3.2.63 \(\int \frac {A+B x^2+C x^4+D x^6}{x^8 (a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=334 \[ -\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}+\frac {128 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^8 \sqrt {a+b x^2}}+\frac {64 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac {16 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {8 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}} \]

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Rubi [A]  time = 0.48, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1803, 12, 271, 192, 191} \begin {gather*} \frac {128 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^8 \sqrt {a+b x^2}}+\frac {64 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac {16 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {8 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {A}{7 a x^7 \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^8*(a + b*x^2)^(9/2)),x]

[Out]

-A/(7*a*x^7*(a + b*x^2)^(7/2)) + (2*A*b - a*B)/(5*a^2*x^5*(a + b*x^2)^(7/2)) - (24*A*b^2 - a*(12*b*B - 5*a*C))
/(15*a^3*x^3*(a + b*x^2)^(7/2)) + (48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))/(3*a^4*x*(a + b*x^2)^(7/2)) +
(8*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(21*a^5*(a + b*x^2)^(7/2)) + (16*b*(48*A*b^3 - a*(24*b
^2*B - 10*a*b*C + 3*a^2*D))*x)/(35*a^6*(a + b*x^2)^(5/2)) + (64*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D
))*x)/(105*a^7*(a + b*x^2)^(3/2)) + (128*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(105*a^8*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 191

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 192

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 271

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

Rule 1803

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^8 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {14 A b-7 a \left (B+C x^2+D x^4\right )}{x^6 \left (a+b x^2\right )^{9/2}} \, dx}{7 a}\\ &=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}+\frac {\int \frac {12 b (14 A b-7 a B)-5 a \left (-7 a C-7 a D x^2\right )}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{35 a^2}\\ &=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {10 b \left (168 A b^2-84 a b B+35 a^2 C\right )-105 a^3 D}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{105 a^3}\\ &=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}-\frac {\left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{3 a^3}\\ &=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {\left (8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^4}\\ &=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {\left (16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^5}\\ &=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {\left (64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^6}\\ &=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac {\left (128 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^7}\\ &=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac {128 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^8 \sqrt {a+b x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 234, normalized size = 0.70 \begin {gather*} \frac {-a^7 \left (15 A+21 B x^2+35 x^4 \left (C+3 D x^2\right )\right )+14 a^6 b x^2 \left (3 A+6 B x^2+25 C x^4-60 D x^6\right )-56 a^5 b^2 x^4 \left (3 A+15 B x^2-50 C x^4+30 D x^6\right )+112 a^4 b^3 x^6 \left (15 A-60 B x^2+50 C x^4-12 D x^6\right )+128 a^3 b^4 x^8 \left (105 A-105 B x^2+35 C x^4-3 D x^6\right )+256 a^2 b^5 x^{10} \left (105 A-42 B x^2+5 C x^4\right )-3072 a b^6 x^{12} \left (B x^2-7 A\right )+6144 A b^7 x^{14}}{105 a^8 x^7 \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^8*(a + b*x^2)^(9/2)),x]

[Out]

(6144*A*b^7*x^14 - 3072*a*b^6*x^12*(-7*A + B*x^2) + 256*a^2*b^5*x^10*(105*A - 42*B*x^2 + 5*C*x^4) + 14*a^6*b*x
^2*(3*A + 6*B*x^2 + 25*C*x^4 - 60*D*x^6) + 112*a^4*b^3*x^6*(15*A - 60*B*x^2 + 50*C*x^4 - 12*D*x^6) + 128*a^3*b
^4*x^8*(105*A - 105*B*x^2 + 35*C*x^4 - 3*D*x^6) - 56*a^5*b^2*x^4*(3*A + 15*B*x^2 - 50*C*x^4 + 30*D*x^6) - a^7*
(15*A + 21*B*x^2 + 35*x^4*(C + 3*D*x^2)))/(105*a^8*x^7*(a + b*x^2)^(7/2))

