3.621 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=184 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac{x \left (c+d x^2\right )^{3/2} \left (8 a d (a d+3 b c)+3 b^2 c^2\right )}{12 c^2}+\frac{x \sqrt{c+d x^2} \left (8 a d (a d+3 b c)+3 b^2 c^2\right )}{8 c}+\frac{\left (8 a d (a d+3 b c)+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 \sqrt{d}}-\frac{2 a \left (c+d x^2\right )^{5/2} (a d+3 b c)}{3 c^2 x} \]

[Out]

((3*b^2*c^2 + 8*a*d*(3*b*c + a*d))*x*Sqrt[c + d*x^2])/(8*c) + ((3*b^2*c^2 + 8*a*
d*(3*b*c + a*d))*x*(c + d*x^2)^(3/2))/(12*c^2) - (a^2*(c + d*x^2)^(5/2))/(3*c*x^
3) - (2*a*(3*b*c + a*d)*(c + d*x^2)^(5/2))/(3*c^2*x) + ((3*b^2*c^2 + 8*a*d*(3*b*
c + a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*Sqrt[d])

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Rubi [A]  time = 0.344844, antiderivative size = 181, 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 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac{1}{12} x \left (c+d x^2\right )^{3/2} \left (\frac{8 a d (a d+3 b c)}{c^2}+3 b^2\right )+\frac{x \sqrt{c+d x^2} \left (8 a d (a d+3 b c)+3 b^2 c^2\right )}{8 c}+\frac{\left (8 a d (a d+3 b c)+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 \sqrt{d}}-\frac{2 a \left (c+d x^2\right )^{5/2} (a d+3 b c)}{3 c^2 x} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^4,x]

[Out]

((3*b^2*c^2 + 8*a*d*(3*b*c + a*d))*x*Sqrt[c + d*x^2])/(8*c) + ((3*b^2 + (8*a*d*(
3*b*c + a*d))/c^2)*x*(c + d*x^2)^(3/2))/12 - (a^2*(c + d*x^2)^(5/2))/(3*c*x^3) -
(2*a*(3*b*c + a*d)*(c + d*x^2)^(5/2))/(3*c^2*x) + ((3*b^2*c^2 + 8*a*d*(3*b*c +
a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*Sqrt[d])

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Rubi in Sympy [A]  time = 28.3801, size = 170, normalized size = 0.92 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{3 c x^{3}} - \frac{2 a \left (c + d x^{2}\right )^{\frac{5}{2}} \left (a d + 3 b c\right )}{3 c^{2} x} + \frac{\left (8 a d \left (a d + 3 b c\right ) + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 \sqrt{d}} + \frac{x \sqrt{c + d x^{2}} \left (a d \left (a d + 3 b c\right ) + \frac{3 b^{2} c^{2}}{8}\right )}{c} + \frac{x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (8 a d \left (a d + 3 b c\right ) + 3 b^{2} c^{2}\right )}{12 c^{2}} \]

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**4,x)

[Out]

-a**2*(c + d*x**2)**(5/2)/(3*c*x**3) - 2*a*(c + d*x**2)**(5/2)*(a*d + 3*b*c)/(3*
c**2*x) + (8*a*d*(a*d + 3*b*c) + 3*b**2*c**2)*atanh(sqrt(d)*x/sqrt(c + d*x**2))/
(8*sqrt(d)) + x*sqrt(c + d*x**2)*(a*d*(a*d + 3*b*c) + 3*b**2*c**2/8)/c + x*(c +
d*x**2)**(3/2)*(8*a*d*(a*d + 3*b*c) + 3*b**2*c**2)/(12*c**2)

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Mathematica [A]  time = 0.152712, size = 118, normalized size = 0.64 \[ \frac{1}{24} \left (\frac{3 \left (8 a^2 d^2+24 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}+\frac{\sqrt{c+d x^2} \left (-8 a^2 c+3 b x^4 (8 a d+5 b c)-16 a x^2 (2 a d+3 b c)+6 b^2 d x^6\right )}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^4,x]

[Out]

((Sqrt[c + d*x^2]*(-8*a^2*c - 16*a*(3*b*c + 2*a*d)*x^2 + 3*b*(5*b*c + 8*a*d)*x^4
+ 6*b^2*d*x^6))/x^3 + (3*(3*b^2*c^2 + 24*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]
*Sqrt[c + d*x^2]])/Sqrt[d])/24

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Maple [A]  time = 0.018, size = 241, normalized size = 1.3 \[{\frac{x{b}^{2}}{4} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{2}cx}{8}\sqrt{d{x}^{2}+c}}+{\frac{3\,{b}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}-{\frac{{a}^{2}}{3\,c{x}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{a}^{2}d}{3\,{c}^{2}x} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{a}^{2}{d}^{2}x}{3\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}{d}^{2}x}{c}\sqrt{d{x}^{2}+c}}+{a}^{2}{d}^{{\frac{3}{2}}}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) -2\,{\frac{ab \left ( d{x}^{2}+c \right ) ^{5/2}}{cx}}+2\,{\frac{abdx \left ( d{x}^{2}+c \right ) ^{3/2}}{c}}+3\,abdx\sqrt{d{x}^{2}+c}+3\,ab\sqrt{d}c\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^4,x)

