3.372 \(\int x^m (a+b x)^3 (A+B x) \, dx\)
Optimal. Leaf size=96 \[ \frac{a^2 x^{m+2} (a B+3 A b)}{m+2}+\frac{a^3 A x^{m+1}}{m+1}+\frac{b^2 x^{m+4} (3 a B+A b)}{m+4}+\frac{3 a b x^{m+3} (a B+A b)}{m+3}+\frac{b^3 B x^{m+5}}{m+5} \]
[Out]
(a^3*A*x^(1 + m))/(1 + m) + (a^2*(3*A*b + a*B)*x^(2 + m))/(2 + m) + (3*a*b*(A*b + a*B)*x^(3 + m))/(3 + m) + (b
^2*(A*b + 3*a*B)*x^(4 + m))/(4 + m) + (b^3*B*x^(5 + m))/(5 + m)
________________________________________________________________________________________
Rubi [A] time = 0.0516339, antiderivative size = 96, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used =
{76} \[ \frac{a^2 x^{m+2} (a B+3 A b)}{m+2}+\frac{a^3 A x^{m+1}}{m+1}+\frac{b^2 x^{m+4} (3 a B+A b)}{m+4}+\frac{3 a b x^{m+3} (a B+A b)}{m+3}+\frac{b^3 B x^{m+5}}{m+5} \]
Antiderivative was successfully verified.
[In]
Int[x^m*(a + b*x)^3*(A + B*x),x]
[Out]
(a^3*A*x^(1 + m))/(1 + m) + (a^2*(3*A*b + a*B)*x^(2 + m))/(2 + m) + (3*a*b*(A*b + a*B)*x^(3 + m))/(3 + m) + (b
^2*(A*b + 3*a*B)*x^(4 + m))/(4 + m) + (b^3*B*x^(5 + m))/(5 + m)
Rule 76
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Rubi steps
\begin{align*} \int x^m (a+b x)^3 (A+B x) \, dx &=\int \left (a^3 A x^m+a^2 (3 A b+a B) x^{1+m}+3 a b (A b+a B) x^{2+m}+b^2 (A b+3 a B) x^{3+m}+b^3 B x^{4+m}\right ) \, dx\\ &=\frac{a^3 A x^{1+m}}{1+m}+\frac{a^2 (3 A b+a B) x^{2+m}}{2+m}+\frac{3 a b (A b+a B) x^{3+m}}{3+m}+\frac{b^2 (A b+3 a B) x^{4+m}}{4+m}+\frac{b^3 B x^{5+m}}{5+m}\\ \end{align*}
Mathematica [A] time = 0.101666, size = 87, normalized size = 0.91 \[ \frac{x^{m+1} \left (\left (\frac{3 a^2 b x}{m+2}+\frac{a^3}{m+1}+\frac{3 a b^2 x^2}{m+3}+\frac{b^3 x^3}{m+4}\right ) (A b (m+5)-a B (m+1))+B (a+b x)^4\right )}{b (m+5)} \]
Antiderivative was successfully verified.
[In]
Integrate[x^m*(a + b*x)^3*(A + B*x),x]
[Out]
(x^(1 + m)*(B*(a + b*x)^4 + (-(a*B*(1 + m)) + A*b*(5 + m))*(a^3/(1 + m) + (3*a^2*b*x)/(2 + m) + (3*a*b^2*x^2)/
(3 + m) + (b^3*x^3)/(4 + m))))/(b*(5 + m))
________________________________________________________________________________________
Maple [B] time = 0.004, size = 454, normalized size = 4.7 \begin{align*}{\frac{{x}^{1+m} \left ( B{b}^{3}{m}^{4}{x}^{4}+A{b}^{3}{m}^{4}{x}^{3}+3\,Ba{b}^{2}{m}^{4}{x}^{3}+10\,B{b}^{3}{m}^{3}{x}^{4}+3\,Aa{b}^{2}{m}^{4}{x}^{2}+11\,A{b}^{3}{m}^{3}{x}^{3}+3\,B{a}^{2}b{m}^{4}{x}^{2}+33\,Ba{b}^{2}{m}^{3}{x}^{3}+35\,B{b}^{3}{m}^{2}{x}^{4}+3\,A{a}^{2}b{m}^{4}x+36\,Aa{b}^{2}{m}^{3}{x}^{2}+41\,A{b}^{3}{m}^{2}{x}^{3}+B{a}^{3}{m}^{4}x+36\,B{a}^{2}b{m}^{3}{x}^{2}+123\,Ba{b}^{2}{m}^{2}{x}^{3}+50\,B{b}^{3}m{x}^{4}+A{a}^{3}{m}^{4}+39\,A{a}^{2}b{m}^{3}x+147\,Aa{b}^{2}{m}^{2}{x}^{2}+61\,A{b}^{3}m{x}^{3}+13\,B{a}^{3}{m}^{3}x+147\,B{a}^{2}b{m}^{2}{x}^{2}+183\,Ba{b}^{2}m{x}^{3}+24\,B{b}^{3}{x}^{4}+14\,A{a}^{3}{m}^{3}+177\,A{a}^{2}b{m}^{2}x+234\,Aa{b}^{2}m{x}^{2}+30\,A{b}^{3}{x}^{3}+59\,B{a}^{3}{m}^{2}x+234\,B{a}^{2}bm{x}^{2}+90\,B{x}^{3}a{b}^{2}+71\,A{a}^{3}{m}^{2}+321\,A{a}^{2}bmx+120\,aA{b}^{2}{x}^{2}+107\,B{a}^{3}mx+120\,B{x}^{2}{a}^{2}b+154\,A{a}^{3}m+180\,{a}^{2}Abx+60\,{a}^{3}Bx+120\,{a}^{3}A \right ) }{ \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(b*x+a)^3*(B*x+A),x)
