3.645 \(\int \frac{(a+b x^2)^2}{x^5 \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{5/2}}-\frac{a^2 \sqrt{c+d x^2}}{4 c x^4}-\frac{a \sqrt{c+d x^2} (8 b c-3 a d)}{8 c^2 x^2} \]

[Out]

-(a^2*Sqrt[c + d*x^2])/(4*c*x^4) - (a*(8*b*c - 3*a*d)*Sqrt[c + d*x^2])/(8*c^2*x^2) - ((8*b^2*c^2 - 8*a*b*c*d +
3*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(5/2))

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Rubi [A]  time = 0.105793, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {446, 89, 78, 63, 208} \[ -\frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{5/2}}-\frac{a^2 \sqrt{c+d x^2}}{4 c x^4}-\frac{a \sqrt{c+d x^2} (8 b c-3 a d)}{8 c^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(x^5*Sqrt[c + d*x^2]),x]

[Out]

-(a^2*Sqrt[c + d*x^2])/(4*c*x^4) - (a*(8*b*c - 3*a*d)*Sqrt[c + d*x^2])/(8*c^2*x^2) - ((8*b^2*c^2 - 8*a*b*c*d +
3*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(5/2))

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 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 78

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] || LtQ
[p, n]))))

Rule 63

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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^5 \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^3 \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=-\frac{a^2 \sqrt{c+d x^2}}{4 c x^4}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} a (8 b c-3 a d)+2 b^2 c x}{x^2 \sqrt{c+d x}} \, dx,x,x^2\right )}{4 c}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{4 c x^4}-\frac{a (8 b c-3 a d) \sqrt{c+d x^2}}{8 c^2 x^2}+\frac{1}{16} \left (8 b^2-\frac{a d (8 b c-3 a d)}{c^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=-\frac{a^2 \sqrt{c+d x^2}}{4 c x^4}-\frac{a (8 b c-3 a d) \sqrt{c+d x^2}}{8 c^2 x^2}+\frac{\left (8 b^2-\frac{a d (8 b c-3 a d)}{c^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{8 d}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{4 c x^4}-\frac{a (8 b c-3 a d) \sqrt{c+d x^2}}{8 c^2 x^2}-\frac{\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.08643, size = 92, normalized size = 0.87 \[ -\frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{5/2}}-\frac{a \sqrt{c+d x^2} \left (2 a c-3 a d x^2+8 b c x^2\right )}{8 c^2 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/(x^5*Sqrt[c + d*x^2]),x]

[Out]

-(a*Sqrt[c + d*x^2]*(2*a*c + 8*b*c*x^2 - 3*a*d*x^2))/(8*c^2*x^4) - ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTan
h[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(5/2))

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Maple [A]  time = 0.012, size = 157, normalized size = 1.5 \begin{align*} -{{b}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}-{\frac{{a}^{2}}{4\,c{x}^{4}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{a}^{2}d}{8\,{c}^{2}{x}^{2}}\sqrt{d{x}^{2}+c}}-{\frac{3\,{a}^{2}{d}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}-{\frac{ab}{c{x}^{2}}\sqrt{d{x}^{2}+c}}+{abd\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^5/(d*x^2+c)^(1/2),x)

[Out]

-b^2/c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-1/4*a^2*(d*x^2+c)^(1/2)/c/x^4+3/8*a^2*d/c^2/x^2*(d*x^2+c)^(
1/2)-3/8*a^2*d^2/c^(5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-a*b/c/x^2*(d*x^2+c)^(1/2)+a*b*d/c^(3/2)*ln((2*c
+2*c^(1/2)*(d*x^2+c)^(1/2))/x)

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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((b*x^2+a)^2/x^5/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.42033, size = 463, normalized size = 4.37 \begin{align*} \left [\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{c} x^{4} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) - 2 \,{\left (2 \, a^{2} c^{2} +{\left (8 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{16 \, c^{3} x^{4}}, \frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (2 \, a^{2} c^{2} +{\left (8 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{8 \, c^{3} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 77.5791, size = 178, normalized size = 1.68 \begin{align*} - \frac{a^{2}}{4 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} \sqrt{d}}{8 c x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{3 a^{2} d^{\frac{3}{2}}}{8 c^{2} x \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a^{2} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{8 c^{\frac{5}{2}}} - \frac{a b \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{c x} + \frac{a b d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{c^{\frac{3}{2}}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{\sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**5/(d*x**2+c)**(1/2),x)

[Out]

-a**2/(4*sqrt(d)*x**5*sqrt(c/(d*x**2) + 1)) + a**2*sqrt(d)/(8*c*x**3*sqrt(c/(d*x**2) + 1)) + 3*a**2*d**(3/2)/(
8*c**2*x*sqrt(c/(d*x**2) + 1)) - 3*a**2*d**2*asinh(sqrt(c)/(sqrt(d)*x))/(8*c**(5/2)) - a*b*sqrt(d)*sqrt(c/(d*x
**2) + 1)/(c*x) + a*b*d*asinh(sqrt(c)/(sqrt(d)*x))/c**(3/2) - b**2*asinh(sqrt(c)/(sqrt(d)*x))/sqrt(c)

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Giac [A]  time = 1.13172, size = 189, normalized size = 1.78 \begin{align*} \frac{\frac{{\left (8 \, b^{2} c^{2} d - 8 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d^{2} - 8 \, \sqrt{d x^{2} + c} a b c^{2} d^{2} - 3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{3} + 5 \, \sqrt{d x^{2} + c} a^{2} c d^{3}}{c^{2} d^{2} x^{4}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((8*b^2*c^2*d - 8*a*b*c*d^2 + 3*a^2*d^3)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(sqrt(-c)*c^2) - (8*(d*x^2 + c)^
(3/2)*a*b*c*d^2 - 8*sqrt(d*x^2 + c)*a*b*c^2*d^2 - 3*(d*x^2 + c)^(3/2)*a^2*d^3 + 5*sqrt(d*x^2 + c)*a^2*c*d^3)/(
c^2*d^2*x^4))/d