3.31.97 \(\int (a+b x)^m (c+d x)^{-4-m} (e+f x)^3 \, dx\) [3097]

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

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 406, normalized size of antiderivative = 1.00, 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, 47, 37, 72, 71} \begin {gather*} \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 {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}+\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} \end {gather*}

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 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)^{-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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 26.79, size = 1150, normalized size = 2.83 \begin {gather*} \frac {1}{4} (a+b x)^m (c+d x)^{-m} \left (\frac {12 e f^2 \left (1+\frac {b x}{a}\right )^{-m} \left (1+\frac {d x}{c}\right )^m \left (b^3 c^3 \left (2+3 m+m^2\right ) x^3 \left (\frac {c (a+b x)}{a (c+d x)}\right )^m-a b^2 c^2 (1+m) x^2 \left (\frac {c (a+b x)}{a (c+d x)}\right )^m (-c m+2 d (3+m) x)+a^2 b c x \left (\frac {c (a+b x)}{a (c+d x)}\right )^m \left (-2 c^2 m-2 c d m (3+m) x+d^2 \left (6+5 m+m^2\right ) x^2\right )+a^3 \left (-2 d^3 x^3+2 c^3 \left (-1+\left (\frac {c (a+b x)}{a (c+d x)}\right )^m\right )+2 c^2 d x \left (-3+3 \left (\frac {c (a+b x)}{a (c+d x)}\right )^m+m \left (\frac {c (a+b x)}{a (c+d x)}\right )^m\right )+c d^2 x^2 \left (-6+6 \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+m^2 \left (\frac {c (a+b x)}{a (c+d x)}\right )^m\right )\right )\right )}{c (b c-a d)^3 (1+m) (2+m) (3+m) (c+d x)^3}+\frac {f^3 x^4 \left (1+\frac {b x}{a}\right )^{-m} \left (1+\frac {d x}{c}\right )^m F_1\left (4;-m,4+m;5;-\frac {b x}{a},-\frac {d x}{c}\right )}{c^4}+\frac {6 e^2 f \left ((c+d x) \left (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 \left (d^2 (3+m) x^2 \left (-2-m+2 \left (\frac {a (c+d x)}{c (a+b x)}\right )^m\right )+2 c d (3+m) x \left (-2+m+2 \left (\frac {a (c+d x)}{c (a+b x)}\right )^m\right )+2 c^2 \left (-3+2 m+3 \left (\frac {a (c+d x)}{c (a+b x)}\right )^m+m \left (\frac {a (c+d x)}{c (a+b x)}\right )^m\right )\right )+a^3 \left (2 d^3 m x^3 \left (\frac {a (c+d x)}{c (a+b x)}\right )^m-6 c^3 \left (-1+\left (\frac {a (c+d x)}{c (a+b x)}\right )^m\right )+2 c^2 d x \left (6-6 \left (\frac {a (c+d x)}{c (a+b x)}\right )^m+m \left (2+\left (\frac {a (c+d x)}{c (a+b x)}\right )^m\right )\right )+c d^2 x^2 \left (6+m^2-6 \left (\frac {a (c+d x)}{c (a+b x)}\right )^m+m \left (5+4 \left (\frac {a (c+d x)}{c (a+b x)}\right )^m\right )\right )\right )\right ) \Gamma (1-m)+m (3 c+d x) \left (b^3 c^3 \left (2+3 m+m^2\right ) x^3+a b^2 c^2 (1+m) x^2 (c m-2 d (3+m) x)+a^2 b c x \left (-2 c^2 m-2 c d m (3+m) x+d^2 \left (6+5 m+m^2\right ) x^2\right )+a^3 \left (-2 d^3 x^3 \left (\frac {a (c+d x)}{c (a+b x)}\right )^m-2 c^3 \left (-1+\left (\frac {a (c+d x)}{c (a+b x)}\right )^m\right )-2 c^2 d x \left (-3-m+3 \left (\frac {a (c+d x)}{c (a+b x)}\right )^m\right )-c d^2 x^2 \left (-6-5 m-m^2+6 \left (\frac {a (c+d x)}{c (a+b x)}\right )^m\right )\right )\right ) \Gamma (-m)\right )}{c^2 (b c-a d)^3 m (1+m) (2+m) (3+m) x (c+d x)^3 \Gamma (-m)}-\frac {4 e^3 \left (\frac {d (a+b x)}{-b c+a d}\right )^{-m} \, _2F_1\left (-3-m,-m;-2-m;\frac {b (c+d x)}{b c-a d}\right )}{d (3+m) (c+d x)^3}\right ) \end {gather*}

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
)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (d x +c \right )^{-4-m} \left (f x +e \right )^{3}\, dx\]

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 = 0.00, 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)^(-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 = 0.00, 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)^(-4-m)*(f*x+e)^3,x, algorithm="fricas")

[Out]

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

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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)**(-4-m)*(f*x+e)**3,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)^(-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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^3\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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