3.1.55 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{23}} \, dx\) [55]

Optimal. Leaf size=117 \[ -\frac {a^5 A}{22 x^{22}}-\frac {a^4 (5 A b+a B)}{20 x^{20}}-\frac {5 a^3 b (2 A b+a B)}{18 x^{18}}-\frac {5 a^2 b^2 (A b+a B)}{8 x^{16}}-\frac {5 a b^3 (A b+2 a B)}{14 x^{14}}-\frac {b^4 (A b+5 a B)}{12 x^{12}}-\frac {b^5 B}{10 x^{10}} \]

[Out]

-1/22*a^5*A/x^22-1/20*a^4*(5*A*b+B*a)/x^20-5/18*a^3*b*(2*A*b+B*a)/x^18-5/8*a^2*b^2*(A*b+B*a)/x^16-5/14*a*b^3*(
A*b+2*B*a)/x^14-1/12*b^4*(A*b+5*B*a)/x^12-1/10*b^5*B/x^10

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77} \begin {gather*} -\frac {a^5 A}{22 x^{22}}-\frac {a^4 (a B+5 A b)}{20 x^{20}}-\frac {5 a^3 b (a B+2 A b)}{18 x^{18}}-\frac {5 a^2 b^2 (a B+A b)}{8 x^{16}}-\frac {b^4 (5 a B+A b)}{12 x^{12}}-\frac {5 a b^3 (2 a B+A b)}{14 x^{14}}-\frac {b^5 B}{10 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^23,x]

[Out]

-1/22*(a^5*A)/x^22 - (a^4*(5*A*b + a*B))/(20*x^20) - (5*a^3*b*(2*A*b + a*B))/(18*x^18) - (5*a^2*b^2*(A*b + a*B
))/(8*x^16) - (5*a*b^3*(A*b + 2*a*B))/(14*x^14) - (b^4*(A*b + 5*a*B))/(12*x^12) - (b^5*B)/(10*x^10)

Rule 77

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])

Rule 457

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]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{23}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^{12}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^5 A}{x^{12}}+\frac {a^4 (5 A b+a B)}{x^{11}}+\frac {5 a^3 b (2 A b+a B)}{x^{10}}+\frac {10 a^2 b^2 (A b+a B)}{x^9}+\frac {5 a b^3 (A b+2 a B)}{x^8}+\frac {b^4 (A b+5 a B)}{x^7}+\frac {b^5 B}{x^6}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5 A}{22 x^{22}}-\frac {a^4 (5 A b+a B)}{20 x^{20}}-\frac {5 a^3 b (2 A b+a B)}{18 x^{18}}-\frac {5 a^2 b^2 (A b+a B)}{8 x^{16}}-\frac {5 a b^3 (A b+2 a B)}{14 x^{14}}-\frac {b^4 (A b+5 a B)}{12 x^{12}}-\frac {b^5 B}{10 x^{10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 121, normalized size = 1.03 \begin {gather*} -\frac {462 b^5 x^{10} \left (5 A+6 B x^2\right )+1650 a b^4 x^8 \left (6 A+7 B x^2\right )+2475 a^2 b^3 x^6 \left (7 A+8 B x^2\right )+1925 a^3 b^2 x^4 \left (8 A+9 B x^2\right )+770 a^4 b x^2 \left (9 A+10 B x^2\right )+126 a^5 \left (10 A+11 B x^2\right )}{27720 x^{22}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^23,x]

[Out]

-1/27720*(462*b^5*x^10*(5*A + 6*B*x^2) + 1650*a*b^4*x^8*(6*A + 7*B*x^2) + 2475*a^2*b^3*x^6*(7*A + 8*B*x^2) + 1
925*a^3*b^2*x^4*(8*A + 9*B*x^2) + 770*a^4*b*x^2*(9*A + 10*B*x^2) + 126*a^5*(10*A + 11*B*x^2))/x^22

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 104, normalized size = 0.89

method	result	size
default	\(-\frac {a^{5} A}{22 x^{22}}-\frac {a^{4} \left (5 A b +B a \right )}{20 x^{20}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{18 x^{18}}-\frac {5 a^{2} b^{2} \left (A b +B a \right )}{8 x^{16}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{14 x^{14}}-\frac {b^{4} \left (A b +5 B a \right )}{12 x^{12}}-\frac {b^{5} B}{10 x^{10}}\)	\(104\)
norman	\(\frac {-\frac {a^{5} A}{22}+\left (-\frac {1}{4} a^{4} b A -\frac {1}{20} a^{5} B \right ) x^{2}+\left (-\frac {5}{9} a^{3} b^{2} A -\frac {5}{18} a^{4} b B \right ) x^{4}+\left (-\frac {5}{8} a^{2} b^{3} A -\frac {5}{8} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{14} a \,b^{4} A -\frac {5}{7} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{12} b^{5} A -\frac {5}{12} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{10}}{x^{22}}\)	\(122\)
risch	\(\frac {-\frac {a^{5} A}{22}+\left (-\frac {1}{4} a^{4} b A -\frac {1}{20} a^{5} B \right ) x^{2}+\left (-\frac {5}{9} a^{3} b^{2} A -\frac {5}{18} a^{4} b B \right ) x^{4}+\left (-\frac {5}{8} a^{2} b^{3} A -\frac {5}{8} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{14} a \,b^{4} A -\frac {5}{7} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{12} b^{5} A -\frac {5}{12} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{10}}{x^{22}}\)	\(122\)
gosper	\(-\frac {2772 b^{5} B \,x^{12}+2310 A \,b^{5} x^{10}+11550 B a \,b^{4} x^{10}+9900 a A \,b^{4} x^{8}+19800 B \,a^{2} b^{3} x^{8}+17325 a^{2} A \,b^{3} x^{6}+17325 B \,a^{3} b^{2} x^{6}+15400 a^{3} A \,b^{2} x^{4}+7700 B \,a^{4} b \,x^{4}+6930 a^{4} A b \,x^{2}+1386 B \,a^{5} x^{2}+1260 a^{5} A}{27720 x^{22}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^23,x,method=_RETURNVERBOSE)

