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

Optimal. Leaf size=67 \[ \frac{\left (a+b x^2\right )^7 (A b-2 a B)}{14 b^3}-\frac{a \left (a+b x^2\right )^6 (A b-a B)}{12 b^3}+\frac{B \left (a+b x^2\right )^8}{16 b^3} \]

[Out]

-(a*(A*b - a*B)*(a + b*x^2)^6)/(12*b^3) + ((A*b - 2*a*B)*(a + b*x^2)^7)/(14*b^3) + (B*(a + b*x^2)^8)/(16*b^3)

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Rubi [A]  time = 0.143136, antiderivative size = 67, 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 = {446, 76} \[ \frac{\left (a+b x^2\right )^7 (A b-2 a B)}{14 b^3}-\frac{a \left (a+b x^2\right )^6 (A b-a B)}{12 b^3}+\frac{B \left (a+b x^2\right )^8}{16 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

-(a*(A*b - a*B)*(a + b*x^2)^6)/(12*b^3) + ((A*b - 2*a*B)*(a + b*x^2)^7)/(14*b^3) + (B*(a + b*x^2)^8)/(16*b^3)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

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^3 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^5 (A+B x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a (-A b+a B) (a+b x)^5}{b^2}+\frac{(A b-2 a B) (a+b x)^6}{b^2}+\frac{B (a+b x)^7}{b^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{a (A b-a B) \left (a+b x^2\right )^6}{12 b^3}+\frac{(A b-2 a B) \left (a+b x^2\right )^7}{14 b^3}+\frac{B \left (a+b x^2\right )^8}{16 b^3}\\ \end{align*}

Mathematica [A]  time = 0.015552, size = 114, normalized size = 1.7 \[ a^2 b^2 x^{10} (a B+A b)+\frac{5}{8} a^3 b x^8 (a B+2 A b)+\frac{1}{6} a^4 x^6 (a B+5 A b)+\frac{1}{4} a^5 A x^4+\frac{1}{14} b^4 x^{14} (5 a B+A b)+\frac{5}{12} a b^3 x^{12} (2 a B+A b)+\frac{1}{16} b^5 B x^{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*A*x^4)/4 + (a^4*(5*A*b + a*B)*x^6)/6 + (5*a^3*b*(2*A*b + a*B)*x^8)/8 + a^2*b^2*(A*b + a*B)*x^10 + (5*a*b^
3*(A*b + 2*a*B)*x^12)/12 + (b^4*(A*b + 5*a*B)*x^14)/14 + (b^5*B*x^16)/16

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Maple [B]  time = 0.001, size = 124, normalized size = 1.9 \begin{align*}{\frac{{b}^{5}B{x}^{16}}{16}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{14}}{14}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{12}}{12}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{8}}{8}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{6}}{6}}+{\frac{{a}^{5}A{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*b^5*B*x^16+1/14*(A*b^5+5*B*a*b^4)*x^14+1/12*(5*A*a*b^4+10*B*a^2*b^3)*x^12+1/10*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^10+1/8*(10*A*a^3*b^2+5*B*a^4*b)*x^8+1/6*(5*A*a^4*b+B*a^5)*x^6+1/4*a^5*A*x^4

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Maxima [A]  time = 0.981593, size = 159, normalized size = 2.37 \begin{align*} \frac{1}{16} \, B b^{5} x^{16} + \frac{1}{14} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{14} + \frac{5}{12} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} +{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{10} + \frac{1}{4} \, A a^{5} x^{4} + \frac{5}{8} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \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^3*(b*x^2+a)^5*(B*x^2+A),x, algorithm="maxima")

[Out]

1/16*B*b^5*x^16 + 1/14*(5*B*a*b^4 + A*b^5)*x^14 + 5/12*(2*B*a^2*b^3 + A*a*b^4)*x^12 + (B*a^3*b^2 + A*a^2*b^3)*
x^10 + 1/4*A*a^5*x^4 + 5/8*(B*a^4*b + 2*A*a^3*b^2)*x^8 + 1/6*(B*a^5 + 5*A*a^4*b)*x^6

