3.632 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=223 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}+\frac{x \left (c+d x^2\right )^{5/2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )}{6 c^2}+\frac{5 x \left (c+d x^2\right )^{3/2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )}{24 c}+\frac{5}{16} x \sqrt{c+d x^2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )+\frac{5 c \left (4 a d (2 a d+3 b c)+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 \sqrt{d}}-\frac{2 a \left (c+d x^2\right )^{7/2} (2 a d+3 b c)}{3 c^2 x} \]

[Out]

(5*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*x*Sqrt[c + d*x^2])/16 + (5*(b^2*c^2 + 4*a*d
*(3*b*c + 2*a*d))*x*(c + d*x^2)^(3/2))/(24*c) + ((b^2*c^2 + 4*a*d*(3*b*c + 2*a*d
))*x*(c + d*x^2)^(5/2))/(6*c^2) - (a^2*(c + d*x^2)^(7/2))/(3*c*x^3) - (2*a*(3*b*
c + 2*a*d)*(c + d*x^2)^(7/2))/(3*c^2*x) + (5*c*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))
*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*Sqrt[d])

_______________________________________________________________________________________

Rubi [A]  time = 0.443085, antiderivative size = 219, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}+\frac{1}{6} x \left (c+d x^2\right )^{5/2} \left (\frac{4 a d (2 a d+3 b c)}{c^2}+b^2\right )+\frac{5 x \left (c+d x^2\right )^{3/2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )}{24 c}+\frac{5}{16} x \sqrt{c+d x^2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )+\frac{5 c \left (4 a d (2 a d+3 b c)+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 \sqrt{d}}-\frac{2 a \left (c+d x^2\right )^{7/2} (2 a d+3 b c)}{3 c^2 x} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^4,x]

[Out]

(5*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*x*Sqrt[c + d*x^2])/16 + (5*(b^2*c^2 + 4*a*d
*(3*b*c + 2*a*d))*x*(c + d*x^2)^(3/2))/(24*c) + ((b^2 + (4*a*d*(3*b*c + 2*a*d))/
c^2)*x*(c + d*x^2)^(5/2))/6 - (a^2*(c + d*x^2)^(7/2))/(3*c*x^3) - (2*a*(3*b*c +
2*a*d)*(c + d*x^2)^(7/2))/(3*c^2*x) + (5*c*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*Arc
Tanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*Sqrt[d])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.0122, size = 218, normalized size = 0.98 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{3 c x^{3}} - \frac{2 a \left (c + d x^{2}\right )^{\frac{7}{2}} \left (2 a d + 3 b c\right )}{3 c^{2} x} + \frac{5 c \left (4 a d \left (2 a d + 3 b c\right ) + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{16 \sqrt{d}} + x \sqrt{c + d x^{2}} \left (\frac{5 a d \left (2 a d + 3 b c\right )}{4} + \frac{5 b^{2} c^{2}}{16}\right ) + \frac{5 x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (4 a d \left (2 a d + 3 b c\right ) + b^{2} c^{2}\right )}{24 c} + \frac{x \left (c + d x^{2}\right )^{\frac{5}{2}} \left (4 a d \left (2 a d + 3 b c\right ) + b^{2} c^{2}\right )}{6 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(5/2)/x**4,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.205568, size = 155, normalized size = 0.7 \[ \frac{1}{48} \left (\frac{15 c \left (8 a^2 d^2+12 a b c d+b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}+\frac{\sqrt{c+d x^2} \left (-8 a^2 \left (2 c^2+14 c d x^2-3 d^2 x^4\right )+12 a b x^2 \left (-8 c^2+9 c d x^2+2 d^2 x^4\right )+b^2 x^4 \left (33 c^2+26 c d x^2+8 d^2 x^4\right )\right )}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^4,x]

[Out]

((Sqrt[c + d*x^2]*(-8*a^2*(2*c^2 + 14*c*d*x^2 - 3*d^2*x^4) + 12*a*b*x^2*(-8*c^2
+ 9*c*d*x^2 + 2*d^2*x^4) + b^2*x^4*(33*c^2 + 26*c*d*x^2 + 8*d^2*x^4)))/x^3 + (15
*c*(b^2*c^2 + 12*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d
])/48