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IntegrateAlgebraic [A]  time = 0.72, size = 304, normalized size = 0.91 \begin {gather*} \frac {-15 a^7 A-21 a^7 B x^2-35 a^7 C x^4-105 a^7 D x^6+42 a^6 A b x^2+84 a^6 b B x^4+350 a^6 b C x^6-840 a^6 b D x^8-168 a^5 A b^2 x^4-840 a^5 b^2 B x^6+2800 a^5 b^2 C x^8-1680 a^5 b^2 D x^{10}+1680 a^4 A b^3 x^6-6720 a^4 b^3 B x^8+5600 a^4 b^3 C x^{10}-1344 a^4 b^3 D x^{12}+13440 a^3 A b^4 x^8-13440 a^3 b^4 B x^{10}+4480 a^3 b^4 C x^{12}-384 a^3 b^4 D x^{14}+26880 a^2 A b^5 x^{10}-10752 a^2 b^5 B x^{12}+1280 a^2 b^5 C x^{14}+21504 a A b^6 x^{12}-3072 a b^6 B x^{14}+6144 A b^7 x^{14}}{105 a^8 x^7 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x^2 + C*x^4 + D*x^6)/(x^8*(a + b*x^2)^(9/2)),x]

[Out]

(-15*a^7*A + 42*a^6*A*b*x^2 - 21*a^7*B*x^2 - 168*a^5*A*b^2*x^4 + 84*a^6*b*B*x^4 - 35*a^7*C*x^4 + 1680*a^4*A*b^
3*x^6 - 840*a^5*b^2*B*x^6 + 350*a^6*b*C*x^6 - 105*a^7*D*x^6 + 13440*a^3*A*b^4*x^8 - 6720*a^4*b^3*B*x^8 + 2800*
a^5*b^2*C*x^8 - 840*a^6*b*D*x^8 + 26880*a^2*A*b^5*x^10 - 13440*a^3*b^4*B*x^10 + 5600*a^4*b^3*C*x^10 - 1680*a^5
*b^2*D*x^10 + 21504*a*A*b^6*x^12 - 10752*a^2*b^5*B*x^12 + 4480*a^3*b^4*C*x^12 - 1344*a^4*b^3*D*x^12 + 6144*A*b
^7*x^14 - 3072*a*b^6*B*x^14 + 1280*a^2*b^5*C*x^14 - 384*a^3*b^4*D*x^14)/(105*a^8*x^7*(a + b*x^2)^(7/2))

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fricas [A]  time = 2.90, size = 311, normalized size = 0.93 \begin {gather*} -\frac {{\left (128 \, {\left (3 \, D a^{3} b^{4} - 10 \, C a^{2} b^{5} + 24 \, B a b^{6} - 48 \, A b^{7}\right )} x^{14} + 448 \, {\left (3 \, D a^{4} b^{3} - 10 \, C a^{3} b^{4} + 24 \, B a^{2} b^{5} - 48 \, A a b^{6}\right )} x^{12} + 560 \, {\left (3 \, D a^{5} b^{2} - 10 \, C a^{4} b^{3} + 24 \, B a^{3} b^{4} - 48 \, A a^{2} b^{5}\right )} x^{10} + 280 \, {\left (3 \, D a^{6} b - 10 \, C a^{5} b^{2} + 24 \, B a^{4} b^{3} - 48 \, A a^{3} b^{4}\right )} x^{8} + 15 \, A a^{7} + 35 \, {\left (3 \, D a^{7} - 10 \, C a^{6} b + 24 \, B a^{5} b^{2} - 48 \, A a^{4} b^{3}\right )} x^{6} + 7 \, {\left (5 \, C a^{7} - 12 \, B a^{6} b + 24 \, A a^{5} b^{2}\right )} x^{4} + 21 \, {\left (B a^{7} - 2 \, A a^{6} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{8} b^{4} x^{15} + 4 \, a^{9} b^{3} x^{13} + 6 \, a^{10} b^{2} x^{11} + 4 \, a^{11} b x^{9} + a^{12} x^{7}\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^8/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

