3.125 \(\int x^m \left (a+b x^3\right )^2 \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=71 \[ \frac{a^2 A x^{m+1}}{m+1}+\frac{a x^{m+4} (a B+2 A b)}{m+4}+\frac{b x^{m+7} (2 a B+A b)}{m+7}+\frac{b^2 B x^{m+10}}{m+10} \]

[Out]

(a^2*A*x^(1 + m))/(1 + m) + (a*(2*A*b + a*B)*x^(4 + m))/(4 + m) + (b*(A*b + 2*a*
B)*x^(7 + m))/(7 + m) + (b^2*B*x^(10 + m))/(10 + m)

_______________________________________________________________________________________

Rubi [A]  time = 0.124598, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{a^2 A x^{m+1}}{m+1}+\frac{a x^{m+4} (a B+2 A b)}{m+4}+\frac{b x^{m+7} (2 a B+A b)}{m+7}+\frac{b^2 B x^{m+10}}{m+10} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^3)^2*(A + B*x^3),x]

[Out]

(a^2*A*x^(1 + m))/(1 + m) + (a*(2*A*b + a*B)*x^(4 + m))/(4 + m) + (b*(A*b + 2*a*
B)*x^(7 + m))/(7 + m) + (b^2*B*x^(10 + m))/(10 + m)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.2511, size = 63, normalized size = 0.89 \[ \frac{A a^{2} x^{m + 1}}{m + 1} + \frac{B b^{2} x^{m + 10}}{m + 10} + \frac{a x^{m + 4} \left (2 A b + B a\right )}{m + 4} + \frac{b x^{m + 7} \left (A b + 2 B a\right )}{m + 7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**3+a)**2*(B*x**3+A),x)

[Out]

A*a**2*x**(m + 1)/(m + 1) + B*b**2*x**(m + 10)/(m + 10) + a*x**(m + 4)*(2*A*b +
B*a)/(m + 4) + b*x**(m + 7)*(A*b + 2*B*a)/(m + 7)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0870738, size = 65, normalized size = 0.92 \[ x^m \left (\frac{a^2 A x}{m+1}+\frac{b x^7 (2 a B+A b)}{m+7}+\frac{a x^4 (a B+2 A b)}{m+4}+\frac{b^2 B x^{10}}{m+10}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^3)^2*(A + B*x^3),x]

[Out]

x^m*((a^2*A*x)/(1 + m) + (a*(2*A*b + a*B)*x^4)/(4 + m) + (b*(A*b + 2*a*B)*x^7)/(
7 + m) + (b^2*B*x^10)/(10 + m))

_______________________________________________________________________________________

Maple [B]  time = 0.009, size = 262, normalized size = 3.7 \[{\frac{{x}^{1+m} \left ( B{b}^{2}{m}^{3}{x}^{9}+12\,B{b}^{2}{m}^{2}{x}^{9}+39\,B{b}^{2}m{x}^{9}+A{b}^{2}{m}^{3}{x}^{6}+2\,Bab{m}^{3}{x}^{6}+28\,B{x}^{9}{b}^{2}+15\,A{b}^{2}{m}^{2}{x}^{6}+30\,Bab{m}^{2}{x}^{6}+54\,A{b}^{2}m{x}^{6}+108\,Babm{x}^{6}+2\,Aab{m}^{3}{x}^{3}+40\,A{b}^{2}{x}^{6}+B{a}^{2}{m}^{3}{x}^{3}+80\,B{x}^{6}ab+36\,Aab{m}^{2}{x}^{3}+18\,B{a}^{2}{m}^{2}{x}^{3}+174\,Aabm{x}^{3}+87\,B{a}^{2}m{x}^{3}+A{a}^{2}{m}^{3}+140\,aAb{x}^{3}+70\,B{x}^{3}{a}^{2}+21\,A{a}^{2}{m}^{2}+138\,A{a}^{2}m+280\,A{a}^{2} \right ) }{ \left ( 10+m \right ) \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^3+a)^2*(B*x^3+A),x)

