3.359 \(\int \frac {1}{x^3 (a+\frac {b}{c+d x^2})^{3/2}} \, dx\)

Optimal. Leaf size=146 \[ \frac {3 b d}{2 (a c+b)^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {c+d x^2}{2 x^2 (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {3 b \sqrt {c} d \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{2 (a c+b)^{5/2}} \]

[Out]

-3/2*b*d*arctanh(c^(1/2)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(a*c+b)^(1/2))*c^(1/2)/(a*c+b)^(5/2)+3/2*b*d/(a*c+b
)^2/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/2*(-d*x^2-c)/(a*c+b)/x^2/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)

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Rubi [A]  time = 0.45, antiderivative size = 174, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6722, 1975, 446, 94, 93, 208} \[ \frac {3 b d}{2 (a c+b)^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {c+d x^2}{2 x^2 (a c+b) \sqrt {a+\frac {b}{c+d x^2}}}-\frac {3 b \sqrt {c} d \sqrt {a \left (c+d x^2\right )+b} \tanh ^{-1}\left (\frac {\sqrt {a c+b} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a \left (c+d x^2\right )+b}}\right )}{2 (a c+b)^{5/2} \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b/(c + d*x^2))^(3/2)),x]

[Out]

(3*b*d)/(2*(b + a*c)^2*Sqrt[a + b/(c + d*x^2)]) - (c + d*x^2)/(2*(b + a*c)*x^2*Sqrt[a + b/(c + d*x^2)]) - (3*b
*Sqrt[c]*d*Sqrt[b + a*(c + d*x^2)]*ArcTanh[(Sqrt[b + a*c]*Sqrt[c + d*x^2])/(Sqrt[c]*Sqrt[b + a*(c + d*x^2)])])
/(2*(b + a*c)^(5/2)*Sqrt[c + d*x^2]*Sqrt[a + b/(c + d*x^2)])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 94

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

Rule 208

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

Rule 446

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]]

Rule 1975

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&&  !BinomialMatchQ[{u, v}, x]

Rule 6722

Int[(u_.)*((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[(a + b*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b + a/
v^n)^FracPart[p]), Int[u*v^(n*p)*(b + a/v^n)^p, x], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[p] && ILtQ[n, 0] &
& BinomialQ[v, x] &&  !LinearQ[v, x]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (b+a \left (c+d x^2\right )\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (b+a c+a d x^2\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x^2 (b+a c+a d x)^{3/2}} \, dx,x,x^2\right )}{2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {c+d x^2}{2 (b+a c) x^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (3 b d \sqrt {b+a \left (c+d x^2\right )}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x (b+a c+a d x)^{3/2}} \, dx,x,x^2\right )}{4 (b+a c) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {3 b d}{2 (b+a c)^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {c+d x^2}{2 (b+a c) x^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (3 b c d \sqrt {b+a \left (c+d x^2\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{4 (b+a c)^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {3 b d}{2 (b+a c)^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {c+d x^2}{2 (b+a c) x^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (3 b c d \sqrt {b+a \left (c+d x^2\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{-c-(-b-a c) x^2} \, dx,x,\frac {\sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{2 (b+a c)^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {3 b d}{2 (b+a c)^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {c+d x^2}{2 (b+a c) x^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {3 b \sqrt {c} d \sqrt {b+a \left (c+d x^2\right )} \tanh ^{-1}\left (\frac {\sqrt {b+a c} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {b+a \left (c+d x^2\right )}}\right )}{2 (b+a c)^{5/2} \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ \end {align*}

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Mathematica [A]  time = 0.42, size = 229, normalized size = 1.57 \[ \frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-2 \sqrt {c (a c+b)} \left (c+d x^2\right ) \left (a c \left (c+d x^2\right )+b \left (c-2 d x^2\right )\right )+6 b c d x^2 \log (x) \sqrt {\left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )}-3 b c d x^2 \sqrt {\left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )} \log \left (2 \sqrt {c (a c+b)} \sqrt {\left (c+d x^2\right ) \left (a c+a d x^2+b\right )}+2 a c \left (c+d x^2\right )+b \left (2 c+d x^2\right )\right )\right )}{4 x^2 (a c+b)^2 \sqrt {c (a c+b)} \left (a \left (c+d x^2\right )+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b/(c + d*x^2))^(3/2)),x]

