3.6.35 \(\int \frac {(a+b x^2)^{3/2} (A+B x^2)}{x^9} \, dx\) [535]

Optimal. Leaf size=156 \[ \frac {b (3 A b-8 a B) \sqrt {a+b x^2}}{64 a x^4}+\frac {b^2 (3 A b-8 a B) \sqrt {a+b x^2}}{128 a^2 x^2}+\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac {b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}} \]

[Out]

1/48*(3*A*b-8*B*a)*(b*x^2+a)^(3/2)/a/x^6-1/8*A*(b*x^2+a)^(5/2)/a/x^8-1/128*b^3*(3*A*b-8*B*a)*arctanh((b*x^2+a)
^(1/2)/a^(1/2))/a^(5/2)+1/64*b*(3*A*b-8*B*a)*(b*x^2+a)^(1/2)/a/x^4+1/128*b^2*(3*A*b-8*B*a)*(b*x^2+a)^(1/2)/a^2
/x^2

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Rubi [A]
time = 0.09, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {457, 79, 43, 44, 65, 214} \begin {gather*} -\frac {b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}}+\frac {b^2 \sqrt {a+b x^2} (3 A b-8 a B)}{128 a^2 x^2}+\frac {\left (a+b x^2\right )^{3/2} (3 A b-8 a B)}{48 a x^6}+\frac {b \sqrt {a+b x^2} (3 A b-8 a B)}{64 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^9,x]

[Out]

(b*(3*A*b - 8*a*B)*Sqrt[a + b*x^2])/(64*a*x^4) + (b^2*(3*A*b - 8*a*B)*Sqrt[a + b*x^2])/(128*a^2*x^2) + ((3*A*b
- 8*a*B)*(a + b*x^2)^(3/2))/(48*a*x^6) - (A*(a + b*x^2)^(5/2))/(8*a*x^8) - (b^3*(3*A*b - 8*a*B)*ArcTanh[Sqrt[
a + b*x^2]/Sqrt[a]])/(128*a^(5/2))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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 )^{3/2} \left (A+B x^2\right )}{x^9} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{3/2} (A+B x)}{x^5} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac {\left (-\frac {3 A b}{2}+4 a B\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^4} \, dx,x,x^2\right )}{8 a}\\ &=\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac {(b (3 A b-8 a B)) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,x^2\right )}{32 a}\\ &=\frac {b (3 A b-8 a B) \sqrt {a+b x^2}}{64 a x^4}+\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac {\left (b^2 (3 A b-8 a B)\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{128 a}\\ &=\frac {b (3 A b-8 a B) \sqrt {a+b x^2}}{64 a x^4}+\frac {b^2 (3 A b-8 a B) \sqrt {a+b x^2}}{128 a^2 x^2}+\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac {\left (b^3 (3 A b-8 a B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{256 a^2}\\ &=\frac {b (3 A b-8 a B) \sqrt {a+b x^2}}{64 a x^4}+\frac {b^2 (3 A b-8 a B) \sqrt {a+b x^2}}{128 a^2 x^2}+\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac {\left (b^2 (3 A b-8 a B)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{128 a^2}\\ &=\frac {b (3 A b-8 a B) \sqrt {a+b x^2}}{64 a x^4}+\frac {b^2 (3 A b-8 a B) \sqrt {a+b x^2}}{128 a^2 x^2}+\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac {b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 126, normalized size = 0.81 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-48 a^3 A-72 a^2 A b x^2-64 a^3 B x^2-6 a A b^2 x^4-112 a^2 b B x^4+9 A b^3 x^6-24 a b^2 B x^6\right )}{384 a^2 x^8}+\frac {b^3 (-3 A b+8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^9,x]

[Out]

(Sqrt[a + b*x^2]*(-48*a^3*A - 72*a^2*A*b*x^2 - 64*a^3*B*x^2 - 6*a*A*b^2*x^4 - 112*a^2*b*B*x^4 + 9*A*b^3*x^6 -
24*a*b^2*B*x^6))/(384*a^2*x^8) + (b^3*(-3*A*b + 8*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(128*a^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs. \(2(132)=264\).
time = 0.10, size = 278, normalized size = 1.78

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-9 x^{6} A \,b^{3}+24 x^{6} B a \,b^{2}+6 A a \,b^{2} x^{4}+112 x^{4} B \,a^{2} b +72 x^{2} A \,a^{2} b +64 B \,a^{3} x^{2}+48 A \,a^{3}\right )}{384 x^{8} a^{2}}-\frac {3 b^{4} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) A}{128 a^{\frac {5}{2}}}+\frac {b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) B}{16 a^{\frac {3}{2}}}\)	\(148\)
default	\(B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 a \,x^{8}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )\)	\(278\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(3/2)*(B*x^2+A)/x^9,x,method=_RETURNVERBOSE)

