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

Optimal. Leaf size=133 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{c x}-\frac{\left (b^2 c^2-8 a d (a d+b c)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{3/2}}-\frac{x \sqrt{c+d x^2} \left (b^2 c^2-8 a d (a d+b c)\right )}{8 c d}+\frac{b^2 x \left (c+d x^2\right )^{3/2}}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.086252, antiderivative size = 130, normalized size of antiderivative = 0.98, 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 = {462, 388, 195, 217, 206} \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{c x}-\frac{\left (b^2 c^2-8 a d (a d+b c)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{3/2}}-\frac{1}{8} x \sqrt{c+d x^2} \left (\frac{b^2 c}{d}-\frac{8 a (a d+b c)}{c}\right )+\frac{b^2 x \left (c+d x^2\right )^{3/2}}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b^2*c)/d - (8*a*(b*c + a*d))/c)*x*Sqrt[c + d*x^2])/8 - (a^2*(c + d*x^2)^(3/2))/(c*x) + (b^2*x*(c + d*x^2)^
(3/2))/(4*d) - ((b^2*c^2 - 8*a*d*(b*c + a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

Mathematica [A]  time = 0.0908908, size = 99, normalized size = 0.74 \[ \frac{\left (8 a^2 d^2+8 a b c d-b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{8 d^{3/2}}+\sqrt{c+d x^2} \left (-\frac{a^2}{x}+a b x+\frac{b^2 x \left (c+2 d x^2\right )}{8 d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 163, normalized size = 1.2 \begin{align*}{\frac{{b}^{2}x}{4\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}cx}{8\,d}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}+abx\sqrt{d{x}^{2}+c}+{abc\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}-{\frac{{a}^{2}}{cx} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}dx}{c}\sqrt{d{x}^{2}+c}}+{a}^{2}\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x*(d*x^2+c)^(3/2)/d-1/8*b^2*c/d*x*(d*x^2+c)^(1/2)-1/8*b^2*c^2/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+a*
b*x*(d*x^2+c)^(1/2)+a*b*c/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-a^2*(d*x^2+c)^(3/2)/c/x+a^2*d/c*x*(d*x^2+c)^(1
/2)+a^2*d^(1/2)*ln(x*d^(1/2)+(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*(d*x^2+c)^(1/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.64763, size = 483, normalized size = 3.63 \begin{align*} \left [-\frac{{\left (b^{2} c^{2} - 8 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt{d} x \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) - 2 \,{\left (2 \, b^{2} d^{2} x^{4} - 8 \, a^{2} d^{2} +{\left (b^{2} c d + 8 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{16 \, d^{2} x}, \frac{{\left (b^{2} c^{2} - 8 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt{-d} x \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) +{\left (2 \, b^{2} d^{2} x^{4} - 8 \, a^{2} d^{2} +{\left (b^{2} c d + 8 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{8 \, d^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 6.55167, size = 219, normalized size = 1.65 \begin{align*} - \frac{a^{2} \sqrt{c}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + a^{2} \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )} - \frac{a^{2} d x}{\sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + a b \sqrt{c} x \sqrt{1 + \frac{d x^{2}}{c}} + \frac{a b c \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{\sqrt{d}} + \frac{b^{2} c^{\frac{3}{2}} x}{8 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} \sqrt{c} x^{3}}{8 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 d^{\frac{3}{2}}} + \frac{b^{2} d x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12594, size = 170, normalized size = 1.28 \begin{align*} \frac{2 \, a^{2} c \sqrt{d}}{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c} + \frac{1}{8} \,{\left (2 \, b^{2} x^{2} + \frac{b^{2} c d + 8 \, a b d^{2}}{d^{2}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (b^{2} c^{2} \sqrt{d} - 8 \, a b c d^{\frac{3}{2}} - 8 \, a^{2} d^{\frac{5}{2}}\right )} \log \left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{16 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a^2*c*sqrt(d)/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c) + 1/8*(2*b^2*x^2 + (b^2*c*d + 8*a*b*d^2)/d^2)*sqrt(d*x^2
+ c)*x + 1/16*(b^2*c^2*sqrt(d) - 8*a*b*c*d^(3/2) - 8*a^2*d^(5/2))*log((sqrt(d)*x - sqrt(d*x^2 + c))^2)/d^2