[Out]

-1/22*a^5*A/x^22-1/20*a^4*(5*A*b+B*a)/x^20-5/18*a^3*b*(2*A*b+B*a)/x^18-5/8*a^2*b^2*(A*b+B*a)/x^16-5/14*a*b^3*(
A*b+2*B*a)/x^14-1/12*b^4*(A*b+5*B*a)/x^12-1/10*b^5*B/x^10

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 121, normalized size = 1.03 \begin {gather*} -\frac {2772 \, B b^{5} x^{12} + 2310 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 9900 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 17325 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 1260 \, A a^{5} + 7700 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 1386 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{27720 \, x^{22}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^23,x, algorithm="maxima")

[Out]

-1/27720*(2772*B*b^5*x^12 + 2310*(5*B*a*b^4 + A*b^5)*x^10 + 9900*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 17325*(B*a^3*b^
2 + A*a^2*b^3)*x^6 + 1260*A*a^5 + 7700*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 1386*(B*a^5 + 5*A*a^4*b)*x^2)/x^22

________________________________________________________________________________________

Fricas [A]
time = 1.02, size = 121, normalized size = 1.03 \begin {gather*} -\frac {2772 \, B b^{5} x^{12} + 2310 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 9900 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 17325 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 1260 \, A a^{5} + 7700 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 1386 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{27720 \, x^{22}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^23,x, algorithm="fricas")

[Out]

-1/27720*(2772*B*b^5*x^12 + 2310*(5*B*a*b^4 + A*b^5)*x^10 + 9900*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 17325*(B*a^3*b^
2 + A*a^2*b^3)*x^6 + 1260*A*a^5 + 7700*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 1386*(B*a^5 + 5*A*a^4*b)*x^2)/x^22

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**23,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 1.67, size = 127, normalized size = 1.09 \begin {gather*} -\frac {2772 \, B b^{5} x^{12} + 11550 \, B a b^{4} x^{10} + 2310 \, A b^{5} x^{10} + 19800 \, B a^{2} b^{3} x^{8} + 9900 \, A a b^{4} x^{8} + 17325 \, B a^{3} b^{2} x^{6} + 17325 \, A a^{2} b^{3} x^{6} + 7700 \, B a^{4} b x^{4} + 15400 \, A a^{3} b^{2} x^{4} + 1386 \, B a^{5} x^{2} + 6930 \, A a^{4} b x^{2} + 1260 \, A a^{5}}{27720 \, x^{22}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^23,x, algorithm="giac")

[Out]

-1/27720*(2772*B*b^5*x^12 + 11550*B*a*b^4*x^10 + 2310*A*b^5*x^10 + 19800*B*a^2*b^3*x^8 + 9900*A*a*b^4*x^8 + 17
325*B*a^3*b^2*x^6 + 17325*A*a^2*b^3*x^6 + 7700*B*a^4*b*x^4 + 15400*A*a^3*b^2*x^4 + 1386*B*a^5*x^2 + 6930*A*a^4
*b*x^2 + 1260*A*a^5)/x^22

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 122, normalized size = 1.04 \begin {gather*} -\frac {\frac {A\,a^5}{22}+x^8\,\left (\frac {5\,B\,a^2\,b^3}{7}+\frac {5\,A\,a\,b^4}{14}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{18}+\frac {5\,A\,a^3\,b^2}{9}\right )+x^2\,\left (\frac {B\,a^5}{20}+\frac {A\,b\,a^4}{4}\right )+x^{10}\,\left (\frac {A\,b^5}{12}+\frac {5\,B\,a\,b^4}{12}\right )+x^6\,\left (\frac {5\,B\,a^3\,b^2}{8}+\frac {5\,A\,a^2\,b^3}{8}\right )+\frac {B\,b^5\,x^{12}}{10}}{x^{22}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^23,x)

[Out]

-((A*a^5)/22 + x^8*((5*B*a^2*b^3)/7 + (5*A*a*b^4)/14) + x^4*((5*A*a^3*b^2)/9 + (5*B*a^4*b)/18) + x^2*((B*a^5)/
20 + (A*a^4*b)/4) + x^10*((A*b^5)/12 + (5*B*a*b^4)/12) + x^6*((5*A*a^2*b^3)/8 + (5*B*a^3*b^2)/8) + (B*b^5*x^12
)/10)/x^22

________________________________________________________________________________________