3.3108 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x) \, dx\)
Optimal. Leaf size=268 \[ -\frac{2 b^2 (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+4)-b (c f (m+1)+3 d e))}{d (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4}}{d (m+4) (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b (c f (m+1)+3 d e))}{d (m+3) (m+4) (b c-a d)^2}-\frac{2 b (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+4)-b (c f (m+1)+3 d e))}{d (m+2) (m+3) (m+4) (b c-a d)^3} \]
[Out]
((d*e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/(d*(b*c - a*d)*(4 + m)) - ((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)) - (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)) - (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))
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Rubi [A] time = 0.150459, 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 = {79, 45, 37} \[ \frac{2 b^2 (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+4)+b c f (m+1)+3 b d e)}{d (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4}}{d (m+4) (b c-a d)}+\frac{(a+b x)^{m+1} (c+d x)^{-m-3} (-a d f (m+4)+b c f (m+1)+3 b d e)}{d (m+3) (m+4) (b c-a d)^2}+\frac{2 b (a+b x)^{m+1} (c+d x)^{-m-2} (-a d f (m+4)+b c f (m+1)+3 b d e)}{d (m+2) (m+3) (m+4) (b c-a d)^3} \]
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 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]
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]
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*}
Mathematica [A] time = 0.405592, size = 144, normalized size = 0.54 \[ -\frac{(a+b x)^{m+1} (c+d x)^{-m-4} \left (\frac{(c+d x) \left (2 b (c+d x) (-a d (m+1)+b c (m+2)+b d x)+(m+1) (m+2) (b c-a d)^2\right ) (-a d f (m+4)+b c f (m+1)+3 b d e)}{(m+1) (m+2) (m+3) (b c-a d)^3}-c f+d e\right )}{d (m+4) (a d-b c)} \]
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] time = 0.008, size = 1184, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}
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)
[Out]
-(b*x+a)^(1+m)*(d*x+c)^(-4-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., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \end{align*}
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] time = 2.21562, size = 3607, normalized size = 13.46 \begin{align*} \text{result too large to display} \end{align*}
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*(3*b^4*d^4*e + (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 + (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)*e*m^3 + 2*(15*b^4*c*d^3*e + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*f*m^2
+ 5*(b^4*c^2*d^2 - 4*a*b^3*c*d^3)*f + (3*(b^4*c*d^3 - a*b^3*d^4)*e + 2*(3*b^4*c^2*d^2 - 5*a*b^3*c*d^3 + 2*a^2
*b^2*d^4)*f)*m)*x^4 + (60*b^4*c^2*d^2*e + (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 +
(3*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e + 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 + 20*(b^4*c^3*d - 4*a*b^3*c^2*d^2)*f + (3*(9*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + a^2*b^2*d^4)*e + (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)*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)*e - (a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*f)*m^2 + (60*b^4*c^3*d*e + ((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)*e + (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 + (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)*e + (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 + 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 + ((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)*e + (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)*x^2 + 6*(4*a*b^3*c^4 - 6*a^2*
b^2*c^3*d + 4*a^3*b*c^2*d^2 - a^4*c*d^3)*e - 2*(6*a^2*b^2*c^4 - 4*a^3*b*c^3*d + a^4*c^2*d^2)*f + ((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)*e - (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)*e + (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 + (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)*e + (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 + 6*(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)*e - 10*(6*a^2*b^2*c^3*d - 4*a^3*b*c^2*d^2 + a^4*c*d^3)*f + ((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)*e + (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)*x)*(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)] 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)**(-5-m)*(f*x+e),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 )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \end{align*}
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)