[Out]
x^(1+m)*(B*b^3*m^4*x^4+A*b^3*m^4*x^3+3*B*a*b^2*m^4*x^3+10*B*b^3*m^3*x^4+3*A*a*b^2*m^4*x^2+11*A*b^3*m^3*x^3+3*B
*a^2*b*m^4*x^2+33*B*a*b^2*m^3*x^3+35*B*b^3*m^2*x^4+3*A*a^2*b*m^4*x+36*A*a*b^2*m^3*x^2+41*A*b^3*m^2*x^3+B*a^3*m
^4*x+36*B*a^2*b*m^3*x^2+123*B*a*b^2*m^2*x^3+50*B*b^3*m*x^4+A*a^3*m^4+39*A*a^2*b*m^3*x+147*A*a*b^2*m^2*x^2+61*A
*b^3*m*x^3+13*B*a^3*m^3*x+147*B*a^2*b*m^2*x^2+183*B*a*b^2*m*x^3+24*B*b^3*x^4+14*A*a^3*m^3+177*A*a^2*b*m^2*x+23
4*A*a*b^2*m*x^2+30*A*b^3*x^3+59*B*a^3*m^2*x+234*B*a^2*b*m*x^2+90*B*a*b^2*x^3+71*A*a^3*m^2+321*A*a^2*b*m*x+120*
A*a*b^2*x^2+107*B*a^3*m*x+120*B*a^2*b*x^2+154*A*a^3*m+180*A*a^2*b*x+60*B*a^3*x+120*A*a^3)/(5+m)/(4+m)/(3+m)/(2
+m)/(1+m)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x+a)^3*(B*x+A),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.45756, size = 845, normalized size = 8.8 \begin{align*} \frac{{\left ({\left (B b^{3} m^{4} + 10 \, B b^{3} m^{3} + 35 \, B b^{3} m^{2} + 50 \, B b^{3} m + 24 \, B b^{3}\right )} x^{5} +{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 90 \, B a b^{2} + 30 \, A b^{3} + 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 41 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 61 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{4} + 3 \,{\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 40 \, B a^{2} b + 40 \, A a b^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} m^{3} + 49 \,{\left (B a^{2} b + A a b^{2}\right )} m^{2} + 78 \,{\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{3} +{\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 60 \, B a^{3} + 180 \, A a^{2} b + 13 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 59 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 107 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{2} +{\left (A a^{3} m^{4} + 14 \, A a^{3} m^{3} + 71 \, A a^{3} m^{2} + 154 \, A a^{3} m + 120 \, A a^{3}\right )} x\right )} x^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x+a)^3*(B*x+A),x, algorithm="fricas")
[Out]
((B*b^3*m^4 + 10*B*b^3*m^3 + 35*B*b^3*m^2 + 50*B*b^3*m + 24*B*b^3)*x^5 + ((3*B*a*b^2 + A*b^3)*m^4 + 90*B*a*b^2
+ 30*A*b^3 + 11*(3*B*a*b^2 + A*b^3)*m^3 + 41*(3*B*a*b^2 + A*b^3)*m^2 + 61*(3*B*a*b^2 + A*b^3)*m)*x^4 + 3*((B*
a^2*b + A*a*b^2)*m^4 + 40*B*a^2*b + 40*A*a*b^2 + 12*(B*a^2*b + A*a*b^2)*m^3 + 49*(B*a^2*b + A*a*b^2)*m^2 + 78*
(B*a^2*b + A*a*b^2)*m)*x^3 + ((B*a^3 + 3*A*a^2*b)*m^4 + 60*B*a^3 + 180*A*a^2*b + 13*(B*a^3 + 3*A*a^2*b)*m^3 +
59*(B*a^3 + 3*A*a^2*b)*m^2 + 107*(B*a^3 + 3*A*a^2*b)*m)*x^2 + (A*a^3*m^4 + 14*A*a^3*m^3 + 71*A*a^3*m^2 + 154*A
*a^3*m + 120*A*a^3)*x)*x^m/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)
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Sympy [A] time = 3.51602, size = 2018, normalized size = 21.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m*(b*x+a)**3*(B*x+A),x)
[Out]
Piecewise((-A*a**3/(4*x**4) - A*a**2*b/x**3 - 3*A*a*b**2/(2*x**2) - A*b**3/x - B*a**3/(3*x**3) - 3*B*a**2*b/(2
*x**2) - 3*B*a*b**2/x + B*b**3*log(x), Eq(m, -5)), (-A*a**3/(3*x**3) - 3*A*a**2*b/(2*x**2) - 3*A*a*b**2/x + A*
