c*(a + b*x)))^m))))*Gamma[1 - m] + m*(3*c + d*x)*(b^3*c^3*(2 + 3*m + m^2)*x^3 + a*b^2*c^2*(1 +
m)*x^2*(c*m - 2*d*(3 + m)*x) + a^2*b*c*x*(-2*c^2*m - 2*c*d*m*(3 + m)*x + d^2*(6 + 5*m + m^2)*x^2) + a^3*(-2*d
^3*x^3*((a*(c + d*x))/(c*(a + b*x)))^m - 2*c^3*(-1 + ((a*(c + d*x))/(c*(a + b*x)))^m) - 2*c^2*d*x*(-3 - m + 3*
((a*(c + d*x))/(c*(a + b*x)))^m) - c*d^2*x^2*(-6 - 5*m - m^2 + 6*((a*(c + d*x))/(c*(a + b*x)))^m)))*Gamma[-m])
)/(c^2*(b*c - a*d)^3*m*(1 + m)*(2 + m)*(3 + m)*x*(c + d*x)^3*Gamma[-m]) - (4*e^3*Hypergeometric2F1[-3 - m, -m,
-2 - m, (b*(c + d*x))/(b*c - a*d)])/(d*(3 + m)*((d*(a + b*x))/(-(b*c) + a*d))^m*(c + d*x)^3)))/(4*(c + d*x)^m
)

________________________________________________________________________________________

Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-4-m} \left ( fx+e \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x)

[Out]

int((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x, algorithm="maxima")

[Out]

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x, algorithm="fricas")

[Out]

integral((f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x + e^3)*(b*x + a)^m*(d*x + c)^(-m - 4), 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)**(-4-m)*(f*x+e)**3,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 )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x, algorithm="giac")

[Out]

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 4), x)