\(\int (a+b x)^m (c+d x)^{-4-m} \, dx\) [3100]

Optimal result

Integrand size = 19, antiderivative size = 130 \[ \int (a+b x)^m (c+d x)^{-4-m} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (3+m)}+\frac {2 b (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^2 (2+m) (3+m)}+\frac {2 b^2 (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^3 (1+m) (2+m) (3+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int (a+b x)^m (c+d x)^{-4-m} \, dx=\frac {2 b^2 (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (m+2) (m+3) (b c-a d)^3}+\frac {(a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d)}+\frac {2 b (a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (m+3) (b c-a d)^2} \]

[In]

Int[(a + b*x)^m*(c + d*x)^(-4 - m),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (3+m)}+\frac {(2 b) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{(b c-a d) (3+m)} \\ & = \frac {(a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (3+m)}+\frac {2 b (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^2 (2+m) (3+m)}+\frac {\left (2 b^2\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{(b c-a d)^2 (2+m) (3+m)} \\ & = \frac {(a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (3+m)}+\frac {2 b (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^2 (2+m) (3+m)}+\frac {2 b^2 (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^3 (1+m) (2+m) (3+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int (a+b x)^m (c+d x)^{-4-m} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-3-m} \left (a^2 d^2 \left (2+3 m+m^2\right )-2 a b d (1+m) (c (3+m)+d x)+b^2 \left (c^2 \left (6+5 m+m^2\right )+2 c d (3+m) x+2 d^2 x^2\right )\right )}{(b c-a d)^3 (1+m) (2+m) (3+m)} \]

[In]

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

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-3 - m)*(a^2*d^2*(2 + 3*m + m^2) - 2*a*b*d*(1 + m)*(c*(3 + m) + d*x) + b^2*(c^2*
(6 + 5*m + m^2) + 2*c*d*(3 + m)*x + 2*d^2*x^2)))/((b*c - a*d)^3*(1 + m)*(2 + m)*(3 + m))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(318\) vs. \(2(130)=260\).

Time = 1.64 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.45

method	result	size
gosper	\(-\frac {\left (b x +a \right )^{1+m} \left (d x +c \right )^{-3-m} \left (a^{2} d^{2} m^{2}-2 a b c d \,m^{2}-2 a b \,d^{2} m x +b^{2} c^{2} m^{2}+2 b^{2} c d m x +2 d^{2} x^{2} b^{2}+3 a^{2} d^{2} m -8 a b c d m -2 x a b \,d^{2}+5 b^{2} c^{2} m +6 x \,b^{2} c d +2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}\right )}{a^{3} d^{3} m^{3}-3 a^{2} b c \,d^{2} m^{3}+3 a \,b^{2} c^{2} d \,m^{3}-b^{3} c^{3} m^{3}+6 a^{3} d^{3} m^{2}-18 a^{2} b c \,d^{2} m^{2}+18 a \,b^{2} c^{2} d \,m^{2}-6 b^{3} c^{3} m^{2}+11 a^{3} d^{3} m -33 a^{2} b c \,d^{2} m +33 a \,b^{2} c^{2} d m -11 b^{3} c^{3} m +6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}}\)	\(319\)

[In]

int((b*x+a)^m*(d*x+c)^(-4-m),x,method=_RETURNVERBOSE)

[Out]

-(b*x+a)^(1+m)*(d*x+c)^(-3-m)*(a^2*d^2*m^2-2*a*b*c*d*m^2-2*a*b*d^2*m*x+b^2*c^2*m^2+2*b^2*c*d*m*x+2*b^2*d^2*x^2
+3*a^2*d^2*m-8*a*b*c*d*m-2*a*b*d^2*x+5*b^2*c^2*m+6*b^2*c*d*x+2*a^2*d^2-6*a*b*c*d+6*b^2*c^2)/(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+1
1*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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (130) = 260\).

Time = 0.25 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.90 \[ \int (a+b x)^m (c+d x)^{-4-m} \, dx=\frac {{\left (2 \, b^{3} d^{3} x^{4} + 6 \, a b^{2} c^{3} - 6 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2} + 2 \, {\left (4 \, b^{3} c d^{2} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} m\right )} x^{3} + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} m^{2} + {\left (12 \, b^{3} c^{2} d + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} m^{2} + {\left (7 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} m\right )} x^{2} + {\left (5 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} m + {\left (6 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} m^{2} + {\left (5 \, b^{3} c^{3} - a b^{2} c^{2} d - 7 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} m\right )} x\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4}}{6 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 6 \, a^{3} d^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} m^{3} + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} m^{2} + 11 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} m} \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^m (c+d x)^{-4-m} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

\[ \int (a+b x)^m (c+d x)^{-4-m} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (a+b x)^m (c+d x)^{-4-m} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 3.23 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.06 \[ \int (a+b x)^m (c+d x)^{-4-m} \, dx=-\frac {x\,{\left (a+b\,x\right )}^m\,\left (a^3\,d^3\,m^2+3\,a^3\,d^3\,m+2\,a^3\,d^3-a^2\,b\,c\,d^2\,m^2-7\,a^2\,b\,c\,d^2\,m-6\,a^2\,b\,c\,d^2-a\,b^2\,c^2\,d\,m^2-a\,b^2\,c^2\,d\,m+6\,a\,b^2\,c^2\,d+b^3\,c^3\,m^2+5\,b^3\,c^3\,m+6\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^m\,\left (a^2\,d^2\,m^2+3\,a^2\,d^2\,m+2\,a^2\,d^2-2\,a\,b\,c\,d\,m^2-8\,a\,b\,c\,d\,m-6\,a\,b\,c\,d+b^2\,c^2\,m^2+5\,b^2\,c^2\,m+6\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {2\,b^3\,d^3\,x^4\,{\left (a+b\,x\right )}^m}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^m\,\left (a^2\,d^2\,m^2+a^2\,d^2\,m-2\,a\,b\,c\,d\,m^2-8\,a\,b\,c\,d\,m+b^2\,c^2\,m^2+7\,b^2\,c^2\,m+12\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {2\,b^2\,d^2\,x^3\,{\left (a+b\,x\right )}^m\,\left (4\,b\,c-a\,d\,m+b\,c\,m\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )} \]

[In]

int((a + b*x)^m/(c + d*x)^(m + 4),x)

[Out]

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