3.2.66 \(\int \frac {A+B x^2+C x^4+D x^6}{x^6 (a+b x^2)^{9/2}} \, dx\) [166]

Optimal. Leaf size=281 \[ -\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-a \left (80 b^2 B-24 a b C+3 a^2 D\right )\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {2 \left (192 A b^3-a \left (80 b^2 B-24 a b C+3 a^2 D\right )\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {8 \left (192 A b^3-a \left (80 b^2 B-24 a b C+3 a^2 D\right )\right ) x}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac {16 \left (192 A b^3-a \left (80 b^2 B-24 a b C+3 a^2 D\right )\right ) x}{105 a^7 \sqrt {a+b x^2}} \]

[Out]

-1/5*A/a/x^5/(b*x^2+a)^(7/2)+1/15*(12*A*b-5*B*a)/a^2/x^3/(b*x^2+a)^(7/2)+1/3*(-24*A*b^2+a*(10*B*b-3*C*a))/a^3/
x/(b*x^2+a)^(7/2)-1/21*(192*A*b^3-a*(80*B*b^2-24*C*a*b+3*D*a^2))*x/a^4/(b*x^2+a)^(7/2)-2/35*(192*A*b^3-a*(80*B
*b^2-24*C*a*b+3*D*a^2))*x/a^5/(b*x^2+a)^(5/2)-8/105*(192*A*b^3-a*(80*B*b^2-24*C*a*b+3*D*a^2))*x/a^6/(b*x^2+a)^
(3/2)-16/105*(192*A*b^3-a*(80*B*b^2-24*C*a*b+3*D*a^2))*x/a^7/(b*x^2+a)^(1/2)

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Rubi [A]
time = 0.28, antiderivative size = 275, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1817, 12, 198, 197} \begin {gather*} -\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {16 x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{105 a^7 \sqrt {a+b x^2}}-\frac {8 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac {2 x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^6*(a + b*x^2)^(9/2)),x]

[Out]

-1/5*A/(a*x^5*(a + b*x^2)^(7/2)) + (12*A*b - 5*a*B)/(15*a^2*x^3*(a + b*x^2)^(7/2)) - (24*A*b^2 - a*(10*b*B - 3
*a*C))/(3*a^3*x*(a + b*x^2)^(7/2)) - ((192*A*b^3 - 8*a*b*(10*b*B - 3*a*C) - 3*a^3*D)*x)/(21*a^4*(a + b*x^2)^(7
/2)) - (2*(192*A*b^3 - 8*a*b*(10*b*B - 3*a*C) - 3*a^3*D)*x)/(35*a^5*(a + b*x^2)^(5/2)) - (8*(192*A*b^3 - a*(80
*b^2*B - 24*a*b*C + 3*a^2*D))*x)/(105*a^6*(a + b*x^2)^(3/2)) - (16*(192*A*b^3 - 8*a*b*(10*b*B - 3*a*C) - 3*a^3
*D)*x)/(105*a^7*Sqrt[a + b*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 1817

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient
[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Dist[1/(a*(m + 1)),
Int[x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]

Rubi steps

\begin {align*} \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {12 A b-5 a \left (B+C x^2+D x^4\right )}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{5 a}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}+\frac {\int \frac {10 b (12 A b-5 a B)-3 a \left (-5 a C-5 a D x^2\right )}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{15 a^2}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {8 b \left (120 A b^2-50 a b B+15 a^2 C\right )-15 a^3 D}{\left (a+b x^2\right )^{9/2}} \, dx}{15 a^3}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^3}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {\left (2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^4}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {\left (8 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^5}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {8 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac {\left (16 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^6}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {8 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac {16 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{105 a^7 \sqrt {a+b x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 202, normalized size = 0.72 \begin {gather*} \frac {-3072 A b^6 x^{12}+256 a b^5 x^{10} \left (-42 A+5 B x^2\right )-128 a^2 b^4 x^8 \left (105 A-35 B x^2+3 C x^4\right )+16 a^3 b^3 x^6 \left (-420 A+350 B x^2-84 C x^4+3 D x^6\right )+56 a^4 b^2 x^4 \left (-15 A+50 B x^2-30 C x^4+3 D x^6\right )+14 a^5 b x^2 \left (6 A+25 B x^2-60 C x^4+15 D x^6\right )-7 a^6 \left (3 A+5 x^2 \left (B+3 C x^2-3 D x^4\right )\right )}{105 a^7 x^5 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^6*(a + b*x^2)^(9/2)),x]

