3.619 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=175 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{c x}-\frac{c \left (b^2 c^2-12 a d (2 a d+b c)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 d^{3/2}}-\frac{x \left (c+d x^2\right )^{3/2} \left (b^2 c^2-12 a d (2 a d+b c)\right )}{24 c d}-\frac{x \sqrt{c+d x^2} \left (b^2 c^2-12 a d (2 a d+b c)\right )}{16 d}+\frac{b^2 x \left (c+d x^2\right )^{5/2}}{6 d} \]

[Out]

-((b^2*c^2 - 12*a*d*(b*c + 2*a*d))*x*Sqrt[c + d*x^2])/(16*d) - ((b^2*c^2 - 12*a*
d*(b*c + 2*a*d))*x*(c + d*x^2)^(3/2))/(24*c*d) - (a^2*(c + d*x^2)^(5/2))/(c*x) +
(b^2*x*(c + d*x^2)^(5/2))/(6*d) - (c*(b^2*c^2 - 12*a*d*(b*c + 2*a*d))*ArcTanh[(
Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d^(3/2))

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Rubi [A]  time = 0.29686, antiderivative size = 172, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{1}{24} x \left (c+d x^2\right )^{3/2} \left (\frac{24 a^2 d}{c}+12 a b-\frac{b^2 c}{d}\right )-\frac{a^2 \left (c+d x^2\right )^{5/2}}{c x}-\frac{c \left (b^2 c^2-12 a d (2 a d+b c)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 d^{3/2}}-\frac{x \sqrt{c+d x^2} \left (b^2 c^2-12 a d (2 a d+b c)\right )}{16 d}+\frac{b^2 x \left (c+d x^2\right )^{5/2}}{6 d} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^2,x]

[Out]

-((b^2*c^2 - 12*a*d*(b*c + 2*a*d))*x*Sqrt[c + d*x^2])/(16*d) + ((12*a*b - (b^2*c
)/d + (24*a^2*d)/c)*x*(c + d*x^2)^(3/2))/24 - (a^2*(c + d*x^2)^(5/2))/(c*x) + (b
^2*x*(c + d*x^2)^(5/2))/(6*d) - (c*(b^2*c^2 - 12*a*d*(b*c + 2*a*d))*ArcTanh[(Sqr
t[d]*x)/Sqrt[c + d*x^2]])/(16*d^(3/2))

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Rubi in Sympy [A]  time = 28.0679, size = 155, normalized size = 0.89 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{c x} + \frac{b^{2} x \left (c + d x^{2}\right )^{\frac{5}{2}}}{6 d} - \frac{c \left (- 12 a d \left (2 a d + b c\right ) + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{16 d^{\frac{3}{2}}} - \frac{x \sqrt{c + d x^{2}} \left (- 12 a d \left (2 a d + b c\right ) + b^{2} c^{2}\right )}{16 d} - \frac{x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (- 12 a d \left (2 a d + b c\right ) + b^{2} c^{2}\right )}{24 c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x**2,x)

[Out]

-a**2*(c + d*x**2)**(5/2)/(c*x) + b**2*x*(c + d*x**2)**(5/2)/(6*d) - c*(-12*a*d*
(2*a*d + b*c) + b**2*c**2)*atanh(sqrt(d)*x/sqrt(c + d*x**2))/(16*d**(3/2)) - x*s
qrt(c + d*x**2)*(-12*a*d*(2*a*d + b*c) + b**2*c**2)/(16*d) - x*(c + d*x**2)**(3/
2)*(-12*a*d*(2*a*d + b*c) + b**2*c**2)/(24*c*d)

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Mathematica [A]  time = 0.23786, size = 135, normalized size = 0.77 \[ \sqrt{c+d x^2} \left (\frac{x \left (8 a^2 d^2+20 a b c d+b^2 c^2\right )}{16 d}-\frac{a^2 c}{x}+\frac{1}{24} b x^3 (12 a d+7 b c)+\frac{1}{6} b^2 d x^5\right )-\frac{c \left (-24 a^2 d^2-12 a b c d+b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{16 d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^2,x]

[Out]

Sqrt[c + d*x^2]*(-((a^2*c)/x) + ((b^2*c^2 + 20*a*b*c*d + 8*a^2*d^2)*x)/(16*d) +
(b*(7*b*c + 12*a*d)*x^3)/24 + (b^2*d*x^5)/6) - (c*(b^2*c^2 - 12*a*b*c*d - 24*a^2
*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(16*d^(3/2))

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Maple [A]  time = 0.016, size = 221, normalized size = 1.3 \[{\frac{{b}^{2}x}{6\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}cx}{24\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}{c}^{2}x}{16\,d}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}{c}^{3}}{16}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}-{\frac{{a}^{2}}{cx} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}dx}{c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}dx}{2}\sqrt{d{x}^{2}+c}}+{\frac{3\,{a}^{2}c}{2}\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }+{\frac{abx}{2} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,abcx}{4}\sqrt{d{x}^{2}+c}}+{\frac{3\,ab{c}^{2}}{4}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^2,x)

[Out]