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 298, normalized size = 1.3 \[{\frac{x{b}^{2}}{6} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{b}^{2}cx}{24} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}{c}^{2}x}{16}\sqrt{d{x}^{2}+c}}+{\frac{5\,{b}^{2}{c}^{3}}{16}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}-{\frac{{a}^{2}}{3\,c{x}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{4\,{a}^{2}d}{3\,{c}^{2}x} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{4\,{a}^{2}{d}^{2}x}{3\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{2}{d}^{2}x}{3\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{d}^{2}x}{2}\sqrt{d{x}^{2}+c}}+{\frac{5\,{a}^{2}c}{2}{d}^{{\frac{3}{2}}}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }-2\,{\frac{ab \left ( d{x}^{2}+c \right ) ^{7/2}}{cx}}+2\,{\frac{abdx \left ( d{x}^{2}+c \right ) ^{5/2}}{c}}+{\frac{5\,abdx}{2} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{15\,cabdx}{4}\sqrt{d{x}^{2}+c}}+{\frac{15\,ab{c}^{2}}{4}\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^4,x)

[Out]

1/6*x*b^2*(d*x^2+c)^(5/2)+5/24*b^2*c*x*(d*x^2+c)^(3/2)+5/16*b^2*c^2*x*(d*x^2+c)^
(1/2)+5/16*b^2*c^3/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-1/3*a^2*(d*x^2+c)^(7/2)
/c/x^3-4/3*a^2*d/c^2/x*(d*x^2+c)^(7/2)+4/3*a^2*d^2/c^2*x*(d*x^2+c)^(5/2)+5/3*a^2
*d^2/c*x*(d*x^2+c)^(3/2)+5/2*a^2*d^2*x*(d*x^2+c)^(1/2)+5/2*a^2*d^(3/2)*c*ln(x*d^
(1/2)+(d*x^2+c)^(1/2))-2*a*b/c/x*(d*x^2+c)^(7/2)+2*a*b*d/c*x*(d*x^2+c)^(5/2)+5/2
*a*b*d*x*(d*x^2+c)^(3/2)+15/4*a*b*d*c*x*(d*x^2+c)^(1/2)+15/4*a*b*d^(1/2)*c^2*ln(
x*d^(1/2)+(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*(d*x^2 + c)^(5/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.354301, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (b^{2} c^{3} + 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} x^{3} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) + 2 \,{\left (8 \, b^{2} d^{2} x^{8} + 2 \,{\left (13 \, b^{2} c d + 12 \, a b d^{2}\right )} x^{6} + 3 \,{\left (11 \, b^{2} c^{2} + 36 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} - 16 \, a^{2} c^{2} - 16 \,{\left (6 \, a b c^{2} + 7 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{d}}{96 \, \sqrt{d} x^{3}}, \frac{15 \,{\left (b^{2} c^{3} + 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} x^{3} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) +{\left (8 \, b^{2} d^{2} x^{8} + 2 \,{\left (13 \, b^{2} c d + 12 \, a b d^{2}\right )} x^{6} + 3 \,{\left (11 \, b^{2} c^{2} + 36 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} - 16 \, a^{2} c^{2} - 16 \,{\left (6 \, a b c^{2} + 7 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{48 \, \sqrt{-d} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)/x^4,x, algorithm="fricas")

[Out]

[1/96*(15*(b^2*c^3 + 12*a*b*c^2*d + 8*a^2*c*d^2)*x^3*log(-2*sqrt(d*x^2 + c)*d*x
- (2*d*x^2 + c)*sqrt(d)) + 2*(8*b^2*d^2*x^8 + 2*(13*b^2*c*d + 12*a*b*d^2)*x^6 +
3*(11*b^2*c^2 + 36*a*b*c*d + 8*a^2*d^2)*x^4 - 16*a^2*c^2 - 16*(6*a*b*c^2 + 7*a^2
*c*d)*x^2)*sqrt(d*x^2 + c)*sqrt(d))/(sqrt(d)*x^3), 1/48*(15*(b^2*c^3 + 12*a*b*c^
2*d + 8*a^2*c*d^2)*x^3*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (8*b^2*d^2*x^8 + 2*(
13*b^2*c*d + 12*a*b*d^2)*x^6 + 3*(11*b^2*c^2 + 36*a*b*c*d + 8*a^2*d^2)*x^4 - 16*
a^2*c^2 - 16*(6*a*b*c^2 + 7*a^2*c*d)*x^2)*sqrt(d*x^2 + c)*sqrt(-d))/(sqrt(-d)*x^
3)]