[Out]

(-3072*A*b^6*x^12 + 256*a*b^5*x^10*(-42*A + 5*B*x^2) - 128*a^2*b^4*x^8*(105*A - 35*B*x^2 + 3*C*x^4) + 16*a^3*b
^3*x^6*(-420*A + 350*B*x^2 - 84*C*x^4 + 3*D*x^6) + 56*a^4*b^2*x^4*(-15*A + 50*B*x^2 - 30*C*x^4 + 3*D*x^6) + 14
*a^5*b*x^2*(6*A + 25*B*x^2 - 60*C*x^4 + 15*D*x^6) - 7*a^6*(3*A + 5*x^2*(B + 3*C*x^2 - 3*D*x^4)))/(105*a^7*x^5*
(a + b*x^2)^(7/2))

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Maple [A]
time = 0.27, size = 446, normalized size = 1.59

method	result	size
gosper	\(-\frac {3072 A \,b^{6} x^{12}-1280 B a \,b^{5} x^{12}+384 C \,a^{2} b^{4} x^{12}-48 D a^{3} b^{3} x^{12}+10752 A a \,b^{5} x^{10}-4480 B \,a^{2} b^{4} x^{10}+1344 C \,a^{3} b^{3} x^{10}-168 D a^{4} b^{2} x^{10}+13440 A \,a^{2} b^{4} x^{8}-5600 B \,a^{3} b^{3} x^{8}+1680 C \,a^{4} b^{2} x^{8}-210 D a^{5} b \,x^{8}+6720 A \,a^{3} b^{3} x^{6}-2800 B \,a^{4} b^{2} x^{6}+840 C \,a^{5} b \,x^{6}-105 D a^{6} x^{6}+840 A \,a^{4} b^{2} x^{4}-350 B \,a^{5} b \,x^{4}+105 C \,a^{6} x^{4}-84 A \,a^{5} b \,x^{2}+35 B \,a^{6} x^{2}+21 A \,a^{6}}{105 x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{7}}\)	\(253\)
trager	\(-\frac {3072 A \,b^{6} x^{12}-1280 B a \,b^{5} x^{12}+384 C \,a^{2} b^{4} x^{12}-48 D a^{3} b^{3} x^{12}+10752 A a \,b^{5} x^{10}-4480 B \,a^{2} b^{4} x^{10}+1344 C \,a^{3} b^{3} x^{10}-168 D a^{4} b^{2} x^{10}+13440 A \,a^{2} b^{4} x^{8}-5600 B \,a^{3} b^{3} x^{8}+1680 C \,a^{4} b^{2} x^{8}-210 D a^{5} b \,x^{8}+6720 A \,a^{3} b^{3} x^{6}-2800 B \,a^{4} b^{2} x^{6}+840 C \,a^{5} b \,x^{6}-105 D a^{6} x^{6}+840 A \,a^{4} b^{2} x^{4}-350 B \,a^{5} b \,x^{4}+105 C \,a^{6} x^{4}-84 A \,a^{5} b \,x^{2}+35 B \,a^{6} x^{2}+21 A \,a^{6}}{105 x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{7}}\)	\(253\)
default	\(D \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )+A \left (-\frac {1}{5 a \,x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {12 b \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {10 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )}{3 a}\right )}{5 a}\right )+B \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {10 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )}{3 a}\right )+C \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )\)	\(446\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^6+C*x^4+B*x^2+A)/x^6/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)

[Out]

D*(1/7*x/a/(b*x^2+a)^(7/2)+6/7/a*(1/5*x/a/(b*x^2+a)^(5/2)+4/5/a*(1/3*x/a/(b*x^2+a)^(3/2)+2/3*x/a^2/(b*x^2+a)^(
1/2))))+A*(-1/5/a/x^5/(b*x^2+a)^(7/2)-12/5*b/a*(-1/3/a/x^3/(b*x^2+a)^(7/2)-10/3*b/a*(-1/a/x/(b*x^2+a)^(7/2)-8*
b/a*(1/7*x/a/(b*x^2+a)^(7/2)+6/7/a*(1/5*x/a/(b*x^2+a)^(5/2)+4/5/a*(1/3*x/a/(b*x^2+a)^(3/2)+2/3*x/a^2/(b*x^2+a)
^(1/2)))))))+B*(-1/3/a/x^3/(b*x^2+a)^(7/2)-10/3*b/a*(-1/a/x/(b*x^2+a)^(7/2)-8*b/a*(1/7*x/a/(b*x^2+a)^(7/2)+6/7
/a*(1/5*x/a/(b*x^2+a)^(5/2)+4/5/a*(1/3*x/a/(b*x^2+a)^(3/2)+2/3*x/a^2/(b*x^2+a)^(1/2))))))+C*(-1/a/x/(b*x^2+a)^
(7/2)-8*b/a*(1/7*x/a/(b*x^2+a)^(7/2)+6/7/a*(1/5*x/a/(b*x^2+a)^(5/2)+4/5/a*(1/3*x/a/(b*x^2+a)^(3/2)+2/3*x/a^2/(
b*x^2+a)^(1/2)))))

