3.622 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=181 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{4 c x^4}+\frac{\left (c+d x^2\right )^{3/2} \left (3 a d (a d+8 b c)+8 b^2 c^2\right )}{24 c^2}+\frac{\sqrt{c+d x^2} \left (3 a d (a d+8 b c)+8 b^2 c^2\right )}{8 c}-\frac{\left (3 a d (a d+8 b c)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 \sqrt{c}}-\frac{a \left (c+d x^2\right )^{5/2} (a d+8 b c)}{8 c^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.49698, antiderivative size = 178, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{4 c x^4}+\frac{1}{24} \left (c+d x^2\right )^{3/2} \left (\frac{3 a d (a d+8 b c)}{c^2}+8 b^2\right )+\frac{\sqrt{c+d x^2} \left (3 a d (a d+8 b c)+8 b^2 c^2\right )}{8 c}-\frac{\left (3 a d (a d+8 b c)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 \sqrt{c}}-\frac{a \left (c+d x^2\right )^{5/2} (a d+8 b c)}{8 c^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 32.8158, size = 167, normalized size = 0.92 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{4 c x^{4}} - \frac{a \left (c + d x^{2}\right )^{\frac{5}{2}} \left (a d + 8 b c\right )}{8 c^{2} x^{2}} + \frac{\sqrt{c + d x^{2}} \left (3 a d \left (a d + 8 b c\right ) + 8 b^{2} c^{2}\right )}{8 c} + \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (3 a d \left (a d + 8 b c\right ) + 8 b^{2} c^{2}\right )}{24 c^{2}} - \frac{\left (3 a d \left (a d + 8 b c\right ) + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{8 \sqrt{c}} \]

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**5,x)

[Out]

-a**2*(c + d*x**2)**(5/2)/(4*c*x**4) - a*(c + d*x**2)**(5/2)*(a*d + 8*b*c)/(8*c*
*2*x**2) + sqrt(c + d*x**2)*(3*a*d*(a*d + 8*b*c) + 8*b**2*c**2)/(8*c) + (c + d*x
**2)**(3/2)*(3*a*d*(a*d + 8*b*c) + 8*b**2*c**2)/(24*c**2) - (3*a*d*(a*d + 8*b*c)
+ 8*b**2*c**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(8*sqrt(c))

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Mathematica [A]  time = 0.283613, size = 150, normalized size = 0.83 \[ \frac{1}{24} \left (-\frac{3 \left (3 a^2 d^2+24 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{\sqrt{c}}+\frac{3 \log (x) \left (3 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{\sqrt{c}}+\frac{\sqrt{c+d x^2} \left (-3 a^2 \left (2 c+5 d x^2\right )-24 a b x^2 \left (c-2 d x^2\right )+8 b^2 x^4 \left (4 c+d x^2\right )\right )}{x^4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^5,x]

[Out]