[Out]

x^(1+m)*(B*b^2*m^3*x^9+12*B*b^2*m^2*x^9+39*B*b^2*m*x^9+A*b^2*m^3*x^6+2*B*a*b*m^3
*x^6+28*B*b^2*x^9+15*A*b^2*m^2*x^6+30*B*a*b*m^2*x^6+54*A*b^2*m*x^6+108*B*a*b*m*x
^6+2*A*a*b*m^3*x^3+40*A*b^2*x^6+B*a^2*m^3*x^3+80*B*a*b*x^6+36*A*a*b*m^2*x^3+18*B
*a^2*m^2*x^3+174*A*a*b*m*x^3+87*B*a^2*m*x^3+A*a^2*m^3+140*A*a*b*x^3+70*B*a^2*x^3
+21*A*a^2*m^2+138*A*a^2*m+280*A*a^2)/(10+m)/(7+m)/(4+m)/(1+m)

_______________________________________________________________________________________

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^3 + A)*(b*x^3 + a)^2*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.242579, size = 290, normalized size = 4.08 \[ \frac{{\left ({\left (B b^{2} m^{3} + 12 \, B b^{2} m^{2} + 39 \, B b^{2} m + 28 \, B b^{2}\right )} x^{10} +{\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 80 \, B a b + 40 \, A b^{2} + 15 \,{\left (2 \, B a b + A b^{2}\right )} m^{2} + 54 \,{\left (2 \, B a b + A b^{2}\right )} m\right )} x^{7} +{\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 70 \, B a^{2} + 140 \, A a b + 18 \,{\left (B a^{2} + 2 \, A a b\right )} m^{2} + 87 \,{\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{4} +{\left (A a^{2} m^{3} + 21 \, A a^{2} m^{2} + 138 \, A a^{2} m + 280 \, A a^{2}\right )} x\right )} x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^2*x^m,x, algorithm="fricas")

[Out]

((B*b^2*m^3 + 12*B*b^2*m^2 + 39*B*b^2*m + 28*B*b^2)*x^10 + ((2*B*a*b + A*b^2)*m^
3 + 80*B*a*b + 40*A*b^2 + 15*(2*B*a*b + A*b^2)*m^2 + 54*(2*B*a*b + A*b^2)*m)*x^7
+ ((B*a^2 + 2*A*a*b)*m^3 + 70*B*a^2 + 140*A*a*b + 18*(B*a^2 + 2*A*a*b)*m^2 + 87
*(B*a^2 + 2*A*a*b)*m)*x^4 + (A*a^2*m^3 + 21*A*a^2*m^2 + 138*A*a^2*m + 280*A*a^2)
*x)*x^m/(m^4 + 22*m^3 + 159*m^2 + 418*m + 280)

_______________________________________________________________________________________

Sympy [A]  time = 10.5193, size = 1057, normalized size = 14.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**3+a)**2*(B*x**3+A),x)

[Out]

