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

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)

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Rubi [A]  time = 0.0400606, 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, Rules used = {448} \[ \frac{a^2 A x^{m+1}}{m+1}+\frac{a x^{m+4} (a B+2 A b)}{m+4}+\frac{b x^{m+7} (2 a B+A b)}{m+7}+\frac{b^2 B x^{m+10}}{m+10} \]

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)

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int x^m \left (a+b x^3\right )^2 \left (A+B x^3\right ) \, dx &=\int \left (a^2 A x^m+a (2 A b+a B) x^{3+m}+b (A b+2 a B) x^{6+m}+b^2 B x^{9+m}\right ) \, dx\\ &=\frac{a^2 A x^{1+m}}{1+m}+\frac{a (2 A b+a B) x^{4+m}}{4+m}+\frac{b (A b+2 a B) x^{7+m}}{7+m}+\frac{b^2 B x^{10+m}}{10+m}\\ \end{align*}

Mathematica [A]  time = 0.0513311, size = 66, normalized size = 0.93 \[ x^{m+1} \left (\frac{a^2 A}{m+1}+\frac{b x^6 (2 a B+A b)}{m+7}+\frac{a x^3 (a B+2 A b)}{m+4}+\frac{b^2 B x^9}{m+10}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^(1 + m)*((a^2*A)/(1 + m) + (a*(2*A*b + a*B)*x^3)/(4 + m) + (b*(A*b + 2*a*B)*x^6)/(7 + m) + (b^2*B*x^9)/(10 +
m))

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Maple [B]  time = 0.008, size = 262, normalized size = 3.7 \begin{align*}{\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{b}^{2}{x}^{9}+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}^{2}A \right ) }{ \left ( 10+m \right ) \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \end{align*}

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+3
6*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)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.56837, size = 497, normalized size = 7. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^2*(B*x^3+A),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)

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Sympy [A]  time = 3.78817, size = 1057, normalized size = 14.89 \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**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*l
og(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 + 15
9*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 + 1
59*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))

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Giac [B]  time = 1.12885, size = 448, normalized size = 6.31 \begin{align*} \frac{B b^{2} m^{3} x^{10} x^{m} + 12 \, B b^{2} m^{2} x^{10} x^{m} + 39 \, B b^{2} m x^{10} x^{m} + 2 \, B a b m^{3} x^{7} x^{m} + A b^{2} m^{3} x^{7} x^{m} + 28 \, B b^{2} x^{10} x^{m} + 30 \, B a b m^{2} x^{7} x^{m} + 15 \, A b^{2} m^{2} x^{7} x^{m} + 108 \, B a b m x^{7} x^{m} + 54 \, A b^{2} m x^{7} x^{m} + B a^{2} m^{3} x^{4} x^{m} + 2 \, A a b m^{3} x^{4} x^{m} + 80 \, B a b x^{7} x^{m} + 40 \, A b^{2} x^{7} x^{m} + 18 \, B a^{2} m^{2} x^{4} x^{m} + 36 \, A a b m^{2} x^{4} x^{m} + 87 \, B a^{2} m x^{4} x^{m} + 174 \, A a b m x^{4} x^{m} + A a^{2} m^{3} x x^{m} + 70 \, B a^{2} x^{4} x^{m} + 140 \, A a b x^{4} x^{m} + 21 \, A a^{2} m^{2} x x^{m} + 138 \, A a^{2} m x x^{m} + 280 \, A a^{2} x x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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