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

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Maple [A]  time = 0.018, size = 256, normalized size = 1.4 \[ -{\frac{{a}^{2}}{4\,c{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}d}{8\,{c}^{2}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}{d}^{2}}{8\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{a}^{2}{d}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{3\,{a}^{2}{d}^{2}}{8\,c}\sqrt{d{x}^{2}+c}}+{\frac{{b}^{2}}{3} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{b}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{{\frac{3}{2}}}+{b}^{2}\sqrt{d{x}^{2}+c}c-{\frac{ab}{c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{abd}{c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-3\,abd\sqrt{c}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) +3\,abd\sqrt{d{x}^{2}+c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^5,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242509, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (8 \, b^{2} c^{2} + 24 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) + 2 \,{\left (8 \, b^{2} d x^{6} + 16 \,{\left (2 \, b^{2} c + 3 \, a b d\right )} x^{4} - 6 \, a^{2} c - 3 \,{\left (8 \, a b c + 5 \, a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c}}{48 \, \sqrt{c} x^{4}}, -\frac{3 \,{\left (8 \, b^{2} c^{2} + 24 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (8 \, b^{2} d x^{6} + 16 \,{\left (2 \, b^{2} c + 3 \, a b d\right )} x^{4} - 6 \, a^{2} c - 3 \,{\left (8 \, a b c + 5 \, a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{24 \, \sqrt{-c} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 89.0402, size = 332, normalized size = 1.83 \[ - \frac{a^{2} c^{2}}{4 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a^{2} c \sqrt{d}}{8 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a^{2} d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} - \frac{a^{2} d^{\frac{3}{2}}}{8 x \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a^{2} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{8 \sqrt{c}} - 3 a b \sqrt{c} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )} - \frac{a b c \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{x} + \frac{2 a b c \sqrt{d}}{x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{2 a b d^{\frac{3}{2}} x}{\sqrt{\frac{c}{d x^{2}} + 1}} - b^{2} c^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )} + \frac{b^{2} c^{2}}{\sqrt{d} x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{b^{2} c \sqrt{d} x}{\sqrt{\frac{c}{d x^{2}} + 1}} + b^{2} d \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x**5,x)

[Out]

-a**2*c**2/(4*sqrt(d)*x**5*sqrt(c/(d*x**2) + 1)) - 3*a**2*c*sqrt(d)/(8*x**3*sqrt
(c/(d*x**2) + 1)) - a**2*d**(3/2)*sqrt(c/(d*x**2) + 1)/(2*x) - a**2*d**(3/2)/(8*
x*sqrt(c/(d*x**2) + 1)) - 3*a**2*d**2*asinh(sqrt(c)/(sqrt(d)*x))/(8*sqrt(c)) - 3
*a*b*sqrt(c)*d*asinh(sqrt(c)/(sqrt(d)*x)) - a*b*c*sqrt(d)*sqrt(c/(d*x**2) + 1)/x
+ 2*a*b*c*sqrt(d)/(x*sqrt(c/(d*x**2) + 1)) + 2*a*b*d**(3/2)*x/sqrt(c/(d*x**2) +
1) - b**2*c**(3/2)*asinh(sqrt(c)/(sqrt(d)*x)) + b**2*c**2/(sqrt(d)*x*sqrt(c/(d*
x**2) + 1)) + b**2*c*sqrt(d)*x/sqrt(c/(d*x**2) + 1) + b**2*d*Piecewise((sqrt(c)*
x**2/2, Eq(d, 0)), ((c + d*x**2)**(3/2)/(3*d), True))

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GIAC/XCAS [A]  time = 0.242229, size = 246, normalized size = 1.36 \[ \frac{8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} d + 24 \, \sqrt{d x^{2} + c} b^{2} c d + 48 \, \sqrt{d x^{2} + c} a b d^{2} + \frac{3 \,{\left (8 \, b^{2} c^{2} d + 24 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{3 \,{\left (8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d^{2} - 8 \, \sqrt{d x^{2} + c} a b c^{2} d^{2} + 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{3} - 3 \, \sqrt{d x^{2} + c} a^{2} c d^{3}\right )}}{d^{2} x^{4}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x^5,x, algorithm="giac")

[Out]

1/24*(8*(d*x^2 + c)^(3/2)*b^2*d + 24*sqrt(d*x^2 + c)*b^2*c*d + 48*sqrt(d*x^2 + c
)*a*b*d^2 + 3*(8*b^2*c^2*d + 24*a*b*c*d^2 + 3*a^2*d^3)*arctan(sqrt(d*x^2 + c)/sq
rt(-c))/sqrt(-c) - 3*(8*(d*x^2 + c)^(3/2)*a*b*c*d^2 - 8*sqrt(d*x^2 + c)*a*b*c^2*
d^2 + 5*(d*x^2 + c)^(3/2)*a^2*d^3 - 3*sqrt(d*x^2 + c)*a^2*c*d^3)/(d^2*x^4))/d