3.1.53 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{21}} \, dx\) [53]

Optimal. Leaf size=117 \[ -\frac {a^5 A}{20 x^{20}}-\frac {a^4 (5 A b+a B)}{18 x^{18}}-\frac {5 a^3 b (2 A b+a B)}{16 x^{16}}-\frac {5 a^2 b^2 (A b+a B)}{7 x^{14}}-\frac {5 a b^3 (A b+2 a B)}{12 x^{12}}-\frac {b^4 (A b+5 a B)}{10 x^{10}}-\frac {b^5 B}{8 x^8} \]

[Out]

-1/20*a^5*A/x^20-1/18*a^4*(5*A*b+B*a)/x^18-5/16*a^3*b*(2*A*b+B*a)/x^16-5/7*a^2*b^2*(A*b+B*a)/x^14-5/12*a*b^3*(
A*b+2*B*a)/x^12-1/10*b^4*(A*b+5*B*a)/x^10-1/8*b^5*B/x^8

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Rubi [A]
time = 0.06, 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}{20 x^{20}}-\frac {a^4 (a B+5 A b)}{18 x^{18}}-\frac {5 a^3 b (a B+2 A b)}{16 x^{16}}-\frac {5 a^2 b^2 (a B+A b)}{7 x^{14}}-\frac {b^4 (5 a B+A b)}{10 x^{10}}-\frac {5 a b^3 (2 a B+A b)}{12 x^{12}}-\frac {b^5 B}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^21,x]

[Out]

-1/20*(a^5*A)/x^20 - (a^4*(5*A*b + a*B))/(18*x^18) - (5*a^3*b*(2*A*b + a*B))/(16*x^16) - (5*a^2*b^2*(A*b + a*B
))/(7*x^14) - (5*a*b^3*(A*b + 2*a*B))/(12*x^12) - (b^4*(A*b + 5*a*B))/(10*x^10) - (b^5*B)/(8*x^8)

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^{21}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^{11}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^5 A}{x^{11}}+\frac {a^4 (5 A b+a B)}{x^{10}}+\frac {5 a^3 b (2 A b+a B)}{x^9}+\frac {10 a^2 b^2 (A b+a B)}{x^8}+\frac {5 a b^3 (A b+2 a B)}{x^7}+\frac {b^4 (A b+5 a B)}{x^6}+\frac {b^5 B}{x^5}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5 A}{20 x^{20}}-\frac {a^4 (5 A b+a B)}{18 x^{18}}-\frac {5 a^3 b (2 A b+a B)}{16 x^{16}}-\frac {5 a^2 b^2 (A b+a B)}{7 x^{14}}-\frac {5 a b^3 (A b+2 a B)}{12 x^{12}}-\frac {b^4 (A b+5 a B)}{10 x^{10}}-\frac {b^5 B}{8 x^8}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 121, normalized size = 1.03 \begin {gather*} -\frac {126 b^5 x^{10} \left (4 A+5 B x^2\right )+420 a b^4 x^8 \left (5 A+6 B x^2\right )+600 a^2 b^3 x^6 \left (6 A+7 B x^2\right )+450 a^3 b^2 x^4 \left (7 A+8 B x^2\right )+175 a^4 b x^2 \left (8 A+9 B x^2\right )+28 a^5 \left (9 A+10 B x^2\right )}{5040 x^{20}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^21,x]

[Out]

-1/5040*(126*b^5*x^10*(4*A + 5*B*x^2) + 420*a*b^4*x^8*(5*A + 6*B*x^2) + 600*a^2*b^3*x^6*(6*A + 7*B*x^2) + 450*
a^3*b^2*x^4*(7*A + 8*B*x^2) + 175*a^4*b*x^2*(8*A + 9*B*x^2) + 28*a^5*(9*A + 10*B*x^2))/x^20

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

method	result	size
default	\(-\frac {a^{5} A}{20 x^{20}}-\frac {a^{4} \left (5 A b +B a \right )}{18 x^{18}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{16 x^{16}}-\frac {5 a^{2} b^{2} \left (A b +B a \right )}{7 x^{14}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{12 x^{12}}-\frac {b^{4} \left (A b +5 B a \right )}{10 x^{10}}-\frac {b^{5} B}{8 x^{8}}\)	\(104\)
norman	\(\frac {-\frac {a^{5} A}{20}+\left (-\frac {5}{18} a^{4} b A -\frac {1}{18} a^{5} B \right ) x^{2}+\left (-\frac {5}{8} a^{3} b^{2} A -\frac {5}{16} a^{4} b B \right ) x^{4}+\left (-\frac {5}{7} a^{2} b^{3} A -\frac {5}{7} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{12} a \,b^{4} A -\frac {5}{6} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{10} b^{5} A -\frac {1}{2} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{8}}{x^{20}}\)	\(122\)
risch	\(\frac {-\frac {a^{5} A}{20}+\left (-\frac {5}{18} a^{4} b A -\frac {1}{18} a^{5} B \right ) x^{2}+\left (-\frac {5}{8} a^{3} b^{2} A -\frac {5}{16} a^{4} b B \right ) x^{4}+\left (-\frac {5}{7} a^{2} b^{3} A -\frac {5}{7} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{12} a \,b^{4} A -\frac {5}{6} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{10} b^{5} A -\frac {1}{2} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{8}}{x^{20}}\)	\(122\)
gosper	\(-\frac {630 b^{5} B \,x^{12}+504 A \,b^{5} x^{10}+2520 B a \,b^{4} x^{10}+2100 a A \,b^{4} x^{8}+4200 B \,a^{2} b^{3} x^{8}+3600 a^{2} A \,b^{3} x^{6}+3600 B \,a^{3} b^{2} x^{6}+3150 a^{3} A \,b^{2} x^{4}+1575 B \,a^{4} b \,x^{4}+1400 a^{4} A b \,x^{2}+280 B \,a^{5} x^{2}+252 a^{5} A}{5040 x^{20}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^21,x,method=_RETURNVERBOSE)