b**3*log(x) - B*a**3/(2*x**2) - 3*B*a**2*b/x + 3*B*a*b**2*log(x) + B*b**3*x, Eq(m, -4)), (-A*a**3/(2*x**2) - 3
*A*a**2*b/x + 3*A*a*b**2*log(x) + A*b**3*x - B*a**3/x + 3*B*a**2*b*log(x) + 3*B*a*b**2*x + B*b**3*x**2/2, Eq(m
, -3)), (-A*a**3/x + 3*A*a**2*b*log(x) + 3*A*a*b**2*x + A*b**3*x**2/2 + B*a**3*log(x) + 3*B*a**2*b*x + 3*B*a*b
**2*x**2/2 + B*b**3*x**3/3, Eq(m, -2)), (A*a**3*log(x) + 3*A*a**2*b*x + 3*A*a*b**2*x**2/2 + A*b**3*x**3/3 + B*
a**3*x + 3*B*a**2*b*x**2/2 + B*a*b**2*x**3 + B*b**3*x**4/4, Eq(m, -1)), (A*a**3*m**4*x*x**m/(m**5 + 15*m**4 +
85*m**3 + 225*m**2 + 274*m + 120) + 14*A*a**3*m**3*x*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120)
+ 71*A*a**3*m**2*x*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 154*A*a**3*m*x*x**m/(m**5 + 15*m
**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 120*A*a**3*x*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120
) + 3*A*a**2*b*m**4*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 39*A*a**2*b*m**3*x**2*x**m
/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 177*A*a**2*b*m**2*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 +
225*m**2 + 274*m + 120) + 321*A*a**2*b*m*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 180*
A*a**2*b*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 3*A*a*b**2*m**4*x**3*x**m/(m**5 + 15*
m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 36*A*a*b**2*m**3*x**3*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 2
74*m + 120) + 147*A*a*b**2*m**2*x**3*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 234*A*a*b**2*m
*x**3*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 120*A*a*b**2*x**3*x**m/(m**5 + 15*m**4 + 85*m
**3 + 225*m**2 + 274*m + 120) + A*b**3*m**4*x**4*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 11
*A*b**3*m**3*x**4*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 41*A*b**3*m**2*x**4*x**m/(m**5 +
15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 61*A*b**3*m*x**4*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274
*m + 120) + 30*A*b**3*x**4*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + B*a**3*m**4*x**2*x**m/(m
**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 13*B*a**3*m**3*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m
**2 + 274*m + 120) + 59*B*a**3*m**2*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 107*B*a**3
*m*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 60*B*a**3*x**2*x**m/(m**5 + 15*m**4 + 85*m*
*3 + 225*m**2 + 274*m + 120) + 3*B*a**2*b*m**4*x**3*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) +
36*B*a**2*b*m**3*x**3*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 147*B*a**2*b*m**2*x**3*x**m/
(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 234*B*a**2*b*m*x**3*x**m/(m**5 + 15*m**4 + 85*m**3 + 225
*m**2 + 274*m + 120) + 120*B*a**2*b*x**3*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 3*B*a*b**2
*m**4*x**4*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 33*B*a*b**2*m**3*x**4*x**m/(m**5 + 15*m*
*4 + 85*m**3 + 225*m**2 + 274*m + 120) + 123*B*a*b**2*m**2*x**4*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 27