[Out]

B*(-1/6/a/x^6*(b*x^2+a)^(5/2)-1/6*b/a*(-1/4/a/x^4*(b*x^2+a)^(5/2)+1/4*b/a*(-1/2/a/x^2*(b*x^2+a)^(5/2)+3/2*b/a*
(1/3*(b*x^2+a)^(3/2)+a*((b*x^2+a)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x))))))+A*(-1/8/a/x^8*(b*x^
2+a)^(5/2)-3/8*b/a*(-1/6/a/x^6*(b*x^2+a)^(5/2)-1/6*b/a*(-1/4/a/x^4*(b*x^2+a)^(5/2)+1/4*b/a*(-1/2/a/x^2*(b*x^2+
a)^(5/2)+3/2*b/a*(1/3*(b*x^2+a)^(3/2)+a*((b*x^2+a)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)))))))

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Maxima [A]
time = 0.27, size = 252, normalized size = 1.62 \begin {gather*} \frac {B b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} - \frac {3 \, A b^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {5}{2}}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{3}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} B b^{3}}{16 \, a^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{4}}{128 \, a^{4}} + \frac {3 \, \sqrt {b x^{2} + a} A b^{4}}{128 \, a^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{64 \, a^{3} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{6 \, a x^{6}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{16 \, a^{2} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{8 \, a x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^9,x, algorithm="maxima")

[Out]

1/16*B*b^3*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 3/128*A*b^4*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) - 1/48*(b
*x^2 + a)^(3/2)*B*b^3/a^3 - 1/16*sqrt(b*x^2 + a)*B*b^3/a^2 + 1/128*(b*x^2 + a)^(3/2)*A*b^4/a^4 + 3/128*sqrt(b*
x^2 + a)*A*b^4/a^3 + 1/48*(b*x^2 + a)^(5/2)*B*b^2/(a^3*x^2) - 1/128*(b*x^2 + a)^(5/2)*A*b^3/(a^4*x^2) + 1/24*(
b*x^2 + a)^(5/2)*B*b/(a^2*x^4) - 1/64*(b*x^2 + a)^(5/2)*A*b^2/(a^3*x^4) - 1/6*(b*x^2 + a)^(5/2)*B/(a*x^6) + 1/
16*(b*x^2 + a)^(5/2)*A*b/(a^2*x^6) - 1/8*(b*x^2 + a)^(5/2)*A/(a*x^8)

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Fricas [A]
time = 1.68, size = 271, normalized size = 1.74 \begin {gather*} \left [-\frac {3 \, {\left (8 \, B a b^{3} - 3 \, A b^{4}\right )} \sqrt {a} x^{8} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (8 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{6} + 48 \, A a^{4} + 2 \, {\left (56 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{4} + 8 \, {\left (8 \, B a^{4} + 9 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{768 \, a^{3} x^{8}}, -\frac {3 \, {\left (8 \, B a b^{3} - 3 \, A b^{4}\right )} \sqrt {-a} x^{8} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, {\left (8 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{6} + 48 \, A a^{4} + 2 \, {\left (56 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{4} + 8 \, {\left (8 \, B a^{4} + 9 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{384 \, a^{3} x^{8}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^9,x, algorithm="fricas")

[Out]

[-1/768*(3*(8*B*a*b^3 - 3*A*b^4)*sqrt(a)*x^8*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(3*(8*B*a
^2*b^2 - 3*A*a*b^3)*x^6 + 48*A*a^4 + 2*(56*B*a^3*b + 3*A*a^2*b^2)*x^4 + 8*(8*B*a^4 + 9*A*a^3*b)*x^2)*sqrt(b*x^
2 + a))/(a^3*x^8), -1/384*(3*(8*B*a*b^3 - 3*A*b^4)*sqrt(-a)*x^8*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (3*(8*B*a^2
*b^2 - 3*A*a*b^3)*x^6 + 48*A*a^4 + 2*(56*B*a^3*b + 3*A*a^2*b^2)*x^4 + 8*(8*B*a^4 + 9*A*a^3*b)*x^2)*sqrt(b*x^2
+ a))/(a^3*x^8)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (141) = 282\).
time = 135.61, size = 287, normalized size = 1.84 \begin {gather*} - \frac {A a^{2}}{8 \sqrt {b} x^{9} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 A a \sqrt {b}}{16 x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {13 A b^{\frac {3}{2}}}{64 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {5}{2}}}{128 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 A b^{\frac {7}{2}}}{128 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 A b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{128 a^{\frac {5}{2}}} - \frac {B a^{2}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {11 B a \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 B b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B b^{\frac {5}{2}}}{16 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**9,x)