-1/105*(128*(3*D*a^3*b^4 - 10*C*a^2*b^5 + 24*B*a*b^6 - 48*A*b^7)*x^14 + 448*(3*D*a^4*b^3 - 10*C*a^3*b^4 + 24*B
*a^2*b^5 - 48*A*a*b^6)*x^12 + 560*(3*D*a^5*b^2 - 10*C*a^4*b^3 + 24*B*a^3*b^4 - 48*A*a^2*b^5)*x^10 + 280*(3*D*a
^6*b - 10*C*a^5*b^2 + 24*B*a^4*b^3 - 48*A*a^3*b^4)*x^8 + 15*A*a^7 + 35*(3*D*a^7 - 10*C*a^6*b + 24*B*a^5*b^2 -
48*A*a^4*b^3)*x^6 + 7*(5*C*a^7 - 12*B*a^6*b + 24*A*a^5*b^2)*x^4 + 21*(B*a^7 - 2*A*a^6*b)*x^2)*sqrt(b*x^2 + a)/
(a^8*b^4*x^15 + 4*a^9*b^3*x^13 + 6*a^10*b^2*x^11 + 4*a^11*b*x^9 + a^12*x^7)

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giac [B]  time = 0.71, size = 938, normalized size = 2.81 \begin {gather*} -\frac {{\left ({\left (x^{2} {\left (\frac {{\left (279 \, D a^{21} b^{7} - 790 \, C a^{20} b^{8} + 1686 \, B a^{19} b^{9} - 3072 \, A a^{18} b^{10}\right )} x^{2}}{a^{26} b^{3}} + \frac {7 \, {\left (132 \, D a^{22} b^{6} - 365 \, C a^{21} b^{7} + 768 \, B a^{20} b^{8} - 1386 \, A a^{19} b^{9}\right )}}{a^{26} b^{3}}\right )} + \frac {35 \, {\left (30 \, D a^{23} b^{5} - 80 \, C a^{22} b^{6} + 165 \, B a^{21} b^{7} - 294 \, A a^{20} b^{8}\right )}}{a^{26} b^{3}}\right )} x^{2} + \frac {105 \, {\left (4 \, D a^{24} b^{4} - 10 \, C a^{23} b^{5} + 20 \, B a^{22} b^{6} - 35 \, A a^{21} b^{7}\right )}}{a^{26} b^{3}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, {\left (105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} D a^{3} \sqrt {b} - 420 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} C a^{2} b^{\frac {3}{2}} + 1050 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a b^{\frac {5}{2}} - 2100 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A b^{\frac {7}{2}} - 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} D a^{4} \sqrt {b} + 2730 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} C a^{3} b^{\frac {3}{2}} - 7140 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{2} b^{\frac {5}{2}} + 14700 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a b^{\frac {7}{2}} + 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} D a^{5} \sqrt {b} - 7210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} C a^{4} b^{\frac {3}{2}} + 19950 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{3} b^{\frac {5}{2}} - 42840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{2} b^{\frac {7}{2}} - 2100 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} D a^{6} \sqrt {b} + 9940 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} C a^{5} b^{\frac {3}{2}} - 28560 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{4} b^{\frac {5}{2}} + 64680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{3} b^{\frac {7}{2}} + 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} D a^{7} \sqrt {b} - 7560 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a^{6} b^{\frac {3}{2}} + 21966 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{5} b^{\frac {5}{2}} - 49812 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{4} b^{\frac {7}{2}} - 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} D a^{8} \sqrt {b} + 3010 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{7} b^{\frac {3}{2}} - 8652 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{6} b^{\frac {5}{2}} + 19404 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{5} b^{\frac {7}{2}} + 105 \, D a^{9} \sqrt {b} - 490 \, C a^{8} b^{\frac {3}{2}} + 1386 \, B a^{7} b^{\frac {5}{2}} - 3072 \, A a^{6} b^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7} a^{7}} \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^8/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