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Fricas [A]  time = 1.30656, size = 294, normalized size = 4.39 \begin{align*} \frac{1}{16} x^{16} b^{5} B + \frac{5}{14} x^{14} b^{4} a B + \frac{1}{14} x^{14} b^{5} A + \frac{5}{6} x^{12} b^{3} a^{2} B + \frac{5}{12} x^{12} b^{4} a A + x^{10} b^{2} a^{3} B + x^{10} b^{3} a^{2} A + \frac{5}{8} x^{8} b a^{4} B + \frac{5}{4} x^{8} b^{2} a^{3} A + \frac{1}{6} x^{6} a^{5} B + \frac{5}{6} x^{6} b a^{4} A + \frac{1}{4} x^{4} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*x^16*b^5*B + 5/14*x^14*b^4*a*B + 1/14*x^14*b^5*A + 5/6*x^12*b^3*a^2*B + 5/12*x^12*b^4*a*A + x^10*b^2*a^3*
B + x^10*b^3*a^2*A + 5/8*x^8*b*a^4*B + 5/4*x^8*b^2*a^3*A + 1/6*x^6*a^5*B + 5/6*x^6*b*a^4*A + 1/4*x^4*a^5*A

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Sympy [B]  time = 0.082481, size = 131, normalized size = 1.96 \begin{align*} \frac{A a^{5} x^{4}}{4} + \frac{B b^{5} x^{16}}{16} + x^{14} \left (\frac{A b^{5}}{14} + \frac{5 B a b^{4}}{14}\right ) + x^{12} \left (\frac{5 A a b^{4}}{12} + \frac{5 B a^{2} b^{3}}{6}\right ) + x^{10} \left (A a^{2} b^{3} + B a^{3} b^{2}\right ) + x^{8} \left (\frac{5 A a^{3} b^{2}}{4} + \frac{5 B a^{4} b}{8}\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**3*(b*x**2+a)**5*(B*x**2+A),x)

[Out]

A*a**5*x**4/4 + B*b**5*x**16/16 + x**14*(A*b**5/14 + 5*B*a*b**4/14) + x**12*(5*A*a*b**4/12 + 5*B*a**2*b**3/6)
+ x**10*(A*a**2*b**3 + B*a**3*b**2) + x**8*(5*A*a**3*b**2/4 + 5*B*a**4*b/8) + x**6*(5*A*a**4*b/6 + B*a**5/6)

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Giac [A]  time = 1.15456, size = 166, normalized size = 2.48 \begin{align*} \frac{1}{16} \, B b^{5} x^{16} + \frac{5}{14} \, B a b^{4} x^{14} + \frac{1}{14} \, A b^{5} x^{14} + \frac{5}{6} \, B a^{2} b^{3} x^{12} + \frac{5}{12} \, A a b^{4} x^{12} + B a^{3} b^{2} x^{10} + A a^{2} b^{3} x^{10} + \frac{5}{8} \, B a^{4} b x^{8} + \frac{5}{4} \, A a^{3} b^{2} x^{8} + \frac{1}{6} \, B a^{5} x^{6} + \frac{5}{6} \, A a^{4} b x^{6} + \frac{1}{4} \, A a^{5} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*B*b^5*x^16 + 5/14*B*a*b^4*x^14 + 1/14*A*b^5*x^14 + 5/6*B*a^2*b^3*x^12 + 5/12*A*a*b^4*x^12 + B*a^3*b^2*x^1
0 + A*a^2*b^3*x^10 + 5/8*B*a^4*b*x^8 + 5/4*A*a^3*b^2*x^8 + 1/6*B*a^5*x^6 + 5/6*A*a^4*b*x^6 + 1/4*A*a^5*x^4