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

Optimal. Leaf size=82 \[ -\frac{a^2 \sqrt{c+d x^2}}{c x}-\frac{b (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 d^{3/2}}+\frac{b^2 x \sqrt{c+d x^2}}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.045298, antiderivative size = 82, 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, 388, 217, 206} \[ -\frac{a^2 \sqrt{c+d x^2}}{c x}-\frac{b (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 d^{3/2}}+\frac{b^2 x \sqrt{c+d x^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 388

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

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

Mathematica [A]  time = 0.0637257, size = 76, normalized size = 0.93 \[ \sqrt{c+d x^2} \left (\frac{b^2 x}{2 d}-\frac{a^2}{c x}\right )-\frac{b (b c-4 a d) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{2 d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 88, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}x}{2\,d}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}c}{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}+2\,{\frac{ab\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }{\sqrt{d}}}-{\frac{{a}^{2}}{cx}\sqrt{d{x}^{2}+c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*x*(d*x^2+c)^(1/2)/d-1/2*b^2*c/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+2*a*b*ln(x*d^(1/2)+(d*x^2+c)^(1/2)
)/d^(1/2)-a^2*(d*x^2+c)^(1/2)/c/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^2/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33145, size = 374, normalized size = 4.56 \begin{align*} \left [-\frac{{\left (b^{2} c^{2} - 4 \, a b c d\right )} \sqrt{d} x \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) - 2 \,{\left (b^{2} c d x^{2} - 2 \, a^{2} d^{2}\right )} \sqrt{d x^{2} + c}}{4 \, c d^{2} x}, \frac{{\left (b^{2} c^{2} - 4 \, a b c d\right )} \sqrt{-d} x \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) +{\left (b^{2} c d x^{2} - 2 \, a^{2} d^{2}\right )} \sqrt{d x^{2} + c}}{2 \, c d^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.34884, size = 155, normalized size = 1.89 \begin{align*} - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{c} + 2 a b \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 ) + \frac{b^{2} \sqrt{c} x \sqrt{1 + \frac{d x^{2}}{c}}}{2 d} - \frac{b^{2} c \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 d^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13686, size = 126, normalized size = 1.54 \begin{align*} \frac{\sqrt{d x^{2} + c} b^{2} x}{2 \, d} + \frac{2 \, a^{2} \sqrt{d}}{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c} + \frac{{\left (b^{2} c \sqrt{d} - 4 \, a b d^{\frac{3}{2}}\right )} \log \left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{4 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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