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

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))*Ar
cTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(3/2))

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Rubi [A]  time = 0.234853, 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 \[ \frac{1}{8} x \sqrt{c+d x^2} \left (\frac{8 a^2 d}{c}+8 a b-\frac{b^2 c}{d}\right )-\frac{a^2 \left (c+d x^2\right )^{3/2}}{c x}-\frac{\left (b^2 c^2-8 a d (a d+b c)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{3/2}}+\frac{b^2 x \left (c+d x^2\right )^{3/2}}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.2129, size = 114, normalized size = 0.86 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{c x} + \frac{b^{2} x \left (c + d x^{2}\right )^{\frac{3}{2}}}{4 d} - \frac{\left (- 8 a d \left (a d + b c\right ) + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 d^{\frac{3}{2}}} - \frac{x \sqrt{c + d x^{2}} \left (- 8 a d \left (a d + b c\right ) + b^{2} c^{2}\right )}{8 c d} \]

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**2,x)

[Out]

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

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Mathematica [A]  time = 0.183502, 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 = 0.016, size = 163, normalized size = 1.2 \[{\frac{{b}^{2}x}{4\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}cx}{8\,d}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}-{\frac{{a}^{2}}{cx} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}dx}{c}\sqrt{d{x}^{2}+c}}+{a}^{2}\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) +abx\sqrt{d{x}^{2}+c}+{abc\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}} \]

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^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))+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))

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236915, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} c^{2} - 8 \, a b c d - 8 \, a^{2} d^{2}\right )} x \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) - 2 \,{\left (2 \, b^{2} d x^{4} - 8 \, a^{2} d +{\left (b^{2} c + 8 \, a b d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{d}}{16 \, d^{\frac{3}{2}} x}, -\frac{{\left (b^{2} c^{2} - 8 \, a b c d - 8 \, a^{2} d^{2}\right )} x \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) -{\left (2 \, b^{2} d x^{4} - 8 \, a^{2} d +{\left (b^{2} c + 8 \, a b d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{8 \, \sqrt{-d} d x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 21.7864, size = 219, normalized size = 1.65 \[ - \frac{a^{2} \sqrt{c}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + a^{2} \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )} - \frac{a^{2} d x}{\sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + a b \sqrt{c} x \sqrt{1 + \frac{d x^{2}}{c}} + \frac{a b c \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{\sqrt{d}} + \frac{b^{2} c^{\frac{3}{2}} x}{8 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} \sqrt{c} x^{3}}{8 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 d^{\frac{3}{2}}} + \frac{b^{2} d x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

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/XCAS [A]  time = 0.24256, size = 170, normalized size = 1.28 \[ \frac{2 \, a^{2} c \sqrt{d}}{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c} + \frac{1}{8} \,{\left (2 \, b^{2} x^{2} + \frac{b^{2} c d + 8 \, a b d^{2}}{d^{2}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (b^{2} c^{2} \sqrt{d} - 8 \, a b c d^{\frac{3}{2}} - 8 \, a^{2} d^{\frac{5}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{16 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/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))*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2)/d^2