[Out]

1/4*x*b^2*(d*x^2+c)^(3/2)+3/8*b^2*c*x*(d*x^2+c)^(1/2)+3/8*b^2*c^2/d^(1/2)*ln(x*d
^(1/2)+(d*x^2+c)^(1/2))-1/3*a^2*(d*x^2+c)^(5/2)/c/x^3-2/3*a^2*d/c^2/x*(d*x^2+c)^
(5/2)+2/3*a^2*d^2/c^2*x*(d*x^2+c)^(3/2)+a^2*d^2/c*x*(d*x^2+c)^(1/2)+a^2*d^(3/2)*
ln(x*d^(1/2)+(d*x^2+c)^(1/2))-2*a*b/c/x*(d*x^2+c)^(5/2)+2*a*b*d/c*x*(d*x^2+c)^(3
/2)+3*a*b*d*x*(d*x^2+c)^(1/2)+3*a*b*d^(1/2)*c*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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259439, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (3 \, b^{2} c^{2} + 24 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{3} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) + 2 \,{\left (6 \, b^{2} d x^{6} + 3 \,{\left (5 \, b^{2} c + 8 \, a b d\right )} x^{4} - 8 \, a^{2} c - 16 \,{\left (3 \, a b c + 2 \, a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{d}}{48 \, \sqrt{d} x^{3}}, \frac{3 \,{\left (3 \, b^{2} c^{2} + 24 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{3} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) +{\left (6 \, b^{2} d x^{6} + 3 \,{\left (5 \, b^{2} c + 8 \, a b d\right )} x^{4} - 8 \, a^{2} c - 16 \,{\left (3 \, a b c + 2 \, a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{24 \, \sqrt{-d} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x^4,x, algorithm="fricas")

[Out]

[1/48*(3*(3*b^2*c^2 + 24*a*b*c*d + 8*a^2*d^2)*x^3*log(-2*sqrt(d*x^2 + c)*d*x - (
2*d*x^2 + c)*sqrt(d)) + 2*(6*b^2*d*x^6 + 3*(5*b^2*c + 8*a*b*d)*x^4 - 8*a^2*c - 1
6*(3*a*b*c + 2*a^2*d)*x^2)*sqrt(d*x^2 + c)*sqrt(d))/(sqrt(d)*x^3), 1/24*(3*(3*b^
2*c^2 + 24*a*b*c*d + 8*a^2*d^2)*x^3*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (6*b^2*
d*x^6 + 3*(5*b^2*c + 8*a*b*d)*x^4 - 8*a^2*c - 16*(3*a*b*c + 2*a^2*d)*x^2)*sqrt(d
*x^2 + c)*sqrt(-d))/(sqrt(-d)*x^3)]

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Sympy [A]  time = 33.9108, size = 352, normalized size = 1.91 \[ - \frac{a^{2} \sqrt{c} d}{x \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a^{2} c \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 x^{2}} - \frac{a^{2} d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3} + a^{2} d^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )} - \frac{a^{2} d^{2} x}{\sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{2 a b c^{\frac{3}{2}}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + a b \sqrt{c} d x \sqrt{1 + \frac{d x^{2}}{c}} - \frac{2 a b \sqrt{c} d x}{\sqrt{1 + \frac{d x^{2}}{c}}} + 3 a b c \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )} + \frac{b^{2} c^{\frac{3}{2}} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} + \frac{b^{2} c^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} \sqrt{c} d x^{3}}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 \sqrt{d}} + \frac{b^{2} d^{2} x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.249167, size = 354, normalized size = 1.92 \[ \frac{1}{8} \,{\left (2 \, b^{2} d x^{2} + \frac{5 \, b^{2} c d^{2} + 8 \, a b d^{3}}{d^{2}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (3 \, b^{2} c^{2} \sqrt{d} + 24 \, a b c d^{\frac{3}{2}} + 8 \, a^{2} d^{\frac{5}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{16 \, d} + \frac{4 \,{\left (3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{2} \sqrt{d} + 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c d^{\frac{3}{2}} - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{3} \sqrt{d} - 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c^{2} d^{\frac{3}{2}} + 3 \, a b c^{4} \sqrt{d} + 2 \, a^{2} c^{3} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x^4,x, algorithm="giac")

[Out]

1/8*(2*b^2*d*x^2 + (5*b^2*c*d^2 + 8*a*b*d^3)/d^2)*sqrt(d*x^2 + c)*x - 1/16*(3*b^
2*c^2*sqrt(d) + 24*a*b*c*d^(3/2) + 8*a^2*d^(5/2))*ln((sqrt(d)*x - sqrt(d*x^2 + c
))^2)/d + 4/3*(3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^2*sqrt(d) + 3*(sqrt(d)*x
- sqrt(d*x^2 + c))^4*a^2*c*d^(3/2) - 6*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^3*s
qrt(d) - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c^2*d^(3/2) + 3*a*b*c^4*sqrt(d) +
2*a^2*c^3*d^(3/2))/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^3