3.342 \(\int x^m (a+b x)^3 (A+B x) \, dx\)

Optimal. Leaf size=96 \[ \frac{a^3 A x^{m+1}}{m+1}+\frac{a^2 x^{m+2} (a B+3 A b)}{m+2}+\frac{b^2 x^{m+4} (3 a B+A b)}{m+4}+\frac{3 a b x^{m+3} (a B+A b)}{m+3}+\frac{b^3 B x^{m+5}}{m+5} \]

[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)

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Rubi [A]  time = 0.134228, 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 \[ \frac{a^3 A x^{m+1}}{m+1}+\frac{a^2 x^{m+2} (a B+3 A b)}{m+2}+\frac{b^2 x^{m+4} (3 a B+A b)}{m+4}+\frac{3 a b x^{m+3} (a B+A b)}{m+3}+\frac{b^3 B x^{m+5}}{m+5} \]

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)

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Rubi in Sympy [A]  time = 17.4506, size = 87, normalized size = 0.91 \[ \frac{A a^{3} x^{m + 1}}{m + 1} + \frac{B b^{3} x^{m + 5}}{m + 5} + \frac{a^{2} x^{m + 2} \left (3 A b + B a\right )}{m + 2} + \frac{3 a b x^{m + 3} \left (A b + B a\right )}{m + 3} + \frac{b^{2} x^{m + 4} \left (A b + 3 B a\right )}{m + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x+a)**3*(B*x+A),x)

[Out]

A*a**3*x**(m + 1)/(m + 1) + B*b**3*x**(m + 5)/(m + 5) + a**2*x**(m + 2)*(3*A*b +
B*a)/(m + 2) + 3*a*b*x**(m + 3)*(A*b + B*a)/(m + 3) + b**2*x**(m + 4)*(A*b + 3*
B*a)/(m + 4)

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Mathematica [A]  time = 0.115422, size = 88, normalized size = 0.92 \[ x^m \left (\frac{a^3 A x}{m+1}+\frac{a^2 x^2 (a B+3 A b)}{m+2}+\frac{b^2 x^4 (3 a B+A b)}{m+4}+\frac{3 a b x^3 (a B+A b)}{m+3}+\frac{b^3 B x^5}{m+5}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x)^3*(A + B*x),x]

[Out]

x^m*((a^3*A*x)/(1 + m) + (a^2*(3*A*b + a*B)*x^2)/(2 + m) + (3*a*b*(A*b + a*B)*x^
3)/(3 + m) + (b^2*(A*b + 3*a*B)*x^4)/(4 + m) + (b^3*B*x^5)/(5 + m))

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Maple [B]  time = 0.007, size = 454, normalized size = 4.7 \[{\frac{{x}^{1+m} \left ( B{b}^{3}{m}^{4}{x}^{4}+A{b}^{3}{m}^{4}{x}^{3}+3\,Ba{b}^{2}{m}^{4}{x}^{3}+10\,B{b}^{3}{m}^{3}{x}^{4}+3\,Aa{b}^{2}{m}^{4}{x}^{2}+11\,A{b}^{3}{m}^{3}{x}^{3}+3\,B{a}^{2}b{m}^{4}{x}^{2}+33\,Ba{b}^{2}{m}^{3}{x}^{3}+35\,B{b}^{3}{m}^{2}{x}^{4}+3\,A{a}^{2}b{m}^{4}x+36\,Aa{b}^{2}{m}^{3}{x}^{2}+41\,A{b}^{3}{m}^{2}{x}^{3}+B{a}^{3}{m}^{4}x+36\,B{a}^{2}b{m}^{3}{x}^{2}+123\,Ba{b}^{2}{m}^{2}{x}^{3}+50\,B{b}^{3}m{x}^{4}+A{a}^{3}{m}^{4}+39\,A{a}^{2}b{m}^{3}x+147\,Aa{b}^{2}{m}^{2}{x}^{2}+61\,A{b}^{3}m{x}^{3}+13\,B{a}^{3}{m}^{3}x+147\,B{a}^{2}b{m}^{2}{x}^{2}+183\,Ba{b}^{2}m{x}^{3}+24\,B{b}^{3}{x}^{4}+14\,A{a}^{3}{m}^{3}+177\,A{a}^{2}b{m}^{2}x+234\,Aa{b}^{2}m{x}^{2}+30\,A{b}^{3}{x}^{3}+59\,B{a}^{3}{m}^{2}x+234\,B{a}^{2}bm{x}^{2}+90\,B{x}^{3}a{b}^{2}+71\,A{a}^{3}{m}^{2}+321\,A{a}^{2}bmx+120\,aA{b}^{2}{x}^{2}+107\,B{a}^{3}mx+120\,B{x}^{2}{a}^{2}b+154\,A{a}^{3}m+180\,{a}^{2}Abx+60\,{a}^{3}Bx+120\,{a}^{3}A \right ) }{ \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