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Maxima [A]
time = 0.29, size = 398, normalized size = 1.42 \begin {gather*} \frac {16 \, D x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, D x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, D x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {D x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {128 \, C b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, C b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, C b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, C b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} + \frac {256 \, B b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, B b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, B b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, B b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} - \frac {1024 \, A b^{3} x}{35 \, \sqrt {b x^{2} + a} a^{7}} - \frac {512 \, A b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{6}} - \frac {384 \, A b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{5}} - \frac {64 \, A b^{3} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4}} - \frac {C}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} + \frac {10 \, B b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {8 \, A b^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x} - \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} + \frac {4 \, A b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{3}} - \frac {A}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

16/35*D*x/(sqrt(b*x^2 + a)*a^4) + 8/35*D*x/((b*x^2 + a)^(3/2)*a^3) + 6/35*D*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*D*
x/((b*x^2 + a)^(7/2)*a) - 128/35*C*b*x/(sqrt(b*x^2 + a)*a^5) - 64/35*C*b*x/((b*x^2 + a)^(3/2)*a^4) - 48/35*C*b
*x/((b*x^2 + a)^(5/2)*a^3) - 8/7*C*b*x/((b*x^2 + a)^(7/2)*a^2) + 256/21*B*b^2*x/(sqrt(b*x^2 + a)*a^6) + 128/21
*B*b^2*x/((b*x^2 + a)^(3/2)*a^5) + 32/7*B*b^2*x/((b*x^2 + a)^(5/2)*a^4) + 80/21*B*b^2*x/((b*x^2 + a)^(7/2)*a^3
) - 1024/35*A*b^3*x/(sqrt(b*x^2 + a)*a^7) - 512/35*A*b^3*x/((b*x^2 + a)^(3/2)*a^6) - 384/35*A*b^3*x/((b*x^2 +
a)^(5/2)*a^5) - 64/7*A*b^3*x/((b*x^2 + a)^(7/2)*a^4) - C/((b*x^2 + a)^(7/2)*a*x) + 10/3*B*b/((b*x^2 + a)^(7/2)
*a^2*x) - 8*A*b^2/((b*x^2 + a)^(7/2)*a^3*x) - 1/3*B/((b*x^2 + a)^(7/2)*a*x^3) + 4/5*A*b/((b*x^2 + a)^(7/2)*a^2
*x^3) - 1/5*A/((b*x^2 + a)^(7/2)*a*x^5)

