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

Optimal. Leaf size=143 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{4 c x^4}+\frac{\sqrt{c+d x^2} \left (a d (8 b c-a d)+8 b^2 c^2\right )}{8 c^2}-\frac{\left (a d (8 b c-a d)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{3/2}}-\frac{a \left (c+d x^2\right )^{3/2} (8 b c-a d)}{8 c^2 x^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.406551, antiderivative size = 140, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{4 c x^4}+\frac{1}{8} \sqrt{c+d x^2} \left (\frac{a d (8 b c-a d)}{c^2}+8 b^2\right )-\frac{\left (a d (8 b c-a d)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{3/2}}-\frac{a \left (c+d x^2\right )^{3/2} (8 b c-a d)}{8 c^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.889, size = 128, normalized size = 0.9 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{4 c x^{4}} + \frac{a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - 8 b c\right )}{8 c^{2} x^{2}} + \frac{\sqrt{c + d x^{2}} \left (- a d \left (a d - 8 b c\right ) + 8 b^{2} c^{2}\right )}{8 c^{2}} - \frac{\left (- a d \left (a d - 8 b c\right ) + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{8 c^{\frac{3}{2}}} \]

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**5,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.297864, size = 130, normalized size = 0.91 \[ \frac{\left (a^2 d^2-8 a b c d-8 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{8 c^{3/2}}-\frac{\log (x) \left (a^2 d^2-8 a b c d-8 b^2 c^2\right )}{8 c^{3/2}}+\sqrt{c+d x^2} \left (-\frac{a^2}{4 x^4}-\frac{a (a d+8 b c)}{8 c x^2}+b^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 207, normalized size = 1.5 \[ -{\frac{{a}^{2}}{4\,c{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}d}{8\,{c}^{2}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}{d}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-{\frac{{a}^{2}{d}^{2}}{8\,{c}^{2}}\sqrt{d{x}^{2}+c}}-\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) \sqrt{c}{b}^{2}+\sqrt{d{x}^{2}+c}{b}^{2}-{\frac{ab}{c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{abd\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{abd}{c}\sqrt{d{x}^{2}+c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^5,x)

[Out]

-1/4*a^2*(d*x^2+c)^(3/2)/c/x^4+1/8*a^2*d/c^2/x^2*(d*x^2+c)^(3/2)+1/8*a^2*d^2/c^(
3/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-1/8*a^2*d^2/c^2*(d*x^2+c)^(1/2)-ln((2
*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)*c^(1/2)*b^2+(d*x^2+c)^(1/2)*b^2-a*b/c/x^2*(d*x^
2+c)^(3/2)-a*b*d/c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+a*b*d/c*(d*x^2+c)
^(1/2)

_______________________________________________________________________________________

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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.233486, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (8 \, b^{2} c^{2} + 8 \, a b c d - a^{2} d^{2}\right )} x^{4} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \,{\left (8 \, b^{2} c x^{4} - 2 \, a^{2} c -{\left (8 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c}}{16 \, c^{\frac{3}{2}} x^{4}}, -\frac{{\left (8 \, b^{2} c^{2} + 8 \, a b c d - a^{2} d^{2}\right )} x^{4} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (8 \, b^{2} c x^{4} - 2 \, a^{2} c -{\left (8 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{8 \, \sqrt{-c} c x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 66.9338, size = 219, normalized size = 1.53 \[ - \frac{a^{2} c}{4 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a^{2} \sqrt{d}}{8 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a^{2} d^{\frac{3}{2}}}{8 c x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{8 c^{\frac{3}{2}}} - \frac{a b \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{x} - \frac{a b d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{\sqrt{c}} - b^{2} \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )} + \frac{b^{2} c}{\sqrt{d} x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{b^{2} \sqrt{d} x}{\sqrt{\frac{c}{d x^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**5,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.239792, size = 207, normalized size = 1.45 \[ \frac{8 \, \sqrt{d x^{2} + c} b^{2} d + \frac{{\left (8 \, b^{2} c^{2} d + 8 \, a b c d^{2} - a^{2} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d^{2} - 8 \, \sqrt{d x^{2} + c} a b c^{2} d^{2} +{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{3} + \sqrt{d x^{2} + c} a^{2} c d^{3}}{c d^{2} x^{4}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(8*sqrt(d*x^2 + c)*b^2*d + (8*b^2*c^2*d + 8*a*b*c*d^2 - a^2*d^3)*arctan(sqrt
(d*x^2 + c)/sqrt(-c))/(sqrt(-c)*c) - (8*(d*x^2 + c)^(3/2)*a*b*c*d^2 - 8*sqrt(d*x
^2 + c)*a*b*c^2*d^2 + (d*x^2 + c)^(3/2)*a^2*d^3 + sqrt(d*x^2 + c)*a^2*c*d^3)/(c*
d^2*x^4))/d