-1/105*((x^2*((279*D*a^21*b^7 - 790*C*a^20*b^8 + 1686*B*a^19*b^9 - 3072*A*a^18*b^10)*x^2/(a^26*b^3) + 7*(132*D
*a^22*b^6 - 365*C*a^21*b^7 + 768*B*a^20*b^8 - 1386*A*a^19*b^9)/(a^26*b^3)) + 35*(30*D*a^23*b^5 - 80*C*a^22*b^6
+ 165*B*a^21*b^7 - 294*A*a^20*b^8)/(a^26*b^3))*x^2 + 105*(4*D*a^24*b^4 - 10*C*a^23*b^5 + 20*B*a^22*b^6 - 35*A
*a^21*b^7)/(a^26*b^3))*x/(b*x^2 + a)^(7/2) + 2/105*(105*(sqrt(b)*x - sqrt(b*x^2 + a))^12*D*a^3*sqrt(b) - 420*(
sqrt(b)*x - sqrt(b*x^2 + a))^12*C*a^2*b^(3/2) + 1050*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a*b^(5/2) - 2100*(sqrt
(b)*x - sqrt(b*x^2 + a))^12*A*b^(7/2) - 630*(sqrt(b)*x - sqrt(b*x^2 + a))^10*D*a^4*sqrt(b) + 2730*(sqrt(b)*x -
sqrt(b*x^2 + a))^10*C*a^3*b^(3/2) - 7140*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2*b^(5/2) + 14700*(sqrt(b)*x -
sqrt(b*x^2 + a))^10*A*a*b^(7/2) + 1575*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D*a^5*sqrt(b) - 7210*(sqrt(b)*x - sqrt(
b*x^2 + a))^8*C*a^4*b^(3/2) + 19950*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^3*b^(5/2) - 42840*(sqrt(b)*x - sqrt(b*
x^2 + a))^8*A*a^2*b^(7/2) - 2100*(sqrt(b)*x - sqrt(b*x^2 + a))^6*D*a^6*sqrt(b) + 9940*(sqrt(b)*x - sqrt(b*x^2
+ a))^6*C*a^5*b^(3/2) - 28560*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^4*b^(5/2) + 64680*(sqrt(b)*x - sqrt(b*x^2 +
a))^6*A*a^3*b^(7/2) + 1575*(sqrt(b)*x - sqrt(b*x^2 + a))^4*D*a^7*sqrt(b) - 7560*(sqrt(b)*x - sqrt(b*x^2 + a))^
4*C*a^6*b^(3/2) + 21966*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^5*b^(5/2) - 49812*(sqrt(b)*x - sqrt(b*x^2 + a))^4*
A*a^4*b^(7/2) - 630*(sqrt(b)*x - sqrt(b*x^2 + a))^2*D*a^8*sqrt(b) + 3010*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^7
*b^(3/2) - 8652*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^6*b^(5/2) + 19404*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^5*b^
(7/2) + 105*D*a^9*sqrt(b) - 490*C*a^8*b^(3/2) + 1386*B*a^7*b^(5/2) - 3072*A*a^6*b^(7/2))/(((sqrt(b)*x - sqrt(b
*x^2 + a))^2 - a)^7*a^7)

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maple [A]  time = 0.01, size = 301, normalized size = 0.90 \begin {gather*} -\frac {-6144 A \,b^{7} x^{14}+3072 B a \,b^{6} x^{14}-1280 C \,a^{2} b^{5} x^{14}+384 D a^{3} b^{4} x^{14}-21504 A a \,b^{6} x^{12}+10752 B \,a^{2} b^{5} x^{12}-4480 C \,a^{3} b^{4} x^{12}+1344 D a^{4} b^{3} x^{12}-26880 A \,a^{2} b^{5} x^{10}+13440 B \,a^{3} b^{4} x^{10}-5600 C \,a^{4} b^{3} x^{10}+1680 D a^{5} b^{2} x^{10}-13440 A \,a^{3} b^{4} x^{8}+6720 B \,a^{4} b^{3} x^{8}-2800 C \,a^{5} b^{2} x^{8}+840 D a^{6} b \,x^{8}-1680 A \,a^{4} b^{3} x^{6}+840 B \,a^{5} b^{2} x^{6}-350 C \,a^{6} b \,x^{6}+105 D a^{7} x^{6}+168 A \,a^{5} b^{2} x^{4}-84 B \,a^{6} b \,x^{4}+35 C \,a^{7} x^{4}-42 A \,a^{6} b \,x^{2}+21 B \,a^{7} x^{2}+15 A \,a^{7}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{8} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^6+C*x^4+B*x^2+A)/x^8/(b*x^2+a)^(9/2),x)

[Out]