4*m + 120) + 183*B*a*b**2*m*x**4*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 90*B*a*b**2*x**4*x
**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + B*b**3*m**4*x**5*x**m/(m**5 + 15*m**4 + 85*m**3 + 22
5*m**2 + 274*m + 120) + 10*B*b**3*m**3*x**5*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 35*B*b*
*3*m**2*x**5*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 50*B*b**3*m*x**5*x**m/(m**5 + 15*m**4
+ 85*m**3 + 225*m**2 + 274*m + 120) + 24*B*b**3*x**5*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120),
True))
________________________________________________________________________________________
Giac [B] time = 1.20589, size = 801, normalized size = 8.34 \begin{align*} \frac{B b^{3} m^{4} x^{5} x^{m} + 3 \, B a b^{2} m^{4} x^{4} x^{m} + A b^{3} m^{4} x^{4} x^{m} + 10 \, B b^{3} m^{3} x^{5} x^{m} + 3 \, B a^{2} b m^{4} x^{3} x^{m} + 3 \, A a b^{2} m^{4} x^{3} x^{m} + 33 \, B a b^{2} m^{3} x^{4} x^{m} + 11 \, A b^{3} m^{3} x^{4} x^{m} + 35 \, B b^{3} m^{2} x^{5} x^{m} + B a^{3} m^{4} x^{2} x^{m} + 3 \, A a^{2} b m^{4} x^{2} x^{m} + 36 \, B a^{2} b m^{3} x^{3} x^{m} + 36 \, A a b^{2} m^{3} x^{3} x^{m} + 123 \, B a b^{2} m^{2} x^{4} x^{m} + 41 \, A b^{3} m^{2} x^{4} x^{m} + 50 \, B b^{3} m x^{5} x^{m} + A a^{3} m^{4} x x^{m} + 13 \, B a^{3} m^{3} x^{2} x^{m} + 39 \, A a^{2} b m^{3} x^{2} x^{m} + 147 \, B a^{2} b m^{2} x^{3} x^{m} + 147 \, A a b^{2} m^{2} x^{3} x^{m} + 183 \, B a b^{2} m x^{4} x^{m} + 61 \, A b^{3} m x^{4} x^{m} + 24 \, B b^{3} x^{5} x^{m} + 14 \, A a^{3} m^{3} x x^{m} + 59 \, B a^{3} m^{2} x^{2} x^{m} + 177 \, A a^{2} b m^{2} x^{2} x^{m} + 234 \, B a^{2} b m x^{3} x^{m} + 234 \, A a b^{2} m x^{3} x^{m} + 90 \, B a b^{2} x^{4} x^{m} + 30 \, A b^{3} x^{4} x^{m} + 71 \, A a^{3} m^{2} x x^{m} + 107 \, B a^{3} m x^{2} x^{m} + 321 \, A a^{2} b m x^{2} x^{m} + 120 \, B a^{2} b x^{3} x^{m} + 120 \, A a b^{2} x^{3} x^{m} + 154 \, A a^{3} m x x^{m} + 60 \, B a^{3} x^{2} x^{m} + 180 \, A a^{2} b x^{2} x^{m} + 120 \, A a^{3} x x^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x+a)^3*(B*x+A),x, algorithm="giac")
[Out]
(B*b^3*m^4*x^5*x^m + 3*B*a*b^2*m^4*x^4*x^m + A*b^3*m^4*x^4*x^m + 10*B*b^3*m^3*x^5*x^m + 3*B*a^2*b*m^4*x^3*x^m
+ 3*A*a*b^2*m^4*x^3*x^m + 33*B*a*b^2*m^3*x^4*x^m + 11*A*b^3*m^3*x^4*x^m + 35*B*b^3*m^2*x^5*x^m + B*a^3*m^4*x^2
*x^m + 3*A*a^2*b*m^4*x^2*x^m + 36*B*a^2*b*m^3*x^3*x^m + 36*A*a*b^2*m^3*x^3*x^m + 123*B*a*b^2*m^2*x^4*x^m + 41*
A*b^3*m^2*x^4*x^m + 50*B*b^3*m*x^5*x^m + A*a^3*m^4*x*x^m + 13*B*a^3*m^3*x^2*x^m + 39*A*a^2*b*m^3*x^2*x^m + 147
*B*a^2*b*m^2*x^3*x^m + 147*A*a*b^2*m^2*x^3*x^m + 183*B*a*b^2*m*x^4*x^m + 61*A*b^3*m*x^4*x^m + 24*B*b^3*x^5*x^m
+ 14*A*a^3*m^3*x*x^m + 59*B*a^3*m^2*x^2*x^m + 177*A*a^2*b*m^2*x^2*x^m + 234*B*a^2*b*m*x^3*x^m + 234*A*a*b^2*m
*x^3*x^m + 90*B*a*b^2*x^4*x^m + 30*A*b^3*x^4*x^m + 71*A*a^3*m^2*x*x^m + 107*B*a^3*m*x^2*x^m + 321*A*a^2*b*m*x^
2*x^m + 120*B*a^2*b*x^3*x^m + 120*A*a*b^2*x^3*x^m + 154*A*a^3*m*x*x^m + 60*B*a^3*x^2*x^m + 180*A*a^2*b*x^2*x^m
+ 120*A*a^3*x*x^m)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)