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Fricas [A]
time = 5.53, size = 270, normalized size = 0.96 \begin {gather*} \frac {{\left (16 \, {\left (3 \, D a^{3} b^{3} - 24 \, C a^{2} b^{4} + 80 \, B a b^{5} - 192 \, A b^{6}\right )} x^{12} + 56 \, {\left (3 \, D a^{4} b^{2} - 24 \, C a^{3} b^{3} + 80 \, B a^{2} b^{4} - 192 \, A a b^{5}\right )} x^{10} + 70 \, {\left (3 \, D a^{5} b - 24 \, C a^{4} b^{2} + 80 \, B a^{3} b^{3} - 192 \, A a^{2} b^{4}\right )} x^{8} - 21 \, A a^{6} + 35 \, {\left (3 \, D a^{6} - 24 \, C a^{5} b + 80 \, B a^{4} b^{2} - 192 \, A a^{3} b^{3}\right )} x^{6} - 35 \, {\left (3 \, C a^{6} - 10 \, B a^{5} b + 24 \, A a^{4} b^{2}\right )} x^{4} - 7 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{7} b^{4} x^{13} + 4 \, a^{8} b^{3} x^{11} + 6 \, a^{9} b^{2} x^{9} + 4 \, a^{10} b x^{7} + a^{11} x^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/105*(16*(3*D*a^3*b^3 - 24*C*a^2*b^4 + 80*B*a*b^5 - 192*A*b^6)*x^12 + 56*(3*D*a^4*b^2 - 24*C*a^3*b^3 + 80*B*a
^2*b^4 - 192*A*a*b^5)*x^10 + 70*(3*D*a^5*b - 24*C*a^4*b^2 + 80*B*a^3*b^3 - 192*A*a^2*b^4)*x^8 - 21*A*a^6 + 35*
(3*D*a^6 - 24*C*a^5*b + 80*B*a^4*b^2 - 192*A*a^3*b^3)*x^6 - 35*(3*C*a^6 - 10*B*a^5*b + 24*A*a^4*b^2)*x^4 - 7*(
5*B*a^6 - 12*A*a^5*b)*x^2)*sqrt(b*x^2 + a)/(a^7*b^4*x^13 + 4*a^8*b^3*x^11 + 6*a^9*b^2*x^9 + 4*a^10*b*x^7 + a^1
1*x^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3313 vs. \(2 (277) = 554\).
time = 202.10, size = 3313, normalized size = 11.79 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x**6+C*x**4+B*x**2+A)/x**6/(b*x**2+a)**(9/2),x)

[Out]

