3.30 \(\int x^2 (a+b x^3)^5 (A+B x^3) \, dx\)

Optimal. Leaf size=42 \[ \frac{\left (a+b x^3\right )^6 (A b-a B)}{18 b^2}+\frac{B \left (a+b x^3\right )^7}{21 b^2} \]

[Out]

((A*b - a*B)*(a + b*x^3)^6)/(18*b^2) + (B*(a + b*x^3)^7)/(21*b^2)

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Rubi [A]  time = 0.068837, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 43} \[ \frac{\left (a+b x^3\right )^6 (A b-a B)}{18 b^2}+\frac{B \left (a+b x^3\right )^7}{21 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

((A*b - a*B)*(a + b*x^3)^6)/(18*b^2) + (B*(a + b*x^3)^7)/(21*b^2)

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^2 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int (a+b x)^5 (A+B x) \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{(A b-a B) (a+b x)^5}{b}+\frac{B (a+b x)^6}{b}\right ) \, dx,x,x^3\right )\\ &=\frac{(A b-a B) \left (a+b x^3\right )^6}{18 b^2}+\frac{B \left (a+b x^3\right )^7}{21 b^2}\\ \end{align*}

Mathematica [B]  time = 0.0250203, size = 107, normalized size = 2.55 \[ \frac{1}{126} x^3 \left (105 a^2 b^2 x^9 (a B+A b)+70 a^3 b x^6 (a B+2 A b)+21 a^4 x^3 (a B+5 A b)+42 a^5 A+7 b^4 x^{15} (5 a B+A b)+42 a b^3 x^{12} (2 a B+A b)+6 b^5 B x^{18}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(x^3*(42*a^5*A + 21*a^4*(5*A*b + a*B)*x^3 + 70*a^3*b*(2*A*b + a*B)*x^6 + 105*a^2*b^2*(A*b + a*B)*x^9 + 42*a*b^
3*(A*b + 2*a*B)*x^12 + 7*b^4*(A*b + 5*a*B)*x^15 + 6*b^5*B*x^18))/126

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Maple [B]  time = 0.001, size = 124, normalized size = 3. \begin{align*}{\frac{{b}^{5}B{x}^{21}}{21}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{18}}{18}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{15}}{15}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{12}}{12}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{9}}{9}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{6}}{6}}+{\frac{{a}^{5}A{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^3+a)^5*(B*x^3+A),x)

[Out]

1/21*b^5*B*x^21+1/18*(A*b^5+5*B*a*b^4)*x^18+1/15*(5*A*a*b^4+10*B*a^2*b^3)*x^15+1/12*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^12+1/9*(10*A*a^3*b^2+5*B*a^4*b)*x^9+1/6*(5*A*a^4*b+B*a^5)*x^6+1/3*a^5*A*x^3

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Maxima [B]  time = 1.32201, size = 161, normalized size = 3.83 \begin{align*} \frac{1}{21} \, B b^{5} x^{21} + \frac{1}{18} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{18} + \frac{1}{3} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{15} + \frac{5}{6} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{12} + \frac{5}{9} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{9} + \frac{1}{3} \, A a^{5} x^{3} + \frac{1}{6} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^5*(B*x^3+A),x, algorithm="maxima")

[Out]

1/21*B*b^5*x^21 + 1/18*(5*B*a*b^4 + A*b^5)*x^18 + 1/3*(2*B*a^2*b^3 + A*a*b^4)*x^15 + 5/6*(B*a^3*b^2 + A*a^2*b^
3)*x^12 + 5/9*(B*a^4*b + 2*A*a^3*b^2)*x^9 + 1/3*A*a^5*x^3 + 1/6*(B*a^5 + 5*A*a^4*b)*x^6