[Out]

-A*a**2/(8*sqrt(b)*x**9*sqrt(a/(b*x**2) + 1)) - 5*A*a*sqrt(b)/(16*x**7*sqrt(a/(b*x**2) + 1)) - 13*A*b**(3/2)/(
64*x**5*sqrt(a/(b*x**2) + 1)) + A*b**(5/2)/(128*a*x**3*sqrt(a/(b*x**2) + 1)) + 3*A*b**(7/2)/(128*a**2*x*sqrt(a
/(b*x**2) + 1)) - 3*A*b**4*asinh(sqrt(a)/(sqrt(b)*x))/(128*a**(5/2)) - B*a**2/(6*sqrt(b)*x**7*sqrt(a/(b*x**2)
+ 1)) - 11*B*a*sqrt(b)/(24*x**5*sqrt(a/(b*x**2) + 1)) - 17*B*b**(3/2)/(48*x**3*sqrt(a/(b*x**2) + 1)) - B*b**(5
/2)/(16*a*x*sqrt(a/(b*x**2) + 1)) + B*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(3/2))

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Giac [A]
time = 1.36, size = 194, normalized size = 1.24 \begin {gather*} -\frac {\frac {3 \, {\left (8 \, B a b^{4} - 3 \, A b^{5}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {24 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a b^{4} + 40 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} b^{4} - 88 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{3} b^{4} + 24 \, \sqrt {b x^{2} + a} B a^{4} b^{4} - 9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{5} + 33 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a b^{5} + 33 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{2} b^{5} - 9 \, \sqrt {b x^{2} + a} A a^{3} b^{5}}{a^{2} b^{4} x^{8}}}{384 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^9,x, algorithm="giac")

[Out]

-1/384*(3*(8*B*a*b^4 - 3*A*b^5)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (24*(b*x^2 + a)^(7/2)*B*a*b^
4 + 40*(b*x^2 + a)^(5/2)*B*a^2*b^4 - 88*(b*x^2 + a)^(3/2)*B*a^3*b^4 + 24*sqrt(b*x^2 + a)*B*a^4*b^4 - 9*(b*x^2
+ a)^(7/2)*A*b^5 + 33*(b*x^2 + a)^(5/2)*A*a*b^5 + 33*(b*x^2 + a)^(3/2)*A*a^2*b^5 - 9*sqrt(b*x^2 + a)*A*a^3*b^5
)/(a^2*b^4*x^8))/b

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Mupad [B]
time = 1.72, size = 169, normalized size = 1.08 \begin {gather*} \frac {3\,A\,a\,\sqrt {b\,x^2+a}}{128\,x^8}-\frac {B\,{\left (b\,x^2+a\right )}^{3/2}}{6\,x^6}-\frac {11\,A\,{\left (b\,x^2+a\right )}^{3/2}}{128\,x^8}+\frac {B\,a\,\sqrt {b\,x^2+a}}{16\,x^6}-\frac {11\,A\,{\left (b\,x^2+a\right )}^{5/2}}{128\,a\,x^8}+\frac {3\,A\,{\left (b\,x^2+a\right )}^{7/2}}{128\,a^2\,x^8}-\frac {B\,{\left (b\,x^2+a\right )}^{5/2}}{16\,a\,x^6}+\frac {A\,b^4\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,3{}\mathrm {i}}{128\,a^{5/2}}-\frac {B\,b^3\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{16\,a^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^(3/2))/x^9,x)

[Out]

(A*b^4*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*3i)/(128*a^(5/2)) - (B*(a + b*x^2)^(3/2))/(6*x^6) - (11*A*(a + b*x
^2)^(3/2))/(128*x^8) - (B*b^3*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*1i)/(16*a^(3/2)) + (3*A*a*(a + b*x^2)^(1/2)
)/(128*x^8) + (B*a*(a + b*x^2)^(1/2))/(16*x^6) - (11*A*(a + b*x^2)^(5/2))/(128*a*x^8) + (3*A*(a + b*x^2)^(7/2)
)/(128*a^2*x^8) - (B*(a + b*x^2)^(5/2))/(16*a*x^6)

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