Piecewise((-A*a**2/(9*x**9) - A*a*b/(3*x**6) - A*b**2/(3*x**3) - B*a**2/(6*x**6)
- 2*B*a*b/(3*x**3) + B*b**2*log(x), Eq(m, -10)), (-A*a**2/(6*x**6) - 2*A*a*b/(3
*x**3) + A*b**2*log(x) - B*a**2/(3*x**3) + 2*B*a*b*log(x) + B*b**2*x**3/3, Eq(m,
-7)), (-A*a**2/(3*x**3) + 2*A*a*b*log(x) + A*b**2*x**3/3 + B*a**2*log(x) + 2*B*
a*b*x**3/3 + B*b**2*x**6/6, Eq(m, -4)), (A*a**2*log(x) + 2*A*a*b*x**3/3 + A*b**2
*x**6/6 + B*a**2*x**3/3 + B*a*b*x**6/3 + B*b**2*x**9/9, Eq(m, -1)), (A*a**2*m**3
*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 21*A*a**2*m**2*x*x**m/(m**4
+ 22*m**3 + 159*m**2 + 418*m + 280) + 138*A*a**2*m*x*x**m/(m**4 + 22*m**3 + 159*
m**2 + 418*m + 280) + 280*A*a**2*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280
) + 2*A*a*b*m**3*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 36*A*a*b*
m**2*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 174*A*a*b*m*x**4*x**m
/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 140*A*a*b*x**4*x**m/(m**4 + 22*m**3
+ 159*m**2 + 418*m + 280) + A*b**2*m**3*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 +
418*m + 280) + 15*A*b**2*m**2*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280
) + 54*A*b**2*m*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 40*A*b**2*
x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + B*a**2*m**3*x**4*x**m/(m**
4 + 22*m**3 + 159*m**2 + 418*m + 280) + 18*B*a**2*m**2*x**4*x**m/(m**4 + 22*m**3
+ 159*m**2 + 418*m + 280) + 87*B*a**2*m*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 +
418*m + 280) + 70*B*a**2*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 2
*B*a*b*m**3*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 30*B*a*b*m**2*
x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 108*B*a*b*m*x**7*x**m/(m**
4 + 22*m**3 + 159*m**2 + 418*m + 280) + 80*B*a*b*x**7*x**m/(m**4 + 22*m**3 + 159
*m**2 + 418*m + 280) + B*b**2*m**3*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m
+ 280) + 12*B*b**2*m**2*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) +
39*B*b**2*m*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 28*B*b**2*x**
10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221644, size = 513, normalized size = 7.23 \[ \frac{B b^{2} m^{3} x^{10} e^{\left (m{\rm ln}\left (x\right )\right )} + 12 \, B b^{2} m^{2} x^{10} e^{\left (m{\rm ln}\left (x\right )\right )} + 39 \, B b^{2} m x^{10} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, B a b m^{3} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + A b^{2} m^{3} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 28 \, B b^{2} x^{10} e^{\left (m{\rm ln}\left (x\right )\right )} + 30 \, B a b m^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 15 \, A b^{2} m^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 108 \, B a b m x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 54 \, A b^{2} m x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + B a^{2} m^{3} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, A a b m^{3} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 80 \, B a b x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 40 \, A b^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 18 \, B a^{2} m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 36 \, A a b m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 87 \, B a^{2} m x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 174 \, A a b m x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + A a^{2} m^{3} x e^{\left (m{\rm ln}\left (x\right )\right )} + 70 \, B a^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 140 \, A a b x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 21 \, A a^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 138 \, A a^{2} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 280 \, A a^{2} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^2*x^m,x, algorithm="giac")

[Out]

(B*b^2*m^3*x^10*e^(m*ln(x)) + 12*B*b^2*m^2*x^10*e^(m*ln(x)) + 39*B*b^2*m*x^10*e^
(m*ln(x)) + 2*B*a*b*m^3*x^7*e^(m*ln(x)) + A*b^2*m^3*x^7*e^(m*ln(x)) + 28*B*b^2*x
^10*e^(m*ln(x)) + 30*B*a*b*m^2*x^7*e^(m*ln(x)) + 15*A*b^2*m^2*x^7*e^(m*ln(x)) +
108*B*a*b*m*x^7*e^(m*ln(x)) + 54*A*b^2*m*x^7*e^(m*ln(x)) + B*a^2*m^3*x^4*e^(m*ln
(x)) + 2*A*a*b*m^3*x^4*e^(m*ln(x)) + 80*B*a*b*x^7*e^(m*ln(x)) + 40*A*b^2*x^7*e^(
m*ln(x)) + 18*B*a^2*m^2*x^4*e^(m*ln(x)) + 36*A*a*b*m^2*x^4*e^(m*ln(x)) + 87*B*a^
2*m*x^4*e^(m*ln(x)) + 174*A*a*b*m*x^4*e^(m*ln(x)) + A*a^2*m^3*x*e^(m*ln(x)) + 70
*B*a^2*x^4*e^(m*ln(x)) + 140*A*a*b*x^4*e^(m*ln(x)) + 21*A*a^2*m^2*x*e^(m*ln(x))
+ 138*A*a^2*m*x*e^(m*ln(x)) + 280*A*a^2*x*e^(m*ln(x)))/(m^4 + 22*m^3 + 159*m^2 +
418*m + 280)