\(\int \frac {(a^2+2 a b x^2+b^2 x^4)^{3/2}}{x^5} \, dx\) [565]

Optimal result

Integrand size = 26, antiderivative size = 164 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^5} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left (a+b x^2\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {b^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1126, 272, 45} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^5} \, dx=-\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {3 a b^2 \log (x) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left (a+b x^2\right )} \]

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^5,x]

[Out]

-1/4*(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(x^4*(a + b*x^2)) - (3*a^2*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*x^
2*(a + b*x^2)) + (b^3*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*(a + b*x^2)) + (3*a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4]*Log[x])/(a + b*x^2)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rule 1126

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^3}{x^5} \, dx}{b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^3}{x^3} \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \left (b^6+\frac {a^3 b^3}{x^3}+\frac {3 a^2 b^4}{x^2}+\frac {3 a b^5}{x}\right ) \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )} \\ & = -\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left (a+b x^2\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {b^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(612\) vs. \(2(164)=328\).

Time = 0.81 (sec) , antiderivative size = 612, normalized size of antiderivative = 3.73 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^5} \, dx=\frac {4 a^4 \sqrt {a^2}+28 a^3 \sqrt {a^2} b x^2+35 \left (a^2\right )^{3/2} b^2 x^4+3 a \sqrt {a^2} b^3 x^6-8 \sqrt {a^2} b^4 x^8-4 a^4 \sqrt {\left (a+b x^2\right )^2}-24 a^3 b x^2 \sqrt {\left (a+b x^2\right )^2}-11 a^2 b^2 x^4 \sqrt {\left (a+b x^2\right )^2}+8 a b^3 x^6 \sqrt {\left (a+b x^2\right )^2}-24 a b^2 x^4 \left (a^2+a b x^2-\sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}\right ) \text {arctanh}\left (\frac {b x^2}{\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}}\right )-24 b^2 x^4 \left (\left (a^2\right )^{3/2}+a \sqrt {a^2} b x^2-a^2 \sqrt {\left (a+b x^2\right )^2}\right ) \log \left (x^2\right )+12 \left (a^2\right )^{3/2} b^2 x^4 \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+12 a \sqrt {a^2} b^3 x^6 \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )-12 a^2 b^2 x^4 \sqrt {\left (a+b x^2\right )^2} \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+12 \left (a^2\right )^{3/2} b^2 x^4 \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+12 a \sqrt {a^2} b^3 x^6 \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )-12 a^2 b^2 x^4 \sqrt {\left (a+b x^2\right )^2} \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )}{16 x^4 \left (a^2+a b x^2-\sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}\right )} \]

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^5,x]

[Out]

(4*a^4*Sqrt[a^2] + 28*a^3*Sqrt[a^2]*b*x^2 + 35*(a^2)^(3/2)*b^2*x^4 + 3*a*Sqrt[a^2]*b^3*x^6 - 8*Sqrt[a^2]*b^4*x
^8 - 4*a^4*Sqrt[(a + b*x^2)^2] - 24*a^3*b*x^2*Sqrt[(a + b*x^2)^2] - 11*a^2*b^2*x^4*Sqrt[(a + b*x^2)^2] + 8*a*b
^3*x^6*Sqrt[(a + b*x^2)^2] - 24*a*b^2*x^4*(a^2 + a*b*x^2 - Sqrt[a^2]*Sqrt[(a + b*x^2)^2])*ArcTanh[(b*x^2)/(Sqr
t[a^2] - Sqrt[(a + b*x^2)^2])] - 24*b^2*x^4*((a^2)^(3/2) + a*Sqrt[a^2]*b*x^2 - a^2*Sqrt[(a + b*x^2)^2])*Log[x^
2] + 12*(a^2)^(3/2)*b^2*x^4*Log[Sqrt[a^2] - b*x^2 - Sqrt[(a + b*x^2)^2]] + 12*a*Sqrt[a^2]*b^3*x^6*Log[Sqrt[a^2
] - b*x^2 - Sqrt[(a + b*x^2)^2]] - 12*a^2*b^2*x^4*Sqrt[(a + b*x^2)^2]*Log[Sqrt[a^2] - b*x^2 - Sqrt[(a + b*x^2)
^2]] + 12*(a^2)^(3/2)*b^2*x^4*Log[Sqrt[a^2] + b*x^2 - Sqrt[(a + b*x^2)^2]] + 12*a*Sqrt[a^2]*b^3*x^6*Log[Sqrt[a
^2] + b*x^2 - Sqrt[(a + b*x^2)^2]] - 12*a^2*b^2*x^4*Sqrt[(a + b*x^2)^2]*Log[Sqrt[a^2] + b*x^2 - Sqrt[(a + b*x^
2)^2]])/(16*x^4*(a^2 + a*b*x^2 - Sqrt[a^2]*Sqrt[(a + b*x^2)^2]))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.29

method	result	size
pseudoelliptic	\(-\frac {\operatorname {csgn}\left (b \,x^{2}+a \right ) \left (-2 b^{3} x^{6}-6 b^{2} a \ln \left (x^{2}\right ) x^{4}+6 a^{2} b \,x^{2}+a^{3}\right )}{4 x^{4}}\)	\(48\)
default	\(\frac {{\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}} \left (2 b^{3} x^{6}+12 \ln \left (x \right ) x^{4} a \,b^{2}-6 a^{2} b \,x^{2}-a^{3}\right )}{4 \left (b \,x^{2}+a \right )^{3} x^{4}}\)	\(60\)
risch	\(\frac {b^{3} x^{2} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{2 b \,x^{2}+2 a}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {3}{2} a^{2} b \,x^{2}-\frac {1}{4} a^{3}\right )}{\left (b \,x^{2}+a \right ) x^{4}}+\frac {3 a \,b^{2} \ln \left (x \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b \,x^{2}+a}\)	\(97\)

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.24 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^5} \, dx=\frac {2 \, b^{3} x^{6} + 12 \, a b^{2} x^{4} \log \left (x\right ) - 6 \, a^{2} b x^{2} - a^{3}}{4 \, x^{4}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^5,x, algorithm="fricas")

[Out]

1/4*(2*b^3*x^6 + 12*a*b^2*x^4*log(x) - 6*a^2*b*x^2 - a^3)/x^4

Sympy [F]

\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^5} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx \]

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/x**5,x)

[Out]

Integral(((a + b*x**2)**2)**(3/2)/x**5, x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.21 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^5} \, dx=\frac {1}{2} \, b^{3} x^{2} + 3 \, a b^{2} \log \left (x\right ) - \frac {3 \, a^{2} b}{2 \, x^{2}} - \frac {a^{3}}{4 \, x^{4}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^5,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.53 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^5} \, dx=\frac {1}{2} \, b^{3} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{2} \, a b^{2} \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {9 \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{4 \, x^{4}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^5,x, algorithm="giac")

[Out]

1/2*b^3*x^2*sgn(b*x^2 + a) + 3/2*a*b^2*log(x^2)*sgn(b*x^2 + a) - 1/4*(9*a*b^2*x^4*sgn(b*x^2 + a) + 6*a^2*b*x^2
*sgn(b*x^2 + a) + a^3*sgn(b*x^2 + a))/x^4

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}}{x^5} \,d x \]

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2)/x^5,x)

[Out]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2)/x^5, x)