1/6*b^2*x*(d*x^2+c)^(5/2)/d-1/24*b^2*c/d*x*(d*x^2+c)^(3/2)-1/16*b^2*c^2/d*x*(d*x
^2+c)^(1/2)-1/16*b^2*c^3/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-a^2*(d*x^2+c)^(5/
2)/c/x+a^2*d/c*x*(d*x^2+c)^(3/2)+3/2*a^2*d*x*(d*x^2+c)^(1/2)+3/2*a^2*d^(1/2)*c*l
n(x*d^(1/2)+(d*x^2+c)^(1/2))+1/2*a*b*x*(d*x^2+c)^(3/2)+3/4*a*b*c*x*(d*x^2+c)^(1/
2)+3/4*a*b*c^2/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*(d*x^2 + c)^(3/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280115, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{2} c^{3} - 12 \, a b c^{2} d - 24 \, a^{2} c d^{2}\right )} x \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) - 2 \,{\left (8 \, b^{2} d^{2} x^{6} + 2 \,{\left (7 \, b^{2} c d + 12 \, a b d^{2}\right )} x^{4} - 48 \, a^{2} c d + 3 \,{\left (b^{2} c^{2} + 20 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{d}}{96 \, d^{\frac{3}{2}} x}, -\frac{3 \,{\left (b^{2} c^{3} - 12 \, a b c^{2} d - 24 \, a^{2} c d^{2}\right )} x \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) -{\left (8 \, b^{2} d^{2} x^{6} + 2 \,{\left (7 \, b^{2} c d + 12 \, a b d^{2}\right )} x^{4} - 48 \, a^{2} c d + 3 \,{\left (b^{2} c^{2} + 20 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{48 \, \sqrt{-d} d x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x^2,x, algorithm="fricas")

[Out]

[-1/96*(3*(b^2*c^3 - 12*a*b*c^2*d - 24*a^2*c*d^2)*x*log(-2*sqrt(d*x^2 + c)*d*x -
(2*d*x^2 + c)*sqrt(d)) - 2*(8*b^2*d^2*x^6 + 2*(7*b^2*c*d + 12*a*b*d^2)*x^4 - 48
*a^2*c*d + 3*(b^2*c^2 + 20*a*b*c*d + 8*a^2*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(d))/(d
^(3/2)*x), -1/48*(3*(b^2*c^3 - 12*a*b*c^2*d - 24*a^2*c*d^2)*x*arctan(sqrt(-d)*x/
sqrt(d*x^2 + c)) - (8*b^2*d^2*x^6 + 2*(7*b^2*c*d + 12*a*b*d^2)*x^4 - 48*a^2*c*d
+ 3*(b^2*c^2 + 20*a*b*c*d + 8*a^2*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(-d))/(sqrt(-d)*
d*x)]

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Sympy [A]  time = 56.6956, size = 367, normalized size = 2.1 \[ - \frac{a^{2} c^{\frac{3}{2}}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{a^{2} \sqrt{c} d x \sqrt{1 + \frac{d x^{2}}{c}}}{2} - \frac{a^{2} \sqrt{c} d x}{\sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a^{2} c \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2} + a b c^{\frac{3}{2}} x \sqrt{1 + \frac{d x^{2}}{c}} + \frac{a b c^{\frac{3}{2}} x}{4 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a b \sqrt{c} d x^{3}}{4 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{4 \sqrt{d}} + \frac{a b d^{2} x^{5}}{2 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{b^{2} c^{\frac{5}{2}} x}{16 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 b^{2} c^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{11 b^{2} \sqrt{c} d x^{5}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{16 d^{\frac{3}{2}}} + \frac{b^{2} d^{2} x^{7}}{6 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x**2,x)

[Out]

-a**2*c**(3/2)/(x*sqrt(1 + d*x**2/c)) + a**2*sqrt(c)*d*x*sqrt(1 + d*x**2/c)/2 -
a**2*sqrt(c)*d*x/sqrt(1 + d*x**2/c) + 3*a**2*c*sqrt(d)*asinh(sqrt(d)*x/sqrt(c))/
2 + a*b*c**(3/2)*x*sqrt(1 + d*x**2/c) + a*b*c**(3/2)*x/(4*sqrt(1 + d*x**2/c)) +
3*a*b*sqrt(c)*d*x**3/(4*sqrt(1 + d*x**2/c)) + 3*a*b*c**2*asinh(sqrt(d)*x/sqrt(c)
)/(4*sqrt(d)) + a*b*d**2*x**5/(2*sqrt(c)*sqrt(1 + d*x**2/c)) + b**2*c**(5/2)*x/(
16*d*sqrt(1 + d*x**2/c)) + 17*b**2*c**(3/2)*x**3/(48*sqrt(1 + d*x**2/c)) + 11*b*
*2*sqrt(c)*d*x**5/(24*sqrt(1 + d*x**2/c)) - b**2*c**3*asinh(sqrt(d)*x/sqrt(c))/(
16*d**(3/2)) + b**2*d**2*x**7/(6*sqrt(c)*sqrt(1 + d*x**2/c))

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GIAC/XCAS [A]  time = 0.249414, size = 234, normalized size = 1.34 \[ \frac{2 \, a^{2} c^{2} \sqrt{d}}{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c} + \frac{1}{48} \,{\left (2 \,{\left (4 \, b^{2} d x^{2} + \frac{7 \, b^{2} c d^{4} + 12 \, a b d^{5}}{d^{4}}\right )} x^{2} + \frac{3 \,{\left (b^{2} c^{2} d^{3} + 20 \, a b c d^{4} + 8 \, a^{2} d^{5}\right )}}{d^{4}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (b^{2} c^{3} \sqrt{d} - 12 \, a b c^{2} d^{\frac{3}{2}} - 24 \, a^{2} c d^{\frac{5}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{32 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x^2,x, algorithm="giac")

[Out]

2*a^2*c^2*sqrt(d)/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c) + 1/48*(2*(4*b^2*d*x^2 +
(7*b^2*c*d^4 + 12*a*b*d^5)/d^4)*x^2 + 3*(b^2*c^2*d^3 + 20*a*b*c*d^4 + 8*a^2*d^5
)/d^4)*sqrt(d*x^2 + c)*x + 1/32*(b^2*c^3*sqrt(d) - 12*a*b*c^2*d^(3/2) - 24*a^2*c
*d^(5/2))*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2)/d^2