3.1.50 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{18}} \, dx\) [50]

Optimal. Leaf size=117 \[ -\frac {a^5 A}{17 x^{17}}-\frac {a^4 (5 A b+a B)}{15 x^{15}}-\frac {5 a^3 b (2 A b+a B)}{13 x^{13}}-\frac {10 a^2 b^2 (A b+a B)}{11 x^{11}}-\frac {5 a b^3 (A b+2 a B)}{9 x^9}-\frac {b^4 (A b+5 a B)}{7 x^7}-\frac {b^5 B}{5 x^5} \]

[Out]

-1/17*a^5*A/x^17-1/15*a^4*(5*A*b+B*a)/x^15-5/13*a^3*b*(2*A*b+B*a)/x^13-10/11*a^2*b^2*(A*b+B*a)/x^11-5/9*a*b^3*
(A*b+2*B*a)/x^9-1/7*b^4*(A*b+5*B*a)/x^7-1/5*b^5*B/x^5

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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}{17 x^{17}}-\frac {a^4 (a B+5 A b)}{15 x^{15}}-\frac {5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac {10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac {b^4 (5 a B+A b)}{7 x^7}-\frac {5 a b^3 (2 a B+A b)}{9 x^9}-\frac {b^5 B}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^18,x]

[Out]

-1/17*(a^5*A)/x^17 - (a^4*(5*A*b + a*B))/(15*x^15) - (5*a^3*b*(2*A*b + a*B))/(13*x^13) - (10*a^2*b^2*(A*b + a*
B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(9*x^9) - (b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(5*x^5)

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^{18}} \, dx &=\int \left (\frac {a^5 A}{x^{18}}+\frac {a^4 (5 A b+a B)}{x^{16}}+\frac {5 a^3 b (2 A b+a B)}{x^{14}}+\frac {10 a^2 b^2 (A b+a B)}{x^{12}}+\frac {5 a b^3 (A b+2 a B)}{x^{10}}+\frac {b^4 (A b+5 a B)}{x^8}+\frac {b^5 B}{x^6}\right ) \, dx\\ &=-\frac {a^5 A}{17 x^{17}}-\frac {a^4 (5 A b+a B)}{15 x^{15}}-\frac {5 a^3 b (2 A b+a B)}{13 x^{13}}-\frac {10 a^2 b^2 (A b+a B)}{11 x^{11}}-\frac {5 a b^3 (A b+2 a B)}{9 x^9}-\frac {b^4 (A b+5 a B)}{7 x^7}-\frac {b^5 B}{5 x^5}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 117, normalized size = 1.00 \begin {gather*} -\frac {a^5 A}{17 x^{17}}-\frac {a^4 (5 A b+a B)}{15 x^{15}}-\frac {5 a^3 b (2 A b+a B)}{13 x^{13}}-\frac {10 a^2 b^2 (A b+a B)}{11 x^{11}}-\frac {5 a b^3 (A b+2 a B)}{9 x^9}-\frac {b^4 (A b+5 a B)}{7 x^7}-\frac {b^5 B}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^18,x]

[Out]

-1/17*(a^5*A)/x^17 - (a^4*(5*A*b + a*B))/(15*x^15) - (5*a^3*b*(2*A*b + a*B))/(13*x^13) - (10*a^2*b^2*(A*b + a*
B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(9*x^9) - (b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(5*x^5)

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

method	result	size
default	\(-\frac {a^{5} A}{17 x^{17}}-\frac {a^{4} \left (5 A b +B a \right )}{15 x^{15}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{13 x^{13}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{11 x^{11}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{9 x^{9}}-\frac {b^{4} \left (A b +5 B a \right )}{7 x^{7}}-\frac {b^{5} B}{5 x^{5}}\)	\(104\)
norman	\(\frac {-\frac {a^{5} A}{17}+\left (-\frac {1}{3} a^{4} b A -\frac {1}{15} a^{5} B \right ) x^{2}+\left (-\frac {10}{13} a^{3} b^{2} A -\frac {5}{13} a^{4} b B \right ) x^{4}+\left (-\frac {10}{11} a^{2} b^{3} A -\frac {10}{11} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{9} a \,b^{4} A -\frac {10}{9} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{7} b^{5} A -\frac {5}{7} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{5}}{x^{17}}\)	\(122\)
risch	\(\frac {-\frac {a^{5} A}{17}+\left (-\frac {1}{3} a^{4} b A -\frac {1}{15} a^{5} B \right ) x^{2}+\left (-\frac {10}{13} a^{3} b^{2} A -\frac {5}{13} a^{4} b B \right ) x^{4}+\left (-\frac {10}{11} a^{2} b^{3} A -\frac {10}{11} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{9} a \,b^{4} A -\frac {10}{9} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{7} b^{5} A -\frac {5}{7} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{5}}{x^{17}}\)	\(122\)
gosper	\(-\frac {153153 b^{5} B \,x^{12}+109395 A \,b^{5} x^{10}+546975 B a \,b^{4} x^{10}+425425 a A \,b^{4} x^{8}+850850 B \,a^{2} b^{3} x^{8}+696150 a^{2} A \,b^{3} x^{6}+696150 B \,a^{3} b^{2} x^{6}+589050 a^{3} A \,b^{2} x^{4}+294525 B \,a^{4} b \,x^{4}+255255 a^{4} A b \,x^{2}+51051 B \,a^{5} x^{2}+45045 a^{5} A}{765765 x^{17}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^18,x,method=_RETURNVERBOSE)