_______________________________________________________________________________________

Sympy [A]  time = 74.2006, size = 490, normalized size = 2.2 \[ - \frac{2 a^{2} c^{\frac{3}{2}} d}{x \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{a^{2} \sqrt{c} d^{2} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} - \frac{2 a^{2} \sqrt{c} d^{2} x}{\sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a^{2} c^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 x^{2}} - \frac{a^{2} c d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3} + \frac{5 a^{2} c d^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2} - \frac{2 a b c^{\frac{5}{2}}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + 2 a b c^{\frac{3}{2}} d x \sqrt{1 + \frac{d x^{2}}{c}} - \frac{7 a b c^{\frac{3}{2}} d x}{4 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a b \sqrt{c} d^{2} x^{3}}{4 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{15 a b c^{2} \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{4} + \frac{a b d^{3} x^{5}}{2 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{b^{2} c^{\frac{5}{2}} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} + \frac{3 b^{2} c^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{35 b^{2} c^{\frac{3}{2}} d x^{3}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 b^{2} \sqrt{c} d^{2} x^{5}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{16 \sqrt{d}} + \frac{b^{2} d^{3} x^{7}}{6 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(5/2)/x**4,x)

[Out]

-2*a**2*c**(3/2)*d/(x*sqrt(1 + d*x**2/c)) + a**2*sqrt(c)*d**2*x*sqrt(1 + d*x**2/
c)/2 - 2*a**2*sqrt(c)*d**2*x/sqrt(1 + d*x**2/c) - a**2*c**2*sqrt(d)*sqrt(c/(d*x*
*2) + 1)/(3*x**2) - a**2*c*d**(3/2)*sqrt(c/(d*x**2) + 1)/3 + 5*a**2*c*d**(3/2)*a
sinh(sqrt(d)*x/sqrt(c))/2 - 2*a*b*c**(5/2)/(x*sqrt(1 + d*x**2/c)) + 2*a*b*c**(3/
2)*d*x*sqrt(1 + d*x**2/c) - 7*a*b*c**(3/2)*d*x/(4*sqrt(1 + d*x**2/c)) + 3*a*b*sq
rt(c)*d**2*x**3/(4*sqrt(1 + d*x**2/c)) + 15*a*b*c**2*sqrt(d)*asinh(sqrt(d)*x/sqr
t(c))/4 + a*b*d**3*x**5/(2*sqrt(c)*sqrt(1 + d*x**2/c)) + b**2*c**(5/2)*x*sqrt(1
+ d*x**2/c)/2 + 3*b**2*c**(5/2)*x/(16*sqrt(1 + d*x**2/c)) + 35*b**2*c**(3/2)*d*x
**3/(48*sqrt(1 + d*x**2/c)) + 17*b**2*sqrt(c)*d**2*x**5/(24*sqrt(1 + d*x**2/c))
+ 5*b**2*c**3*asinh(sqrt(d)*x/sqrt(c))/(16*sqrt(d)) + b**2*d**3*x**7/(6*sqrt(c)*
sqrt(1 + d*x**2/c))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.245588, size = 414, normalized size = 1.86 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, b^{2} d^{2} x^{2} + \frac{13 \, b^{2} c d^{5} + 12 \, a b d^{6}}{d^{4}}\right )} x^{2} + \frac{3 \,{\left (11 \, b^{2} c^{2} d^{4} + 36 \, a b c d^{5} + 8 \, a^{2} d^{6}\right )}}{d^{4}}\right )} \sqrt{d x^{2} + c} x - \frac{5 \,{\left (b^{2} c^{3} \sqrt{d} + 12 \, a b c^{2} d^{\frac{3}{2}} + 8 \, a^{2} c d^{\frac{5}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{32 \, d} + \frac{2 \,{\left (6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{3} \sqrt{d} + 9 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac{3}{2}} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{4} \sqrt{d} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c^{3} d^{\frac{3}{2}} + 6 \, a b c^{5} \sqrt{d} + 7 \, a^{2} c^{4} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)/x^4,x, algorithm="giac")

[Out]

1/48*(2*(4*b^2*d^2*x^2 + (13*b^2*c*d^5 + 12*a*b*d^6)/d^4)*x^2 + 3*(11*b^2*c^2*d^
4 + 36*a*b*c*d^5 + 8*a^2*d^6)/d^4)*sqrt(d*x^2 + c)*x - 5/32*(b^2*c^3*sqrt(d) + 1
2*a*b*c^2*d^(3/2) + 8*a^2*c*d^(5/2))*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2)/d + 2/3
*(6*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^3*sqrt(d) + 9*(sqrt(d)*x - sqrt(d*x^2
+ c))^4*a^2*c^2*d^(3/2) - 12*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^4*sqrt(d) - 1
2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c^3*d^(3/2) + 6*a*b*c^5*sqrt(d) + 7*a^2*c^
4*d^(3/2))/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^3