A*(-7*a**8*b**(73/2)*sqrt(a/(b*x**2) + 1)/(35*a**13*b**36*x**4 + 210*a**12*b**37*x**6 + 525*a**11*b**38*x**8 +
700*a**10*b**39*x**10 + 525*a**9*b**40*x**12 + 210*a**8*b**41*x**14 + 35*a**7*b**42*x**16) + 14*a**7*b**(75/2
)*x**2*sqrt(a/(b*x**2) + 1)/(35*a**13*b**36*x**4 + 210*a**12*b**37*x**6 + 525*a**11*b**38*x**8 + 700*a**10*b**
39*x**10 + 525*a**9*b**40*x**12 + 210*a**8*b**41*x**14 + 35*a**7*b**42*x**16) - 231*a**6*b**(77/2)*x**4*sqrt(a
/(b*x**2) + 1)/(35*a**13*b**36*x**4 + 210*a**12*b**37*x**6 + 525*a**11*b**38*x**8 + 700*a**10*b**39*x**10 + 52
5*a**9*b**40*x**12 + 210*a**8*b**41*x**14 + 35*a**7*b**42*x**16) - 2772*a**5*b**(79/2)*x**6*sqrt(a/(b*x**2) +
1)/(35*a**13*b**36*x**4 + 210*a**12*b**37*x**6 + 525*a**11*b**38*x**8 + 700*a**10*b**39*x**10 + 525*a**9*b**40
*x**12 + 210*a**8*b**41*x**14 + 35*a**7*b**42*x**16) - 9240*a**4*b**(81/2)*x**8*sqrt(a/(b*x**2) + 1)/(35*a**13
*b**36*x**4 + 210*a**12*b**37*x**6 + 525*a**11*b**38*x**8 + 700*a**10*b**39*x**10 + 525*a**9*b**40*x**12 + 210
*a**8*b**41*x**14 + 35*a**7*b**42*x**16) - 14784*a**3*b**(83/2)*x**10*sqrt(a/(b*x**2) + 1)/(35*a**13*b**36*x**
4 + 210*a**12*b**37*x**6 + 525*a**11*b**38*x**8 + 700*a**10*b**39*x**10 + 525*a**9*b**40*x**12 + 210*a**8*b**4
1*x**14 + 35*a**7*b**42*x**16) - 12672*a**2*b**(85/2)*x**12*sqrt(a/(b*x**2) + 1)/(35*a**13*b**36*x**4 + 210*a*
*12*b**37*x**6 + 525*a**11*b**38*x**8 + 700*a**10*b**39*x**10 + 525*a**9*b**40*x**12 + 210*a**8*b**41*x**14 +
35*a**7*b**42*x**16) - 5632*a*b**(87/2)*x**14*sqrt(a/(b*x**2) + 1)/(35*a**13*b**36*x**4 + 210*a**12*b**37*x**6
+ 525*a**11*b**38*x**8 + 700*a**10*b**39*x**10 + 525*a**9*b**40*x**12 + 210*a**8*b**41*x**14 + 35*a**7*b**42*
x**16) - 1024*b**(89/2)*x**16*sqrt(a/(b*x**2) + 1)/(35*a**13*b**36*x**4 + 210*a**12*b**37*x**6 + 525*a**11*b**
38*x**8 + 700*a**10*b**39*x**10 + 525*a**9*b**40*x**12 + 210*a**8*b**41*x**14 + 35*a**7*b**42*x**16)) + B*(-7*
a**6*b**(51/2)*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a*
*8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 63*a**5*b**(53/2)*x**2*sqrt(a/(b*x**2) + 1)/(21*
a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 2
1*a**6*b**30*x**12) + 630*a**4*b**(55/2)*x**4*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4
+ 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 1680*a**3*b**(57/
2)*x**6*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**2
8*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 2016*a**2*b**(59/2)*x**8*sqrt(a/(b*x**2) + 1)/(21*a**11
*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**
6*b**30*x**12) + 1152*a*b**(61/2)*x**10*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210
*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 256*b**(63/2)*x**12*sqr
t(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 10
5*a**7*b**29*x**10 + 21*a**6*b**30*x**12)) + C*(-35*a**4*b**(33/2)*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a
**8*b**17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 280*a**3*b**(35/2)*x**2*sqr
t(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b
**20*x**8) - 560*a**2*b**(37/2)*x**4*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x**2 + 210*a**7*b**1
8*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 448*a*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16
+ 140*a**8*b**17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 128*b**(41/2)*x**8*s
qrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5
*b**20*x**8)) + D*(35*a**14*x/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525
*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8
*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)
) + 175*a**13*b*x**3/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2
)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 +
b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 371*a
**12*b**2*x**5/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2
*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**
2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 429*a**11*b
**3*x**7/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*
sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sq...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (252) = 504\).
time = 2.56, size = 592, normalized size = 2.11 \begin {gather*} \frac {{\left ({\left (x^{2} {\left (\frac {{\left (48 \, D a^{18} b^{6} - 279 \, C a^{17} b^{7} + 790 \, B a^{16} b^{8} - 1686 \, A a^{15} b^{9}\right )} x^{2}}{a^{22} b^{3}} + \frac {7 \, {\left (24 \, D a^{19} b^{5} - 132 \, C a^{18} b^{6} + 365 \, B a^{17} b^{7} - 768 \, A a^{16} b^{8}\right )}}{a^{22} b^{3}}\right )} + \frac {35 \, {\left (6 \, D a^{20} b^{4} - 30 \, C a^{19} b^{5} + 80 \, B a^{18} b^{6} - 165 \, A a^{17} b^{7}\right )}}{a^{22} b^{3}}\right )} x^{2} + \frac {105 \, {\left (D a^{21} b^{3} - 4 \, C a^{20} b^{4} + 10 \, B a^{19} b^{5} - 20 \, A a^{18} b^{6}\right )}}{a^{22} b^{3}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, {\left (15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} C a^{2} \sqrt {b} - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a b^{\frac {3}{2}} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A b^{\frac {5}{2}} - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} C a^{3} \sqrt {b} + 270 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {3}{2}} - 720 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a b^{\frac {5}{2}} + 90 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a^{4} \sqrt {b} - 430 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {3}{2}} + 1260 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {5}{2}} - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{5} \sqrt {b} + 290 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {3}{2}} - 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {5}{2}} + 15 \, C a^{6} \sqrt {b} - 70 \, B a^{5} b^{\frac {3}{2}} + 198 \, A a^{4} b^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/105*((x^2*((48*D*a^18*b^6 - 279*C*a^17*b^7 + 790*B*a^16*b^8 - 1686*A*a^15*b^9)*x^2/(a^22*b^3) + 7*(24*D*a^19
*b^5 - 132*C*a^18*b^6 + 365*B*a^17*b^7 - 768*A*a^16*b^8)/(a^22*b^3)) + 35*(6*D*a^20*b^4 - 30*C*a^19*b^5 + 80*B
*a^18*b^6 - 165*A*a^17*b^7)/(a^22*b^3))*x^2 + 105*(D*a^21*b^3 - 4*C*a^20*b^4 + 10*B*a^19*b^5 - 20*A*a^18*b^6)/
(a^22*b^3))*x/(b*x^2 + a)^(7/2) + 2/15*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^2*sqrt(b) - 60*(sqrt(b)*x - sqr
t(b*x^2 + a))^8*B*a*b^(3/2) + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(5/2) - 60*(sqrt(b)*x - sqrt(b*x^2 + a))
^6*C*a^3*sqrt(b) + 270*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2) - 720*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a
*b^(5/2) + 90*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^4*sqrt(b) - 430*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(3/2
) + 1260*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^2*b^(5/2) - 60*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^5*sqrt(b) + 29
0*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3/2) - 840*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^3*b^(5/2) + 15*C*a^6
*sqrt(b) - 70*B*a^5*b^(3/2) + 198*A*a^4*b^(5/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*a^6)