[Out]

-1/17*a^5*A/x^17-1/15*a^4*(5*A*b+B*a)/x^15-5/13*a^3*b*(2*A*b+B*a)/x^13-10/11*a^2*b^2*(A*b+B*a)/x^11-5/9*a*b^3*
(A*b+2*B*a)/x^9-1/7*b^4*(A*b+5*B*a)/x^7-1/5*b^5*B/x^5

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Maxima [A]
time = 0.31, size = 121, normalized size = 1.03 \begin {gather*} -\frac {153153 \, B b^{5} x^{12} + 109395 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 425425 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 696150 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 45045 \, A a^{5} + 294525 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 51051 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{765765 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^18,x, algorithm="maxima")

[Out]

-1/765765*(153153*B*b^5*x^12 + 109395*(5*B*a*b^4 + A*b^5)*x^10 + 425425*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 696150*(
B*a^3*b^2 + A*a^2*b^3)*x^6 + 45045*A*a^5 + 294525*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 51051*(B*a^5 + 5*A*a^4*b)*x^2)
/x^17

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Fricas [A]
time = 0.74, size = 121, normalized size = 1.03 \begin {gather*} -\frac {153153 \, B b^{5} x^{12} + 109395 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 425425 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 696150 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 45045 \, A a^{5} + 294525 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 51051 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{765765 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^18,x, algorithm="fricas")

[Out]

-1/765765*(153153*B*b^5*x^12 + 109395*(5*B*a*b^4 + A*b^5)*x^10 + 425425*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 696150*(
B*a^3*b^2 + A*a^2*b^3)*x^6 + 45045*A*a^5 + 294525*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 51051*(B*a^5 + 5*A*a^4*b)*x^2)
/x^17

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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**18,x)

[Out]

Timed out

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Giac [A]
time = 1.06, size = 127, normalized size = 1.09 \begin {gather*} -\frac {153153 \, B b^{5} x^{12} + 546975 \, B a b^{4} x^{10} + 109395 \, A b^{5} x^{10} + 850850 \, B a^{2} b^{3} x^{8} + 425425 \, A a b^{4} x^{8} + 696150 \, B a^{3} b^{2} x^{6} + 696150 \, A a^{2} b^{3} x^{6} + 294525 \, B a^{4} b x^{4} + 589050 \, A a^{3} b^{2} x^{4} + 51051 \, B a^{5} x^{2} + 255255 \, A a^{4} b x^{2} + 45045 \, A a^{5}}{765765 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^18,x, algorithm="giac")

[Out]

-1/765765*(153153*B*b^5*x^12 + 546975*B*a*b^4*x^10 + 109395*A*b^5*x^10 + 850850*B*a^2*b^3*x^8 + 425425*A*a*b^4
*x^8 + 696150*B*a^3*b^2*x^6 + 696150*A*a^2*b^3*x^6 + 294525*B*a^4*b*x^4 + 589050*A*a^3*b^2*x^4 + 51051*B*a^5*x
^2 + 255255*A*a^4*b*x^2 + 45045*A*a^5)/x^17

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Mupad [B]
time = 0.05, size = 122, normalized size = 1.04 \begin {gather*} -\frac {\frac {A\,a^5}{17}+x^8\,\left (\frac {10\,B\,a^2\,b^3}{9}+\frac {5\,A\,a\,b^4}{9}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{13}+\frac {10\,A\,a^3\,b^2}{13}\right )+x^2\,\left (\frac {B\,a^5}{15}+\frac {A\,b\,a^4}{3}\right )+x^{10}\,\left (\frac {A\,b^5}{7}+\frac {5\,B\,a\,b^4}{7}\right )+x^6\,\left (\frac {10\,B\,a^3\,b^2}{11}+\frac {10\,A\,a^2\,b^3}{11}\right )+\frac {B\,b^5\,x^{12}}{5}}{x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^18,x)

[Out]

-((A*a^5)/17 + x^8*((10*B*a^2*b^3)/9 + (5*A*a*b^4)/9) + x^4*((10*A*a^3*b^2)/13 + (5*B*a^4*b)/13) + x^2*((B*a^5
)/15 + (A*a^4*b)/3) + x^10*((A*b^5)/7 + (5*B*a*b^4)/7) + x^6*((10*A*a^2*b^3)/11 + (10*B*a^3*b^2)/11) + (B*b^5*
x^12)/5)/x^17

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