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

Optimal. Leaf size=111 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{3 c x^3}-\frac{2 a b \left (c+d x^2\right )^{3/2}}{c x}+\frac{b x \sqrt{c+d x^2} (4 a d+b c)}{2 c}+\frac{b (4 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.19783, antiderivative size = 111, 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 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{3 c x^3}-\frac{2 a b \left (c+d x^2\right )^{3/2}}{c x}+\frac{b x \sqrt{c+d x^2} (4 a d+b c)}{2 c}+\frac{b (4 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.6834, size = 99, normalized size = 0.89 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{3 c x^{3}} - \frac{2 a b \left (c + d x^{2}\right )^{\frac{3}{2}}}{c x} + \frac{b \left (4 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 \sqrt{d}} + \frac{b x \sqrt{c + d x^{2}} \left (4 a d + b c\right )}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**4,x)

[Out]

-a**2*(c + d*x**2)**(3/2)/(3*c*x**3) - 2*a*b*(c + d*x**2)**(3/2)/(c*x) + b*(4*a*
d + b*c)*atanh(sqrt(d)*x/sqrt(c + d*x**2))/(2*sqrt(d)) + b*x*sqrt(c + d*x**2)*(4
*a*d + b*c)/(2*c)

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Mathematica [A]  time = 0.143353, size = 91, normalized size = 0.82 \[ \sqrt{c+d x^2} \left (-\frac{a^2}{3 x^3}-\frac{a (a d+6 b c)}{3 c x}+\frac{b^2 x}{2}\right )+\frac{b (4 a d+b c) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{2 \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 122, normalized size = 1.1 \[{\frac{x{b}^{2}}{2}\sqrt{d{x}^{2}+c}}+{\frac{{b}^{2}c}{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}-{\frac{{a}^{2}}{3\,c{x}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-2\,{\frac{ab \left ( d{x}^{2}+c \right ) ^{3/2}}{cx}}+2\,{\frac{dabx\sqrt{d{x}^{2}+c}}{c}}+2\,ab\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233574, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} c^{2} + 4 \, a b c d\right )} x^{3} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) + 2 \,{\left (3 \, b^{2} c x^{4} - 2 \, a^{2} c - 2 \,{\left (6 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{d}}{12 \, c \sqrt{d} x^{3}}, \frac{3 \,{\left (b^{2} c^{2} + 4 \, a b c d\right )} x^{3} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) +{\left (3 \, b^{2} c x^{4} - 2 \, a^{2} c - 2 \,{\left (6 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{6 \, c \sqrt{-d} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^4,x, algorithm="fricas")

[Out]

[1/12*(3*(b^2*c^2 + 4*a*b*c*d)*x^3*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sq
rt(d)) + 2*(3*b^2*c*x^4 - 2*a^2*c - 2*(6*a*b*c + a^2*d)*x^2)*sqrt(d*x^2 + c)*sqr
t(d))/(c*sqrt(d)*x^3), 1/6*(3*(b^2*c^2 + 4*a*b*c*d)*x^3*arctan(sqrt(-d)*x/sqrt(d
*x^2 + c)) + (3*b^2*c*x^4 - 2*a^2*c - 2*(6*a*b*c + a^2*d)*x^2)*sqrt(d*x^2 + c)*s
qrt(-d))/(c*sqrt(-d)*x^3)]

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Sympy [A]  time = 12.9028, size = 170, normalized size = 1.53 \[ - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 x^{2}} - \frac{a^{2} d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3 c} - \frac{2 a b \sqrt{c}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + 2 a b \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )} - \frac{2 a b d x}{\sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{b^{2} \sqrt{c} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} + \frac{b^{2} c \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.249634, size = 254, normalized size = 2.29 \[ \frac{1}{2} \, \sqrt{d x^{2} + c} b^{2} x - \frac{{\left (b^{2} c \sqrt{d} + 4 \, a b d^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{4 \, d} + \frac{2 \,{\left (6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c \sqrt{d} + 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} d^{\frac{3}{2}} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{2} \sqrt{d} + 6 \, a b c^{3} \sqrt{d} + a^{2} c^{2} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^4,x, algorithm="giac")

[Out]

1/2*sqrt(d*x^2 + c)*b^2*x - 1/4*(b^2*c*sqrt(d) + 4*a*b*d^(3/2))*ln((sqrt(d)*x -
sqrt(d*x^2 + c))^2)/d + 2/3*(6*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c*sqrt(d) + 3
*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*d^(3/2) - 12*(sqrt(d)*x - sqrt(d*x^2 + c))^
2*a*b*c^2*sqrt(d) + 6*a*b*c^3*sqrt(d) + a^2*c^2*d^(3/2))/((sqrt(d)*x - sqrt(d*x^
2 + c))^2 - c)^3