-1/105*(-6144*A*b^7*x^14+3072*B*a*b^6*x^14-1280*C*a^2*b^5*x^14+384*D*a^3*b^4*x^14-21504*A*a*b^6*x^12+10752*B*a
^2*b^5*x^12-4480*C*a^3*b^4*x^12+1344*D*a^4*b^3*x^12-26880*A*a^2*b^5*x^10+13440*B*a^3*b^4*x^10-5600*C*a^4*b^3*x
^10+1680*D*a^5*b^2*x^10-13440*A*a^3*b^4*x^8+6720*B*a^4*b^3*x^8-2800*C*a^5*b^2*x^8+840*D*a^6*b*x^8-1680*A*a^4*b
^3*x^6+840*B*a^5*b^2*x^6-350*C*a^6*b*x^6+105*D*a^7*x^6+168*A*a^5*b^2*x^4-84*B*a^6*b*x^4+35*C*a^7*x^4-42*A*a^6*
b*x^2+21*B*a^7*x^2+15*A*a^7)/(b*x^2+a)^(7/2)/x^7/a^8

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maxima [A]  time = 1.52, size = 489, normalized size = 1.46 \begin {gather*} -\frac {128 \, D b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, D b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, D b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, D b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} + \frac {256 \, C b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, C b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, C b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, C b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} - \frac {1024 \, B b^{3} x}{35 \, \sqrt {b x^{2} + a} a^{7}} - \frac {512 \, B b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{6}} - \frac {384 \, B b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{5}} - \frac {64 \, B b^{3} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4}} + \frac {2048 \, A b^{4} x}{35 \, \sqrt {b x^{2} + a} a^{8}} + \frac {1024 \, A b^{4} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{7}} + \frac {768 \, A b^{4} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{6}} + \frac {128 \, A b^{4} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{5}} - \frac {D}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} + \frac {10 \, C b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {8 \, B b^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x} + \frac {16 \, A b^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4} x} - \frac {C}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} + \frac {4 \, B b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{3}} - \frac {8 \, A b^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x^{3}} - \frac {B}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{5}} + \frac {2 \, A b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{5}} - \frac {A}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{7}} \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^8/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

-128/35*D*b*x/(sqrt(b*x^2 + a)*a^5) - 64/35*D*b*x/((b*x^2 + a)^(3/2)*a^4) - 48/35*D*b*x/((b*x^2 + a)^(5/2)*a^3
) - 8/7*D*b*x/((b*x^2 + a)^(7/2)*a^2) + 256/21*C*b^2*x/(sqrt(b*x^2 + a)*a^6) + 128/21*C*b^2*x/((b*x^2 + a)^(3/
2)*a^5) + 32/7*C*b^2*x/((b*x^2 + a)^(5/2)*a^4) + 80/21*C*b^2*x/((b*x^2 + a)^(7/2)*a^3) - 1024/35*B*b^3*x/(sqrt
(b*x^2 + a)*a^7) - 512/35*B*b^3*x/((b*x^2 + a)^(3/2)*a^6) - 384/35*B*b^3*x/((b*x^2 + a)^(5/2)*a^5) - 64/7*B*b^
3*x/((b*x^2 + a)^(7/2)*a^4) + 2048/35*A*b^4*x/(sqrt(b*x^2 + a)*a^8) + 1024/35*A*b^4*x/((b*x^2 + a)^(3/2)*a^7)
+ 768/35*A*b^4*x/((b*x^2 + a)^(5/2)*a^6) + 128/7*A*b^4*x/((b*x^2 + a)^(7/2)*a^5) - D/((b*x^2 + a)^(7/2)*a*x) +
10/3*C*b/((b*x^2 + a)^(7/2)*a^2*x) - 8*B*b^2/((b*x^2 + a)^(7/2)*a^3*x) + 16*A*b^3/((b*x^2 + a)^(7/2)*a^4*x) -
1/3*C/((b*x^2 + a)^(7/2)*a*x^3) + 4/5*B*b/((b*x^2 + a)^(7/2)*a^2*x^3) - 8/5*A*b^2/((b*x^2 + a)^(7/2)*a^3*x^3)
- 1/5*B/((b*x^2 + a)^(7/2)*a*x^5) + 2/5*A*b/((b*x^2 + a)^(7/2)*a^2*x^5) - 1/7*A/((b*x^2 + a)^(7/2)*a*x^7)

