3.1.48 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{16}} \, dx\) [48]

Optimal. Leaf size=117 \[ -\frac {a^5 A}{15 x^{15}}-\frac {a^4 (5 A b+a B)}{13 x^{13}}-\frac {5 a^3 b (2 A b+a B)}{11 x^{11}}-\frac {10 a^2 b^2 (A b+a B)}{9 x^9}-\frac {5 a b^3 (A b+2 a B)}{7 x^7}-\frac {b^4 (A b+5 a B)}{5 x^5}-\frac {b^5 B}{3 x^3} \]

[Out]

-1/15*a^5*A/x^15-1/13*a^4*(5*A*b+B*a)/x^13-5/11*a^3*b*(2*A*b+B*a)/x^11-10/9*a^2*b^2*(A*b+B*a)/x^9-5/7*a*b^3*(A
*b+2*B*a)/x^7-1/5*b^4*(A*b+5*B*a)/x^5-1/3*b^5*B/x^3

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \begin {gather*} -\frac {a^5 A}{15 x^{15}}-\frac {a^4 (a B+5 A b)}{13 x^{13}}-\frac {5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac {10 a^2 b^2 (a B+A b)}{9 x^9}-\frac {b^4 (5 a B+A b)}{5 x^5}-\frac {5 a b^3 (2 a B+A b)}{7 x^7}-\frac {b^5 B}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^16,x]

[Out]

-1/15*(a^5*A)/x^15 - (a^4*(5*A*b + a*B))/(13*x^13) - (5*a^3*b*(2*A*b + a*B))/(11*x^11) - (10*a^2*b^2*(A*b + a*
B))/(9*x^9) - (5*a*b^3*(A*b + 2*a*B))/(7*x^7) - (b^4*(A*b + 5*a*B))/(5*x^5) - (b^5*B)/(3*x^3)

Rule 459

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{16}} \, dx &=\int \left (\frac {a^5 A}{x^{16}}+\frac {a^4 (5 A b+a B)}{x^{14}}+\frac {5 a^3 b (2 A b+a B)}{x^{12}}+\frac {10 a^2 b^2 (A b+a B)}{x^{10}}+\frac {5 a b^3 (A b+2 a B)}{x^8}+\frac {b^4 (A b+5 a B)}{x^6}+\frac {b^5 B}{x^4}\right ) \, dx\\ &=-\frac {a^5 A}{15 x^{15}}-\frac {a^4 (5 A b+a B)}{13 x^{13}}-\frac {5 a^3 b (2 A b+a B)}{11 x^{11}}-\frac {10 a^2 b^2 (A b+a B)}{9 x^9}-\frac {5 a b^3 (A b+2 a B)}{7 x^7}-\frac {b^4 (A b+5 a B)}{5 x^5}-\frac {b^5 B}{3 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 121, normalized size = 1.03 \begin {gather*} -\frac {3003 b^5 x^{10} \left (3 A+5 B x^2\right )+6435 a b^4 x^8 \left (5 A+7 B x^2\right )+7150 a^2 b^3 x^6 \left (7 A+9 B x^2\right )+4550 a^3 b^2 x^4 \left (9 A+11 B x^2\right )+1575 a^4 b x^2 \left (11 A+13 B x^2\right )+231 a^5 \left (13 A+15 B x^2\right )}{45045 x^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^16,x]

[Out]

-1/45045*(3003*b^5*x^10*(3*A + 5*B*x^2) + 6435*a*b^4*x^8*(5*A + 7*B*x^2) + 7150*a^2*b^3*x^6*(7*A + 9*B*x^2) +
4550*a^3*b^2*x^4*(9*A + 11*B*x^2) + 1575*a^4*b*x^2*(11*A + 13*B*x^2) + 231*a^5*(13*A + 15*B*x^2))/x^15

________________________________________________________________________________________

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

method	result	size
default	\(-\frac {a^{5} A}{15 x^{15}}-\frac {a^{4} \left (5 A b +B a \right )}{13 x^{13}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{11 x^{11}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{9 x^{9}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{7 x^{7}}-\frac {b^{4} \left (A b +5 B a \right )}{5 x^{5}}-\frac {b^{5} B}{3 x^{3}}\)	\(104\)
norman	\(\frac {-\frac {a^{5} A}{15}+\left (-\frac {5}{13} a^{4} b A -\frac {1}{13} a^{5} B \right ) x^{2}+\left (-\frac {10}{11} a^{3} b^{2} A -\frac {5}{11} a^{4} b B \right ) x^{4}+\left (-\frac {10}{9} a^{2} b^{3} A -\frac {10}{9} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{7} a \,b^{4} A -\frac {10}{7} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{5} b^{5} A -a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{3}}{x^{15}}\)	\(122\)
risch	\(\frac {-\frac {a^{5} A}{15}+\left (-\frac {5}{13} a^{4} b A -\frac {1}{13} a^{5} B \right ) x^{2}+\left (-\frac {10}{11} a^{3} b^{2} A -\frac {5}{11} a^{4} b B \right ) x^{4}+\left (-\frac {10}{9} a^{2} b^{3} A -\frac {10}{9} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{7} a \,b^{4} A -\frac {10}{7} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{5} b^{5} A -a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{3}}{x^{15}}\)	\(122\)
gosper	\(-\frac {15015 b^{5} B \,x^{12}+9009 A \,b^{5} x^{10}+45045 B a \,b^{4} x^{10}+32175 a A \,b^{4} x^{8}+64350 B \,a^{2} b^{3} x^{8}+50050 a^{2} A \,b^{3} x^{6}+50050 B \,a^{3} b^{2} x^{6}+40950 a^{3} A \,b^{2} x^{4}+20475 B \,a^{4} b \,x^{4}+17325 a^{4} A b \,x^{2}+3465 B \,a^{5} x^{2}+3003 a^{5} A}{45045 x^{15}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^16,x,method=_RETURNVERBOSE)

