3.86 \(\int \frac{d+e x^2}{x^3 (a^2+2 a b x^2+b^2 x^4)^{3/2}} \, dx\)

Optimal. Leaf size=223 \[ -\frac{b d-a e}{4 a^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 b d-a e}{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\log (x) \left (a+b x^2\right ) (3 b d-a e)}{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (3 b d-a e) \log \left (a+b x^2\right )}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{2 a^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.183783, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1250, 446, 77} \[ -\frac{b d-a e}{4 a^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 b d-a e}{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\log (x) \left (a+b x^2\right ) (3 b d-a e)}{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (3 b d-a e) \log \left (a+b x^2\right )}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{2 a^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1250

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dis
t[(a + b*x^2 + c*x^4)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(f*x)^m*(d + e*x^2)^q*(b/2
+ c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p]

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0731758, size = 130, normalized size = 0.58 \[ \frac{a \left (a^2 \left (3 e x^2-2 d\right )+a b \left (2 e x^4-9 d x^2\right )-6 b^2 d x^4\right )+4 x^2 \log (x) \left (a+b x^2\right )^2 (a e-3 b d)+2 x^2 \left (a+b x^2\right )^2 (3 b d-a e) \log \left (a+b x^2\right )}{4 a^4 x^2 \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-6*b^2*d*x^4 + a^2*(-2*d + 3*e*x^2) + a*b*(-9*d*x^2 + 2*e*x^4)) + 4*(-3*b*d + a*e)*x^2*(a + b*x^2)^2*Log[x
] + 2*(3*b*d - a*e)*x^2*(a + b*x^2)^2*Log[a + b*x^2])/(4*a^4*x^2*(a + b*x^2)*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.021, size = 249, normalized size = 1.1 \begin{align*}{\frac{ \left ( 4\,\ln \left ( x \right ){x}^{6}a{b}^{2}e-12\,\ln \left ( x \right ){x}^{6}{b}^{3}d-2\,\ln \left ( b{x}^{2}+a \right ){x}^{6}a{b}^{2}e+6\,\ln \left ( b{x}^{2}+a \right ){x}^{6}{b}^{3}d+8\,\ln \left ( x \right ){x}^{4}{a}^{2}be-24\,\ln \left ( x \right ){x}^{4}a{b}^{2}d-4\,\ln \left ( b{x}^{2}+a \right ){x}^{4}{a}^{2}be+12\,\ln \left ( b{x}^{2}+a \right ){x}^{4}a{b}^{2}d+2\,{x}^{4}{a}^{2}be-6\,{x}^{4}a{b}^{2}d+4\,\ln \left ( x \right ){x}^{2}{a}^{3}e-12\,\ln \left ( x \right ){x}^{2}{a}^{2}bd-2\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{3}e+6\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{2}bd+3\,{x}^{2}{a}^{3}e-9\,{x}^{2}{a}^{2}bd-2\,{a}^{3}d \right ) \left ( b{x}^{2}+a \right ) }{4\,{x}^{2}{a}^{4}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*ln(x)*x^6*a*b^2*e-12*ln(x)*x^6*b^3*d-2*ln(b*x^2+a)*x^6*a*b^2*e+6*ln(b*x^2+a)*x^6*b^3*d+8*ln(x)*x^4*a^2*
b*e-24*ln(x)*x^4*a*b^2*d-4*ln(b*x^2+a)*x^4*a^2*b*e+12*ln(b*x^2+a)*x^4*a*b^2*d+2*x^4*a^2*b*e-6*x^4*a*b^2*d+4*ln
(x)*x^2*a^3*e-12*ln(x)*x^2*a^2*b*d-2*ln(b*x^2+a)*x^2*a^3*e+6*ln(b*x^2+a)*x^2*a^2*b*d+3*x^2*a^3*e-9*x^2*a^2*b*d
-2*a^3*d)*(b*x^2+a)/x^2/a^4/((b*x^2+a)^2)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56555, size = 413, normalized size = 1.85 \begin{align*} -\frac{2 \,{\left (3 \, a b^{2} d - a^{2} b e\right )} x^{4} + 2 \, a^{3} d + 3 \,{\left (3 \, a^{2} b d - a^{3} e\right )} x^{2} - 2 \,{\left ({\left (3 \, b^{3} d - a b^{2} e\right )} x^{6} + 2 \,{\left (3 \, a b^{2} d - a^{2} b e\right )} x^{4} +{\left (3 \, a^{2} b d - a^{3} e\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left ({\left (3 \, b^{3} d - a b^{2} e\right )} x^{6} + 2 \,{\left (3 \, a b^{2} d - a^{2} b e\right )} x^{4} +{\left (3 \, a^{2} b d - a^{3} e\right )} x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*(3*a*b^2*d - a^2*b*e)*x^4 + 2*a^3*d + 3*(3*a^2*b*d - a^3*e)*x^2 - 2*((3*b^3*d - a*b^2*e)*x^6 + 2*(3*a*
b^2*d - a^2*b*e)*x^4 + (3*a^2*b*d - a^3*e)*x^2)*log(b*x^2 + a) + 4*((3*b^3*d - a*b^2*e)*x^6 + 2*(3*a*b^2*d - a
^2*b*e)*x^4 + (3*a^2*b*d - a^3*e)*x^2)*log(x))/(a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x**2)/(x**3*((a + b*x**2)**2)**(3/2)), 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((e*x^2+d)/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x