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mupad [B]  time = 2.84, size = 421, normalized size = 1.26 \begin {gather*} \frac {\frac {61\,B\,b}{35\,a^3}+\frac {78\,B\,b^2\,x^2}{35\,a^4}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {\frac {128\,C\,b}{21\,a^5}+\frac {256\,C\,b^2\,x^2}{21\,a^6}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {C}{3\,a^2}+\frac {19\,C\,b\,x^2}{21\,a^3}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {\frac {167\,A\,b^2}{35\,a^4}+\frac {191\,A\,b^3\,x^2}{35\,a^5}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {\frac {1024\,A\,b^3}{35\,a^7}+\frac {2048\,A\,b^4\,x^2}{35\,a^8}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {512\,B\,b^2}{35\,a^6}+\frac {1024\,B\,b^3\,x^2}{35\,a^7}}{x\,\sqrt {b\,x^2+a}}-\frac {A\,\sqrt {b\,x^2+a}}{7\,a^5\,x^7}-\frac {B\,\sqrt {b\,x^2+a}}{5\,a^5\,x^5}-\frac {{\left (\frac {a}{b\,x^2}+1\right )}^{9/2}\,D\,{{}}_2{\mathrm {F}}_1\left (\frac {9}{2},5;\ 6;\ -\frac {a}{b\,x^2}\right )}{10\,x\,{\left (b\,x^2+a\right )}^{9/2}}+\frac {34\,A\,b\,\sqrt {b\,x^2+a}}{35\,a^6\,x^5}-\frac {B\,b}{7\,a^2\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {32\,C\,b}{21\,a^4\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {C\,b^2\,x}{7\,a^3\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {58\,A\,b^3}{7\,a^6\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {A\,b^2}{7\,a^3\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {27\,B\,b^2}{7\,a^5\,x\,{\left (b\,x^2+a\right )}^{3/2}} \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^8*(a + b*x^2)^(9/2)),x)

[Out]

((61*B*b)/(35*a^3) + (78*B*b^2*x^2)/(35*a^4))/(x^3*(a + b*x^2)^(5/2)) + ((128*C*b)/(21*a^5) + (256*C*b^2*x^2)/
(21*a^6))/(x*(a + b*x^2)^(1/2)) - (C/(3*a^2) + (19*C*b*x^2)/(21*a^3))/(x^3*(a + b*x^2)^(5/2)) - ((167*A*b^2)/(
35*a^4) + (191*A*b^3*x^2)/(35*a^5))/(x^3*(a + b*x^2)^(5/2)) + ((1024*A*b^3)/(35*a^7) + (2048*A*b^4*x^2)/(35*a^
8))/(x*(a + b*x^2)^(1/2)) - ((512*B*b^2)/(35*a^6) + (1024*B*b^3*x^2)/(35*a^7))/(x*(a + b*x^2)^(1/2)) - (A*(a +
b*x^2)^(1/2))/(7*a^5*x^7) - (B*(a + b*x^2)^(1/2))/(5*a^5*x^5) - ((a/(b*x^2) + 1)^(9/2)*D*hypergeom([9/2, 5],
6, -a/(b*x^2)))/(10*x*(a + b*x^2)^(9/2)) + (34*A*b*(a + b*x^2)^(1/2))/(35*a^6*x^5) - (B*b)/(7*a^2*x^3*(a + b*x
^2)^(7/2)) - (32*C*b)/(21*a^4*x*(a + b*x^2)^(3/2)) + (C*b^2*x)/(7*a^3*(a + b*x^2)^(7/2)) - (58*A*b^3)/(7*a^6*x
*(a + b*x^2)^(3/2)) + (A*b^2)/(7*a^3*x^3*(a + b*x^2)^(7/2)) + (27*B*b^2)/(7*a^5*x*(a + b*x^2)^(3/2))

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sympy [F(-1)]  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((D*x**6+C*x**4+B*x**2+A)/x**8/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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