[Out]

(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-2*Sqrt[c*(b + a*c)]*(c + d*x^2)*(b*(c - 2*d*x^2) + a*c*(c + d*x^2)) +
6*b*c*d*x^2*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[x] - 3*b*c*d*x^2*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*
Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/(4*(b +
a*c)^2*Sqrt[c*(b + a*c)]*x^2*(b + a*(c + d*x^2)))

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fricas [A]  time = 0.67, size = 599, normalized size = 4.10 \[ \left [\frac {3 \, {\left (a b d^{2} x^{4} + {\left (a b c + b^{2}\right )} d x^{2}\right )} \sqrt {\frac {c}{a c + b}} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} - 4 \, {\left ({\left (2 \, a^{2} c^{2} + 3 \, a b c + b^{2}\right )} d^{2} x^{4} + 2 \, a^{2} c^{4} + 4 \, a b c^{3} + 2 \, b^{2} c^{2} + {\left (4 \, a^{2} c^{3} + 7 \, a b c^{2} + 3 \, b^{2} c\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} \sqrt {\frac {c}{a c + b}}}{x^{4}}\right ) - 4 \, {\left ({\left (a c - 2 \, b\right )} d^{2} x^{4} + a c^{3} + {\left (2 \, a c^{2} - b c\right )} d x^{2} + b c^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left ({\left (a^{3} c^{2} + 2 \, a^{2} b c + a b^{2}\right )} d x^{4} + {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} + 3 \, a b^{2} c + b^{3}\right )} x^{2}\right )}}, \frac {3 \, {\left (a b d^{2} x^{4} + {\left (a b c + b^{2}\right )} d x^{2}\right )} \sqrt {-\frac {c}{a c + b}} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} \sqrt {-\frac {c}{a c + b}}}{2 \, {\left (a c d x^{2} + a c^{2} + b c\right )}}\right ) - 2 \, {\left ({\left (a c - 2 \, b\right )} d^{2} x^{4} + a c^{3} + {\left (2 \, a c^{2} - b c\right )} d x^{2} + b c^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, {\left ({\left (a^{3} c^{2} + 2 \, a^{2} b c + a b^{2}\right )} d x^{4} + {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} + 3 \, a b^{2} c + b^{3}\right )} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b/(d*x^2+c))^(3/2),x, algorithm="fricas")

[Out]

[1/8*(3*(a*b*d^2*x^4 + (a*b*c + b^2)*d*x^2)*sqrt(c/(a*c + b))*log(((8*a^2*c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2
*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a^2*c^2 + 3*a*b*c + b^2)*d^2*x
^4 + 2*a^2*c^4 + 4*a*b*c^3 + 2*b^2*c^2 + (4*a^2*c^3 + 7*a*b*c^2 + 3*b^2*c)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*
x^2 + c))*sqrt(c/(a*c + b)))/x^4) - 4*((a*c - 2*b)*d^2*x^4 + a*c^3 + (2*a*c^2 - b*c)*d*x^2 + b*c^2)*sqrt((a*d*
x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^2 + 2*a^2*b*c + a*b^2)*d*x^4 + (a^3*c^3 + 3*a^2*b*c^2 + 3*a*b^2*c + b^3)*
x^2), 1/4*(3*(a*b*d^2*x^4 + (a*b*c + b^2)*d*x^2)*sqrt(-c/(a*c + b))*arctan(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 +
2*b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))*sqrt(-c/(a*c + b))/(a*c*d*x^2 + a*c^2 + b*c)) - 2*((a*c - 2*b)*d^
2*x^4 + a*c^3 + (2*a*c^2 - b*c)*d*x^2 + b*c^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^2 + 2*a^2*b*c +
a*b^2)*d*x^4 + (a^3*c^3 + 3*a^2*b*c^2 + 3*a*b^2*c + b^3)*x^2)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b/(d*x^2+c))^(3/2),x, algorithm="giac")

