3.644 \(\int \frac{(a+b x^2)^2}{x^4 \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{a^2 \sqrt{c+d x^2}}{3 c x^3}-\frac{2 a \sqrt{c+d x^2} (3 b c-a d)}{3 c^2 x}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{\sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.0496488, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {462, 451, 217, 206} \[ -\frac{a^2 \sqrt{c+d x^2}}{3 c x^3}-\frac{2 a \sqrt{c+d x^2} (3 b c-a d)}{3 c^2 x}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 462

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

Rule 451

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0707773, size = 72, normalized size = 0.86 \[ \frac{b^2 \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}-\frac{a \sqrt{c+d x^2} \left (a \left (c-2 d x^2\right )+6 b c x^2\right )}{3 c^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 85, normalized size = 1. \begin{align*}{{b}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}-{\frac{{a}^{2}}{3\,c{x}^{3}}\sqrt{d{x}^{2}+c}}+{\frac{2\,{a}^{2}d}{3\,{c}^{2}x}\sqrt{d{x}^{2}+c}}-2\,{\frac{\sqrt{d{x}^{2}+c}ab}{cx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^4/(d*x^2+c)^(1/2),x)

[Out]

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

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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((b*x^2+a)^2/x^4/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.31875, size = 390, normalized size = 4.64 \begin{align*} \left [\frac{3 \, b^{2} c^{2} \sqrt{d} x^{3} \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) - 2 \,{\left (a^{2} c d + 2 \,{\left (3 \, a b c d - a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{6 \, c^{2} d x^{3}}, -\frac{3 \, b^{2} c^{2} \sqrt{-d} x^{3} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) +{\left (a^{2} c d + 2 \,{\left (3 \, a b c d - a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3 \, c^{2} d x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*b^2*c^2*sqrt(d)*x^3*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 2*(a^2*c*d + 2*(3*a*b*c*d - a^2*
d^2)*x^2)*sqrt(d*x^2 + c))/(c^2*d*x^3), -1/3*(3*b^2*c^2*sqrt(-d)*x^3*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (a^2
*c*d + 2*(3*a*b*c*d - a^2*d^2)*x^2)*sqrt(d*x^2 + c))/(c^2*d*x^3)]

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Sympy [A]  time = 2.20859, size = 158, normalized size = 1.88 \begin{align*} - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 c x^{2}} + \frac{2 a^{2} d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3 c^{2}} - \frac{2 a b \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{c} + b^{2} \left (\begin{cases} \frac{\sqrt{- \frac{c}{d}} \operatorname{asin}{\left (x \sqrt{- \frac{d}{c}} \right )}}{\sqrt{c}} & \text{for}\: c > 0 \wedge d < 0 \\\frac{\sqrt{\frac{c}{d}} \operatorname{asinh}{\left (x \sqrt{\frac{d}{c}} \right )}}{\sqrt{c}} & \text{for}\: c > 0 \wedge d > 0 \\\frac{\sqrt{- \frac{c}{d}} \operatorname{acosh}{\left (x \sqrt{- \frac{d}{c}} \right )}}{\sqrt{- c}} & \text{for}\: d > 0 \wedge c < 0 \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**4/(d*x**2+c)**(1/2),x)

[Out]

-a**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(3*c*x**2) + 2*a**2*d**(3/2)*sqrt(c/(d*x**2) + 1)/(3*c**2) - 2*a*b*sqrt(d)*
sqrt(c/(d*x**2) + 1)/c + b**2*Piecewise((sqrt(-c/d)*asin(x*sqrt(-d/c))/sqrt(c), (c > 0) & (d < 0)), (sqrt(c/d)
*asinh(x*sqrt(d/c))/sqrt(c), (c > 0) & (d > 0)), (sqrt(-c/d)*acosh(x*sqrt(-d/c))/sqrt(-c), (d > 0) & (c < 0)))

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Giac [B]  time = 1.16775, size = 211, normalized size = 2.51 \begin{align*} -\frac{b^{2} \log \left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{2 \, \sqrt{d}} + \frac{4 \,{\left (3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b \sqrt{d} - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c \sqrt{d} + 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} d^{\frac{3}{2}} + 3 \, a b c^{2} \sqrt{d} - a^{2} c d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^2*log((sqrt(d)*x - sqrt(d*x^2 + c))^2)/sqrt(d) + 4/3*(3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*sqrt(d) - 6
*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c*sqrt(d) + 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*d^(3/2) + 3*a*b*c^2*sqr
t(d) - a^2*c*d^(3/2))/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^3