3.652 \(\int \frac{1}{x^3 (a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)

Optimal. Leaf size=267 \[ -\frac{3 b}{4 a^4 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{3 a^3 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{8 a^2 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 b}{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a+b x^2}{2 a^5 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 b \log (x) \left (a+b x^2\right )}{a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(-2*b)/(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - b/(8*a^2*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - b/(3*
a^3*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*b)/(4*a^4*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
- (a + b*x^2)/(2*a^5*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (5*b*(a + b*x^2)*Log[x])/(a^6*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4]) + (5*b*(a + b*x^2)*Log[a + b*x^2])/(2*a^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.141949, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1112, 266, 44} \[ -\frac{3 b}{4 a^4 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{3 a^3 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{8 a^2 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 b}{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a+b x^2}{2 a^5 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 b \log (x) \left (a+b x^2\right )}{a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(-2*b)/(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - b/(8*a^2*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - b/(3*
a^3*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*b)/(4*a^4*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
- (a + b*x^2)/(2*a^5*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (5*b*(a + b*x^2)*Log[x])/(a^6*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4]) + (5*b*(a + b*x^2)*Log[a + b*x^2])/(2*a^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

Rule 1112

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]

Rule 266

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{x^3 \left (a b+b^2 x^2\right )^5} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a b+b^2 x\right )^5} \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^5 b^5 x^2}-\frac{5}{a^6 b^4 x}+\frac{1}{a^2 b^3 (a+b x)^5}+\frac{2}{a^3 b^3 (a+b x)^4}+\frac{3}{a^4 b^3 (a+b x)^3}+\frac{4}{a^5 b^3 (a+b x)^2}+\frac{5}{a^6 b^3 (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 b}{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{8 a^2 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{3 a^3 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 b}{4 a^4 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a+b x^2}{2 a^5 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 b \left (a+b x^2\right ) \log (x)}{a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0474817, size = 119, normalized size = 0.45 \[ \frac{-a \left (260 a^2 b^2 x^4+125 a^3 b x^2+12 a^4+210 a b^3 x^6+60 b^4 x^8\right )-120 b x^2 \log (x) \left (a+b x^2\right )^4+60 b x^2 \left (a+b x^2\right )^4 \log \left (a+b x^2\right )}{24 a^6 x^2 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(-(a*(12*a^4 + 125*a^3*b*x^2 + 260*a^2*b^2*x^4 + 210*a*b^3*x^6 + 60*b^4*x^8)) - 120*b*x^2*(a + b*x^2)^4*Log[x]
+ 60*b*x^2*(a + b*x^2)^4*Log[a + b*x^2])/(24*a^6*x^2*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.23, size = 219, normalized size = 0.8 \begin{align*}{\frac{ \left ( 60\,\ln \left ( b{x}^{2}+a \right ){x}^{10}{b}^{5}-120\,{b}^{5}\ln \left ( x \right ){x}^{10}+240\,\ln \left ( b{x}^{2}+a \right ){x}^{8}a{b}^{4}-480\,a{b}^{4}\ln \left ( x \right ){x}^{8}-60\,a{b}^{4}{x}^{8}+360\,\ln \left ( b{x}^{2}+a \right ){x}^{6}{a}^{2}{b}^{3}-720\,{a}^{2}{b}^{3}\ln \left ( x \right ){x}^{6}-210\,{a}^{2}{b}^{3}{x}^{6}+240\,\ln \left ( b{x}^{2}+a \right ){x}^{4}{a}^{3}{b}^{2}-480\,{b}^{2}{a}^{3}\ln \left ( x \right ){x}^{4}-260\,{b}^{2}{a}^{3}{x}^{4}+60\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{4}b-120\,{a}^{4}b\ln \left ( x \right ){x}^{2}-125\,{a}^{4}b{x}^{2}-12\,{a}^{5} \right ) \left ( b{x}^{2}+a \right ) }{24\,{x}^{2}{a}^{6}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/24*(60*ln(b*x^2+a)*x^10*b^5-120*b^5*ln(x)*x^10+240*ln(b*x^2+a)*x^8*a*b^4-480*a*b^4*ln(x)*x^8-60*a*b^4*x^8+36
0*ln(b*x^2+a)*x^6*a^2*b^3-720*a^2*b^3*ln(x)*x^6-210*a^2*b^3*x^6+240*ln(b*x^2+a)*x^4*a^3*b^2-480*b^2*a^3*ln(x)*
x^4-260*b^2*a^3*x^4+60*ln(b*x^2+a)*x^2*a^4*b-120*a^4*b*ln(x)*x^2-125*a^4*b*x^2-12*a^5)*(b*x^2+a)/x^2/a^6/((b*x
^2+a)^2)^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.38873, size = 440, normalized size = 1.65 \begin{align*} -\frac{60 \, a b^{4} x^{8} + 210 \, a^{2} b^{3} x^{6} + 260 \, a^{3} b^{2} x^{4} + 125 \, a^{4} b x^{2} + 12 \, a^{5} - 60 \,{\left (b^{5} x^{10} + 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 120 \,{\left (b^{5} x^{10} + 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (x\right )}{24 \,{\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/24*(60*a*b^4*x^8 + 210*a^2*b^3*x^6 + 260*a^3*b^2*x^4 + 125*a^4*b*x^2 + 12*a^5 - 60*(b^5*x^10 + 4*a*b^4*x^8
+ 6*a^2*b^3*x^6 + 4*a^3*b^2*x^4 + a^4*b*x^2)*log(b*x^2 + a) + 120*(b^5*x^10 + 4*a*b^4*x^8 + 6*a^2*b^3*x^6 + 4*
a^3*b^2*x^4 + a^4*b*x^2)*log(x))/(a^6*b^4*x^10 + 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 + 4*a^9*b*x^4 + a^10*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(1/(x**3*((a + b*x**2)**2)**(5/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x