[Out]

-1/20*a^5*A/x^20-1/18*a^4*(5*A*b+B*a)/x^18-5/16*a^3*b*(2*A*b+B*a)/x^16-5/7*a^2*b^2*(A*b+B*a)/x^14-5/12*a*b^3*(
A*b+2*B*a)/x^12-1/10*b^4*(A*b+5*B*a)/x^10-1/8*b^5*B/x^8

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Maxima [A]
time = 0.29, size = 121, normalized size = 1.03 \begin {gather*} -\frac {630 \, B b^{5} x^{12} + 504 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 2100 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 3600 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 252 \, A a^{5} + 1575 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 280 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{5040 \, x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^21,x, algorithm="maxima")

[Out]

-1/5040*(630*B*b^5*x^12 + 504*(5*B*a*b^4 + A*b^5)*x^10 + 2100*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 3600*(B*a^3*b^2 +
A*a^2*b^3)*x^6 + 252*A*a^5 + 1575*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 280*(B*a^5 + 5*A*a^4*b)*x^2)/x^20

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Fricas [A]
time = 0.79, size = 121, normalized size = 1.03 \begin {gather*} -\frac {630 \, B b^{5} x^{12} + 504 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 2100 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 3600 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 252 \, A a^{5} + 1575 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 280 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{5040 \, x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^21,x, algorithm="fricas")

[Out]

-1/5040*(630*B*b^5*x^12 + 504*(5*B*a*b^4 + A*b^5)*x^10 + 2100*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 3600*(B*a^3*b^2 +
A*a^2*b^3)*x^6 + 252*A*a^5 + 1575*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 280*(B*a^5 + 5*A*a^4*b)*x^2)/x^20

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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**21,x)

[Out]

Timed out

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Giac [A]
time = 0.86, size = 127, normalized size = 1.09 \begin {gather*} -\frac {630 \, B b^{5} x^{12} + 2520 \, B a b^{4} x^{10} + 504 \, A b^{5} x^{10} + 4200 \, B a^{2} b^{3} x^{8} + 2100 \, A a b^{4} x^{8} + 3600 \, B a^{3} b^{2} x^{6} + 3600 \, A a^{2} b^{3} x^{6} + 1575 \, B a^{4} b x^{4} + 3150 \, A a^{3} b^{2} x^{4} + 280 \, B a^{5} x^{2} + 1400 \, A a^{4} b x^{2} + 252 \, A a^{5}}{5040 \, x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^21,x, algorithm="giac")

[Out]

-1/5040*(630*B*b^5*x^12 + 2520*B*a*b^4*x^10 + 504*A*b^5*x^10 + 4200*B*a^2*b^3*x^8 + 2100*A*a*b^4*x^8 + 3600*B*
a^3*b^2*x^6 + 3600*A*a^2*b^3*x^6 + 1575*B*a^4*b*x^4 + 3150*A*a^3*b^2*x^4 + 280*B*a^5*x^2 + 1400*A*a^4*b*x^2 +
252*A*a^5)/x^20

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Mupad [B]
time = 0.05, size = 122, normalized size = 1.04 \begin {gather*} -\frac {\frac {A\,a^5}{20}+x^8\,\left (\frac {5\,B\,a^2\,b^3}{6}+\frac {5\,A\,a\,b^4}{12}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{16}+\frac {5\,A\,a^3\,b^2}{8}\right )+x^2\,\left (\frac {B\,a^5}{18}+\frac {5\,A\,b\,a^4}{18}\right )+x^{10}\,\left (\frac {A\,b^5}{10}+\frac {B\,a\,b^4}{2}\right )+x^6\,\left (\frac {5\,B\,a^3\,b^2}{7}+\frac {5\,A\,a^2\,b^3}{7}\right )+\frac {B\,b^5\,x^{12}}{8}}{x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^21,x)

[Out]

-((A*a^5)/20 + x^8*((5*B*a^2*b^3)/6 + (5*A*a*b^4)/12) + x^4*((5*A*a^3*b^2)/8 + (5*B*a^4*b)/16) + x^2*((B*a^5)/
18 + (5*A*a^4*b)/18) + x^10*((A*b^5)/10 + (B*a*b^4)/2) + x^6*((5*A*a^2*b^3)/7 + (5*B*a^3*b^2)/7) + (B*b^5*x^12
)/8)/x^20

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