[Out]

undef

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maple [B]  time = 0.07, size = 1088, normalized size = 7.45 \[ \frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-3 a^{2} b \,c^{2} d^{2} x^{4} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )+2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c}\, a^{2} d^{3} x^{6}-3 a \,b^{2} c \,d^{2} x^{4} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )-3 a^{2} b \,c^{3} d \,x^{2} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )+6 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c}\, a^{2} c \,d^{2} x^{4}-6 a \,b^{2} c^{2} d \,x^{2} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )+4 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c}\, a b \,d^{2} x^{4}-3 b^{3} c d \,x^{2} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )+4 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c}\, a^{2} c^{2} d \,x^{2}+4 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,c^{2}+b c}\, a b c d \,x^{2}+6 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c}\, a b c d \,x^{2}+4 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,c^{2}+b c}\, b^{2} d \,x^{2}+2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c}\, b^{2} d \,x^{2}-2 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \sqrt {a \,c^{2}+b c}\, a d \,x^{2}-2 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \sqrt {a \,c^{2}+b c}\, a c -2 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \sqrt {a \,c^{2}+b c}\, b \right )}{4 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \left (a c +b \right )^{3} \left (a d \,x^{2}+a c +b \right ) \sqrt {a \,c^{2}+b c}\, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a+b/(d*x^2+c))^(3/2),x)

[Out]

1/4*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*(d*x^2+c)*(2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)
^(1/2)*x^6*a^2*d^3-3*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*b*c+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+
a*c^2+b*c)^(1/2))/x^2)*x^4*a^2*b*c^2*d^2+6*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(1/2)*x
^4*a^2*c*d^2-3*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*b*c+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+
b*c)^(1/2))/x^2)*x^4*a*b^2*c*d^2+4*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(1/2)*x^4*a*b*d
^2-3*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*b*c+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)
)/x^2)*x^2*a^2*b*c^3*d+4*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(1/2)*x^2*a^2*c^2*d-6*ln(
(2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*b*c+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2))/x^2)*x
^2*a*b^2*c^2*d+4*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*(a*c^2+b*c)^(1/2)*x^2*a*b*c*d-2*(a*d^2*x^4+2*a*c*d*x^2+b*d*
x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(1/2)*x^2*a*d+6*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(
1/2)*x^2*a*b*c*d-3*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*b*c+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*
c^2+b*c)^(1/2))/x^2)*x^2*b^3*c*d+4*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*(a*c^2+b*c)^(1/2)*x^2*b^2*d+2*(a*d^2*x^4+
2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(1/2)*x^2*b^2*d-2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(
3/2)*(a*c^2+b*c)^(1/2)*a*c-2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(1/2)*b)/((d*x^2+c)*(
a*d*x^2+a*c+b))^(1/2)/(a*c+b)^3/(a*d*x^2+a*c+b)/x^2/(a*c^2+b*c)^(1/2)

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maxima [A]  time = 2.34, size = 247, normalized size = 1.69 \[ \frac {3 \, b c d \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{4 \, {\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} \sqrt {{\left (a c + b\right )} c}} + \frac {\frac {3 \, {\left (a d x^{2} + a c + b\right )} b c d}{d x^{2} + c} - 2 \, {\left (a b c + b^{2}\right )} d}{2 \, {\left ({\left (a^{2} c^{3} + 2 \, a b c^{2} + b^{2} c\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} + 3 \, a b^{2} c + b^{3}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b/(d*x^2+c))^(3/2),x, algorithm="maxima")

[Out]

3/4*b*c*d*log((c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - sqrt((a*c + b)*c))/(c*sqrt((a*d*x^2 + a*c + b)/(d*x^2
+ c)) + sqrt((a*c + b)*c)))/((a^2*c^2 + 2*a*b*c + b^2)*sqrt((a*c + b)*c)) + 1/2*(3*(a*d*x^2 + a*c + b)*b*c*d/
(d*x^2 + c) - 2*(a*b*c + b^2)*d)/((a^2*c^3 + 2*a*b*c^2 + b^2*c)*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) - (a^3
*c^3 + 3*a^2*b*c^2 + 3*a*b^2*c + b^3)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^3\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/(c + d*x^2))^(3/2)),x)

[Out]

int(1/(x^3*(a + b/(c + d*x^2))^(3/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a+b/(d*x**2+c))**(3/2),x)

[Out]

Integral(1/(x**3*((a*c + a*d*x**2 + b)/(c + d*x**2))**(3/2)), x)

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