[Out]

-1/15*a^5*A/x^15-1/13*a^4*(5*A*b+B*a)/x^13-5/11*a^3*b*(2*A*b+B*a)/x^11-10/9*a^2*b^2*(A*b+B*a)/x^9-5/7*a*b^3*(A
*b+2*B*a)/x^7-1/5*b^4*(A*b+5*B*a)/x^5-1/3*b^5*B/x^3

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 121, normalized size = 1.03 \begin {gather*} -\frac {15015 \, B b^{5} x^{12} + 9009 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 32175 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 50050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 3003 \, A a^{5} + 20475 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 3465 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{45045 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^16,x, algorithm="maxima")

[Out]

-1/45045*(15015*B*b^5*x^12 + 9009*(5*B*a*b^4 + A*b^5)*x^10 + 32175*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 50050*(B*a^3*
b^2 + A*a^2*b^3)*x^6 + 3003*A*a^5 + 20475*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 3465*(B*a^5 + 5*A*a^4*b)*x^2)/x^15

________________________________________________________________________________________

Fricas [A]
time = 0.62, size = 121, normalized size = 1.03 \begin {gather*} -\frac {15015 \, B b^{5} x^{12} + 9009 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 32175 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 50050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 3003 \, A a^{5} + 20475 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 3465 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{45045 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^16,x, algorithm="fricas")

[Out]

-1/45045*(15015*B*b^5*x^12 + 9009*(5*B*a*b^4 + A*b^5)*x^10 + 32175*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 50050*(B*a^3*
b^2 + A*a^2*b^3)*x^6 + 3003*A*a^5 + 20475*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 3465*(B*a^5 + 5*A*a^4*b)*x^2)/x^15

________________________________________________________________________________________

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**16,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 1.49, size = 127, normalized size = 1.09 \begin {gather*} -\frac {15015 \, B b^{5} x^{12} + 45045 \, B a b^{4} x^{10} + 9009 \, A b^{5} x^{10} + 64350 \, B a^{2} b^{3} x^{8} + 32175 \, A a b^{4} x^{8} + 50050 \, B a^{3} b^{2} x^{6} + 50050 \, A a^{2} b^{3} x^{6} + 20475 \, B a^{4} b x^{4} + 40950 \, A a^{3} b^{2} x^{4} + 3465 \, B a^{5} x^{2} + 17325 \, A a^{4} b x^{2} + 3003 \, A a^{5}}{45045 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^16,x, algorithm="giac")

[Out]

-1/45045*(15015*B*b^5*x^12 + 45045*B*a*b^4*x^10 + 9009*A*b^5*x^10 + 64350*B*a^2*b^3*x^8 + 32175*A*a*b^4*x^8 +
50050*B*a^3*b^2*x^6 + 50050*A*a^2*b^3*x^6 + 20475*B*a^4*b*x^4 + 40950*A*a^3*b^2*x^4 + 3465*B*a^5*x^2 + 17325*A
*a^4*b*x^2 + 3003*A*a^5)/x^15

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 121, normalized size = 1.03 \begin {gather*} -\frac {\frac {A\,a^5}{15}+x^8\,\left (\frac {10\,B\,a^2\,b^3}{7}+\frac {5\,A\,a\,b^4}{7}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{11}+\frac {10\,A\,a^3\,b^2}{11}\right )+x^2\,\left (\frac {B\,a^5}{13}+\frac {5\,A\,b\,a^4}{13}\right )+x^{10}\,\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )+x^6\,\left (\frac {10\,B\,a^3\,b^2}{9}+\frac {10\,A\,a^2\,b^3}{9}\right )+\frac {B\,b^5\,x^{12}}{3}}{x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^16,x)

[Out]

-((A*a^5)/15 + x^8*((10*B*a^2*b^3)/7 + (5*A*a*b^4)/7) + x^4*((10*A*a^3*b^2)/11 + (5*B*a^4*b)/11) + x^2*((B*a^5
)/13 + (5*A*a^4*b)/13) + x^10*((A*b^5)/5 + B*a*b^4) + x^6*((10*A*a^2*b^3)/9 + (10*B*a^3*b^2)/9) + (B*b^5*x^12)
/3)/x^15

________________________________________________________________________________________