\(\int (a+b x^2)^5 (A+B x^2) \, dx\) [32]

Optimal result

Integrand size = 17, antiderivative size = 109 \[ \int \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=a^5 A x+\frac {1}{3} a^4 (5 A b+a B) x^3+a^3 b (2 A b+a B) x^5+\frac {10}{7} a^2 b^2 (A b+a B) x^7+\frac {5}{9} a b^3 (A b+2 a B) x^9+\frac {1}{11} b^4 (A b+5 a B) x^{11}+\frac {1}{13} b^5 B x^{13} \]

[Out]

a^5*A*x+1/3*a^4*(5*A*b+B*a)*x^3+a^3*b*(2*A*b+B*a)*x^5+10/7*a^2*b^2*(A*b+B*a)*x^7+5/9*a*b^3*(A*b+2*B*a)*x^9+1/1
1*b^4*(A*b+5*B*a)*x^11+1/13*b^5*B*x^13

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {380} \[ \int \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=a^5 A x+\frac {1}{3} a^4 x^3 (a B+5 A b)+a^3 b x^5 (a B+2 A b)+\frac {10}{7} a^2 b^2 x^7 (a B+A b)+\frac {1}{11} b^4 x^{11} (5 a B+A b)+\frac {5}{9} a b^3 x^9 (2 a B+A b)+\frac {1}{13} b^5 B x^{13} \]

[In]

Int[(a + b*x^2)^5*(A + B*x^2),x]

[Out]

a^5*A*x + (a^4*(5*A*b + a*B)*x^3)/3 + a^3*b*(2*A*b + a*B)*x^5 + (10*a^2*b^2*(A*b + a*B)*x^7)/7 + (5*a*b^3*(A*b
+ 2*a*B)*x^9)/9 + (b^4*(A*b + 5*a*B)*x^11)/11 + (b^5*B*x^13)/13

Rule 380

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 A+a^4 (5 A b+a B) x^2+5 a^3 b (2 A b+a B) x^4+10 a^2 b^2 (A b+a B) x^6+5 a b^3 (A b+2 a B) x^8+b^4 (A b+5 a B) x^{10}+b^5 B x^{12}\right ) \, dx \\ & = a^5 A x+\frac {1}{3} a^4 (5 A b+a B) x^3+a^3 b (2 A b+a B) x^5+\frac {10}{7} a^2 b^2 (A b+a B) x^7+\frac {5}{9} a b^3 (A b+2 a B) x^9+\frac {1}{11} b^4 (A b+5 a B) x^{11}+\frac {1}{13} b^5 B x^{13} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=a^5 A x+\frac {1}{3} a^4 (5 A b+a B) x^3+a^3 b (2 A b+a B) x^5+\frac {10}{7} a^2 b^2 (A b+a B) x^7+\frac {5}{9} a b^3 (A b+2 a B) x^9+\frac {1}{11} b^4 (A b+5 a B) x^{11}+\frac {1}{13} b^5 B x^{13} \]

[In]

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

[Out]

a^5*A*x + (a^4*(5*A*b + a*B)*x^3)/3 + a^3*b*(2*A*b + a*B)*x^5 + (10*a^2*b^2*(A*b + a*B)*x^7)/7 + (5*a*b^3*(A*b
+ 2*a*B)*x^9)/9 + (b^4*(A*b + 5*a*B)*x^11)/11 + (b^5*B*x^13)/13

Maple [A] (verified)

