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

Optimal. Leaf size=42 \[ \frac{\left (a+b x^2\right )^6 (A b-a B)}{12 b^2}+\frac{B \left (a+b x^2\right )^7}{14 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0661224, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {444, 43} \[ \frac{\left (a+b x^2\right )^6 (A b-a B)}{12 b^2}+\frac{B \left (a+b x^2\right )^7}{14 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

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

Mathematica [B]  time = 0.0229003, size = 107, normalized size = 2.55 \[ \frac{1}{84} x^2 \left (105 a^2 b^2 x^6 (a B+A b)+70 a^3 b x^4 (a B+2 A b)+21 a^4 x^2 (a B+5 A b)+42 a^5 A+7 b^4 x^{10} (5 a B+A b)+42 a b^3 x^8 (2 a B+A b)+6 b^5 B x^{12}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(42*a^5*A + 21*a^4*(5*A*b + a*B)*x^2 + 70*a^3*b*(2*A*b + a*B)*x^4 + 105*a^2*b^2*(A*b + a*B)*x^6 + 42*a*b^
3*(A*b + 2*a*B)*x^8 + 7*b^4*(A*b + 5*a*B)*x^10 + 6*b^5*B*x^12))/84

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Maple [B]  time = 0.002, size = 124, normalized size = 3. \begin{align*}{\frac{{b}^{5}B{x}^{14}}{14}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{12}}{12}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{8}}{8}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{4}}{4}}+{\frac{{a}^{5}A{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.980255, size = 161, normalized size = 3.83 \begin{align*} \frac{1}{14} \, B b^{5} x^{14} + \frac{1}{12} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{12} + \frac{1}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac{5}{4} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{8} + \frac{1}{2} \, A a^{5} x^{2} + \frac{5}{6} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + \frac{1}{4} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.26913, size = 296, normalized size = 7.05 \begin{align*} \frac{1}{14} x^{14} b^{5} B + \frac{5}{12} x^{12} b^{4} a B + \frac{1}{12} x^{12} b^{5} A + x^{10} b^{3} a^{2} B + \frac{1}{2} x^{10} b^{4} a A + \frac{5}{4} x^{8} b^{2} a^{3} B + \frac{5}{4} x^{8} b^{3} a^{2} A + \frac{5}{6} x^{6} b a^{4} B + \frac{5}{3} x^{6} b^{2} a^{3} A + \frac{1}{4} x^{4} a^{5} B + \frac{5}{4} x^{4} b a^{4} A + \frac{1}{2} x^{2} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.081339, size = 133, normalized size = 3.17 \begin{align*} \frac{A a^{5} x^{2}}{2} + \frac{B b^{5} x^{14}}{14} + x^{12} \left (\frac{A b^{5}}{12} + \frac{5 B a b^{4}}{12}\right ) + x^{10} \left (\frac{A a b^{4}}{2} + B a^{2} b^{3}\right ) + x^{8} \left (\frac{5 A a^{2} b^{3}}{4} + \frac{5 B a^{3} b^{2}}{4}\right ) + x^{6} \left (\frac{5 A a^{3} b^{2}}{3} + \frac{5 B a^{4} b}{6}\right ) + x^{4} \left (\frac{5 A a^{4} b}{4} + \frac{B a^{5}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.19393, size = 167, normalized size = 3.98 \begin{align*} \frac{1}{14} \, B b^{5} x^{14} + \frac{5}{12} \, B a b^{4} x^{12} + \frac{1}{12} \, A b^{5} x^{12} + B a^{2} b^{3} x^{10} + \frac{1}{2} \, A a b^{4} x^{10} + \frac{5}{4} \, B a^{3} b^{2} x^{8} + \frac{5}{4} \, A a^{2} b^{3} x^{8} + \frac{5}{6} \, B a^{4} b x^{6} + \frac{5}{3} \, A a^{3} b^{2} x^{6} + \frac{1}{4} \, B a^{5} x^{4} + \frac{5}{4} \, A a^{4} b x^{4} + \frac{1}{2} \, A a^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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