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Fricas [B]  time = 1.2861, size = 305, normalized size = 7.26 \begin{align*} \frac{1}{21} x^{21} b^{5} B + \frac{5}{18} x^{18} b^{4} a B + \frac{1}{18} x^{18} b^{5} A + \frac{2}{3} x^{15} b^{3} a^{2} B + \frac{1}{3} x^{15} b^{4} a A + \frac{5}{6} x^{12} b^{2} a^{3} B + \frac{5}{6} x^{12} b^{3} a^{2} A + \frac{5}{9} x^{9} b a^{4} B + \frac{10}{9} x^{9} b^{2} a^{3} A + \frac{1}{6} x^{6} a^{5} B + \frac{5}{6} x^{6} b a^{4} A + \frac{1}{3} x^{3} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^5*(B*x^3+A),x, algorithm="fricas")

[Out]

1/21*x^21*b^5*B + 5/18*x^18*b^4*a*B + 1/18*x^18*b^5*A + 2/3*x^15*b^3*a^2*B + 1/3*x^15*b^4*a*A + 5/6*x^12*b^2*a
^3*B + 5/6*x^12*b^3*a^2*A + 5/9*x^9*b*a^4*B + 10/9*x^9*b^2*a^3*A + 1/6*x^6*a^5*B + 5/6*x^6*b*a^4*A + 1/3*x^3*a
^5*A

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Sympy [B]  time = 0.084238, size = 136, normalized size = 3.24 \begin{align*} \frac{A a^{5} x^{3}}{3} + \frac{B b^{5} x^{21}}{21} + x^{18} \left (\frac{A b^{5}}{18} + \frac{5 B a b^{4}}{18}\right ) + x^{15} \left (\frac{A a b^{4}}{3} + \frac{2 B a^{2} b^{3}}{3}\right ) + x^{12} \left (\frac{5 A a^{2} b^{3}}{6} + \frac{5 B a^{3} b^{2}}{6}\right ) + x^{9} \left (\frac{10 A a^{3} b^{2}}{9} + \frac{5 B a^{4} b}{9}\right ) + x^{6} \left (\frac{5 A a^{4} b}{6} + \frac{B a^{5}}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**3+a)**5*(B*x**3+A),x)

[Out]

A*a**5*x**3/3 + B*b**5*x**21/21 + x**18*(A*b**5/18 + 5*B*a*b**4/18) + x**15*(A*a*b**4/3 + 2*B*a**2*b**3/3) + x
**12*(5*A*a**2*b**3/6 + 5*B*a**3*b**2/6) + x**9*(10*A*a**3*b**2/9 + 5*B*a**4*b/9) + x**6*(5*A*a**4*b/6 + B*a**
5/6)

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Giac [B]  time = 1.17044, size = 169, normalized size = 4.02 \begin{align*} \frac{1}{21} \, B b^{5} x^{21} + \frac{5}{18} \, B a b^{4} x^{18} + \frac{1}{18} \, A b^{5} x^{18} + \frac{2}{3} \, B a^{2} b^{3} x^{15} + \frac{1}{3} \, A a b^{4} x^{15} + \frac{5}{6} \, B a^{3} b^{2} x^{12} + \frac{5}{6} \, A a^{2} b^{3} x^{12} + \frac{5}{9} \, B a^{4} b x^{9} + \frac{10}{9} \, A a^{3} b^{2} x^{9} + \frac{1}{6} \, B a^{5} x^{6} + \frac{5}{6} \, A a^{4} b x^{6} + \frac{1}{3} \, A a^{5} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^5*(B*x^3+A),x, algorithm="giac")

[Out]

1/21*B*b^5*x^21 + 5/18*B*a*b^4*x^18 + 1/18*A*b^5*x^18 + 2/3*B*a^2*b^3*x^15 + 1/3*A*a*b^4*x^15 + 5/6*B*a^3*b^2*
x^12 + 5/6*A*a^2*b^3*x^12 + 5/9*B*a^4*b*x^9 + 10/9*A*a^3*b^2*x^9 + 1/6*B*a^5*x^6 + 5/6*A*a^4*b*x^6 + 1/3*A*a^5
*x^3