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+234*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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22289, size = 512, normalized size = 5.33 \[ \frac{{\left ({\left (B b^{3} m^{4} + 10 \, B b^{3} m^{3} + 35 \, B b^{3} m^{2} + 50 \, B b^{3} m + 24 \, B b^{3}\right )} x^{5} +{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 90 \, B a b^{2} + 30 \, A b^{3} + 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 41 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 61 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{4} + 3 \,{\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 40 \, B a^{2} b + 40 \, A a b^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} m^{3} + 49 \,{\left (B a^{2} b + A a b^{2}\right )} m^{2} + 78 \,{\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{3} +{\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 60 \, B a^{3} + 180 \, A a^{2} b + 13 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 59 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 107 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{2} +{\left (A a^{3} m^{4} + 14 \, A a^{3} m^{3} + 71 \, A a^{3} m^{2} + 154 \, A a^{3} m + 120 \, A a^{3}\right )} x\right )} x^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*x^m,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 + 1
20*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 = 4.7732, size = 2018, normalized size = 21.02 \[ \text{result too large to display} \]

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 + 1
20) + 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) + 15
4*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 + 274*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 + 274*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))

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GIAC/XCAS [A]  time = 0.220019, size = 909, normalized size = 9.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*x^m,x, algorithm="giac")

[Out]

(B*b^3*m^4*x^5*e^(m*ln(x)) + 3*B*a*b^2*m^4*x^4*e^(m*ln(x)) + A*b^3*m^4*x^4*e^(m*
ln(x)) + 10*B*b^3*m^3*x^5*e^(m*ln(x)) + 3*B*a^2*b*m^4*x^3*e^(m*ln(x)) + 3*A*a*b^
2*m^4*x^3*e^(m*ln(x)) + 33*B*a*b^2*m^3*x^4*e^(m*ln(x)) + 11*A*b^3*m^3*x^4*e^(m*l
n(x)) + 35*B*b^3*m^2*x^5*e^(m*ln(x)) + B*a^3*m^4*x^2*e^(m*ln(x)) + 3*A*a^2*b*m^4
*x^2*e^(m*ln(x)) + 36*B*a^2*b*m^3*x^3*e^(m*ln(x)) + 36*A*a*b^2*m^3*x^3*e^(m*ln(x
)) + 123*B*a*b^2*m^2*x^4*e^(m*ln(x)) + 41*A*b^3*m^2*x^4*e^(m*ln(x)) + 50*B*b^3*m
*x^5*e^(m*ln(x)) + A*a^3*m^4*x*e^(m*ln(x)) + 13*B*a^3*m^3*x^2*e^(m*ln(x)) + 39*A
*a^2*b*m^3*x^2*e^(m*ln(x)) + 147*B*a^2*b*m^2*x^3*e^(m*ln(x)) + 147*A*a*b^2*m^2*x
^3*e^(m*ln(x)) + 183*B*a*b^2*m*x^4*e^(m*ln(x)) + 61*A*b^3*m*x^4*e^(m*ln(x)) + 24
*B*b^3*x^5*e^(m*ln(x)) + 14*A*a^3*m^3*x*e^(m*ln(x)) + 59*B*a^3*m^2*x^2*e^(m*ln(x
)) + 177*A*a^2*b*m^2*x^2*e^(m*ln(x)) + 234*B*a^2*b*m*x^3*e^(m*ln(x)) + 234*A*a*b
^2*m*x^3*e^(m*ln(x)) + 90*B*a*b^2*x^4*e^(m*ln(x)) + 30*A*b^3*x^4*e^(m*ln(x)) + 7
1*A*a^3*m^2*x*e^(m*ln(x)) + 107*B*a^3*m*x^2*e^(m*ln(x)) + 321*A*a^2*b*m*x^2*e^(m
*ln(x)) + 120*B*a^2*b*x^3*e^(m*ln(x)) + 120*A*a*b^2*x^3*e^(m*ln(x)) + 154*A*a^3*
m*x*e^(m*ln(x)) + 60*B*a^3*x^2*e^(m*ln(x)) + 180*A*a^2*b*x^2*e^(m*ln(x)) + 120*A
*a^3*x*e^(m*ln(x)))/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)