Time = 2.60 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.07

method	result	size
norman	\(\frac {b^{5} B \,x^{13}}{13}+\left (\frac {1}{11} b^{5} A +\frac {5}{11} a \,b^{4} B \right ) x^{11}+\left (\frac {5}{9} a \,b^{4} A +\frac {10}{9} a^{2} b^{3} B \right ) x^{9}+\left (\frac {10}{7} a^{2} b^{3} A +\frac {10}{7} a^{3} b^{2} B \right ) x^{7}+\left (2 a^{3} b^{2} A +a^{4} b B \right ) x^{5}+\left (\frac {5}{3} a^{4} b A +\frac {1}{3} a^{5} B \right ) x^{3}+a^{5} A x\)	\(117\)
default	\(\frac {b^{5} B \,x^{13}}{13}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{11}}{11}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{9}}{9}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{7}}{7}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{5}}{5}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{3}}{3}+a^{5} A x\)	\(121\)
gosper	\(\frac {1}{13} b^{5} B \,x^{13}+\frac {1}{11} x^{11} b^{5} A +\frac {5}{11} x^{11} a \,b^{4} B +\frac {5}{9} x^{9} a \,b^{4} A +\frac {10}{9} x^{9} a^{2} b^{3} B +\frac {10}{7} x^{7} a^{2} b^{3} A +\frac {10}{7} x^{7} a^{3} b^{2} B +2 A \,a^{3} b^{2} x^{5}+B \,a^{4} b \,x^{5}+\frac {5}{3} x^{3} a^{4} b A +\frac {1}{3} x^{3} a^{5} B +a^{5} A x\)	\(122\)
risch	\(\frac {1}{13} b^{5} B \,x^{13}+\frac {1}{11} x^{11} b^{5} A +\frac {5}{11} x^{11} a \,b^{4} B +\frac {5}{9} x^{9} a \,b^{4} A +\frac {10}{9} x^{9} a^{2} b^{3} B +\frac {10}{7} x^{7} a^{2} b^{3} A +\frac {10}{7} x^{7} a^{3} b^{2} B +2 A \,a^{3} b^{2} x^{5}+B \,a^{4} b \,x^{5}+\frac {5}{3} x^{3} a^{4} b A +\frac {1}{3} x^{3} a^{5} B +a^{5} A x\)	\(122\)
parallelrisch	\(\frac {1}{13} b^{5} B \,x^{13}+\frac {1}{11} x^{11} b^{5} A +\frac {5}{11} x^{11} a \,b^{4} B +\frac {5}{9} x^{9} a \,b^{4} A +\frac {10}{9} x^{9} a^{2} b^{3} B +\frac {10}{7} x^{7} a^{2} b^{3} A +\frac {10}{7} x^{7} a^{3} b^{2} B +2 A \,a^{3} b^{2} x^{5}+B \,a^{4} b \,x^{5}+\frac {5}{3} x^{3} a^{4} b A +\frac {1}{3} x^{3} a^{5} B +a^{5} A x\)	\(122\)

[In]

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

[Out]

1/13*b^5*B*x^13+(1/11*b^5*A+5/11*a*b^4*B)*x^11+(5/9*a*b^4*A+10/9*a^2*b^3*B)*x^9+(10/7*a^2*b^3*A+10/7*a^3*b^2*B
)*x^7+(2*A*a^3*b^2+B*a^4*b)*x^5+(5/3*a^4*b*A+1/3*a^5*B)*x^3+a^5*A*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{13} \, B b^{5} x^{13} + \frac {1}{11} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac {5}{9} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac {10}{7} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{7} + A a^{5} x + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + \frac {1}{3} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \]

[In]

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

[Out]

1/13*B*b^5*x^13 + 1/11*(5*B*a*b^4 + A*b^5)*x^11 + 5/9*(2*B*a^2*b^3 + A*a*b^4)*x^9 + 10/7*(B*a^3*b^2 + A*a^2*b^
3)*x^7 + A*a^5*x + (B*a^4*b + 2*A*a^3*b^2)*x^5 + 1/3*(B*a^5 + 5*A*a^4*b)*x^3