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Mupad [B]
time = 2.40, size = 405, normalized size = 1.44 \begin {gather*} \frac {\frac {61\,A\,b}{35\,a^3}+\frac {78\,A\,b^2\,x^2}{35\,a^4}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {\frac {128\,B\,b}{21\,a^5}+\frac {256\,B\,b^2\,x^2}{21\,a^6}}{x\,\sqrt {b\,x^2+a}}+\frac {x\,D}{{\left (b\,x^2+a\right )}^{9/2}}-\frac {\frac {B}{3\,a^2}+\frac {19\,B\,b\,x^2}{21\,a^3}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {\frac {C}{a^4}+\frac {128\,C\,b\,x^2}{35\,a^5}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {512\,A\,b^2}{35\,a^6}+\frac {1024\,A\,b^3\,x^2}{35\,a^7}}{x\,\sqrt {b\,x^2+a}}-\frac {A\,\sqrt {b\,x^2+a}}{5\,a^5\,x^5}+\frac {18\,b^2\,x^5\,D}{5\,a^2\,{\left (b\,x^2+a\right )}^{9/2}}+\frac {72\,b^3\,x^7\,D}{35\,a^3\,{\left (b\,x^2+a\right )}^{9/2}}+\frac {16\,b^4\,x^9\,D}{35\,a^4\,{\left (b\,x^2+a\right )}^{9/2}}-\frac {A\,b}{7\,a^2\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {32\,B\,b}{21\,a^4\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {B\,b^2\,x}{7\,a^3\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {27\,A\,b^2}{7\,a^5\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {3\,b\,x^3\,D}{a\,{\left (b\,x^2+a\right )}^{9/2}}-\frac {29\,C\,b\,x}{35\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {13\,C\,b\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {C\,b\,x}{7\,a^2\,{\left (b\,x^2+a\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2 + C*x^4 + x^6*D)/(x^6*(a + b*x^2)^(9/2)),x)

[Out]

((61*A*b)/(35*a^3) + (78*A*b^2*x^2)/(35*a^4))/(x^3*(a + b*x^2)^(5/2)) + ((128*B*b)/(21*a^5) + (256*B*b^2*x^2)/
(21*a^6))/(x*(a + b*x^2)^(1/2)) + (x*D)/(a + b*x^2)^(9/2) - (B/(3*a^2) + (19*B*b*x^2)/(21*a^3))/(x^3*(a + b*x^
2)^(5/2)) - (C/a^4 + (128*C*b*x^2)/(35*a^5))/(x*(a + b*x^2)^(1/2)) - ((512*A*b^2)/(35*a^6) + (1024*A*b^3*x^2)/
(35*a^7))/(x*(a + b*x^2)^(1/2)) - (A*(a + b*x^2)^(1/2))/(5*a^5*x^5) + (18*b^2*x^5*D)/(5*a^2*(a + b*x^2)^(9/2))
+ (72*b^3*x^7*D)/(35*a^3*(a + b*x^2)^(9/2)) + (16*b^4*x^9*D)/(35*a^4*(a + b*x^2)^(9/2)) - (A*b)/(7*a^2*x^3*(a
+ b*x^2)^(7/2)) - (32*B*b)/(21*a^4*x*(a + b*x^2)^(3/2)) + (B*b^2*x)/(7*a^3*(a + b*x^2)^(7/2)) + (27*A*b^2)/(7
*a^5*x*(a + b*x^2)^(3/2)) + (3*b*x^3*D)/(a*(a + b*x^2)^(9/2)) - (29*C*b*x)/(35*a^4*(a + b*x^2)^(3/2)) - (13*C*
b*x)/(35*a^3*(a + b*x^2)^(5/2)) - (C*b*x)/(7*a^2*(a + b*x^2)^(7/2))

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