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18 \[ \int \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=A a^{5} x + \frac {B b^{5} x^{13}}{13} + x^{11} \left (\frac {A b^{5}}{11} + \frac {5 B a b^{4}}{11}\right ) + x^{9} \cdot \left (\frac {5 A a b^{4}}{9} + \frac {10 B a^{2} b^{3}}{9}\right ) + x^{7} \cdot \left (\frac {10 A a^{2} b^{3}}{7} + \frac {10 B a^{3} b^{2}}{7}\right ) + x^{5} \cdot \left (2 A a^{3} b^{2} + B a^{4} b\right ) + x^{3} \cdot \left (\frac {5 A a^{4} b}{3} + \frac {B a^{5}}{3}\right ) \]

[In]

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

[Out]

A*a**5*x + B*b**5*x**13/13 + x**11*(A*b**5/11 + 5*B*a*b**4/11) + x**9*(5*A*a*b**4/9 + 10*B*a**2*b**3/9) + x**7
*(10*A*a**2*b**3/7 + 10*B*a**3*b**2/7) + x**5*(2*A*a**3*b**2 + B*a**4*b) + x**3*(5*A*a**4*b/3 + B*a**5/3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{13} \, B b^{5} x^{13} + \frac {1}{11} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac {5}{9} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac {10}{7} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{7} + A a^{5} x + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + \frac {1}{3} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \]

[In]

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

[Out]

1/13*B*b^5*x^13 + 1/11*(5*B*a*b^4 + A*b^5)*x^11 + 5/9*(2*B*a^2*b^3 + A*a*b^4)*x^9 + 10/7*(B*a^3*b^2 + A*a^2*b^
3)*x^7 + A*a^5*x + (B*a^4*b + 2*A*a^3*b^2)*x^5 + 1/3*(B*a^5 + 5*A*a^4*b)*x^3

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{13} \, B b^{5} x^{13} + \frac {5}{11} \, B a b^{4} x^{11} + \frac {1}{11} \, A b^{5} x^{11} + \frac {10}{9} \, B a^{2} b^{3} x^{9} + \frac {5}{9} \, A a b^{4} x^{9} + \frac {10}{7} \, B a^{3} b^{2} x^{7} + \frac {10}{7} \, A a^{2} b^{3} x^{7} + B a^{4} b x^{5} + 2 \, A a^{3} b^{2} x^{5} + \frac {1}{3} \, B a^{5} x^{3} + \frac {5}{3} \, A a^{4} b x^{3} + A a^{5} x \]

[In]

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

[Out]

1/13*B*b^5*x^13 + 5/11*B*a*b^4*x^11 + 1/11*A*b^5*x^11 + 10/9*B*a^2*b^3*x^9 + 5/9*A*a*b^4*x^9 + 10/7*B*a^3*b^2*
x^7 + 10/7*A*a^2*b^3*x^7 + B*a^4*b*x^5 + 2*A*a^3*b^2*x^5 + 1/3*B*a^5*x^3 + 5/3*A*a^4*b*x^3 + A*a^5*x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=x^3\,\left (\frac {B\,a^5}{3}+\frac {5\,A\,b\,a^4}{3}\right )+x^{11}\,\left (\frac {A\,b^5}{11}+\frac {5\,B\,a\,b^4}{11}\right )+\frac {B\,b^5\,x^{13}}{13}+A\,a^5\,x+\frac {10\,a^2\,b^2\,x^7\,\left (A\,b+B\,a\right )}{7}+a^3\,b\,x^5\,\left (2\,A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^9\,\left (A\,b+2\,B\,a\right )}{9} \]

[In]

int((A + B*x^2)*(a + b*x^2)^5,x)

[Out]

x^3*((B*a^5)/3 + (5*A*a^4*b)/3) + x^11*((A*b^5)/11 + (5*B*a*b^4)/11) + (B*b^5*x^13)/13 + A*a^5*x + (10*a^2*b^2
*x^7*(A*b + B*a))/7 + a^3*b*x^5*(2*A*b + B*a) + (5*a*b^3*x^9*(A*b + 2*B*a))/9