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

Optimal. Leaf size=222 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}+\frac{\left (c+d x^2\right )^{5/2} \left (5 a d (3 a d+8 b c)+8 b^2 c^2\right )}{40 c^2}+\frac{\left (c+d x^2\right )^{3/2} \left (5 a d (3 a d+8 b c)+8 b^2 c^2\right )}{24 c}+\frac{1}{8} \sqrt{c+d x^2} \left (5 a d (3 a d+8 b c)+8 b^2 c^2\right )-\frac{1}{8} \sqrt{c} \left (5 a d (3 a d+8 b c)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )-\frac{a \left (c+d x^2\right )^{7/2} (3 a d+8 b c)}{8 c^2 x^2} \]

[Out]

((8*b^2*c^2 + 5*a*d*(8*b*c + 3*a*d))*Sqrt[c + d*x^2])/8 + ((8*b^2*c^2 + 5*a*d*(8
*b*c + 3*a*d))*(c + d*x^2)^(3/2))/(24*c) + ((8*b^2*c^2 + 5*a*d*(8*b*c + 3*a*d))*
(c + d*x^2)^(5/2))/(40*c^2) - (a^2*(c + d*x^2)^(7/2))/(4*c*x^4) - (a*(8*b*c + 3*
a*d)*(c + d*x^2)^(7/2))/(8*c^2*x^2) - (Sqrt[c]*(8*b^2*c^2 + 5*a*d*(8*b*c + 3*a*d
))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/8

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Rubi [A]  time = 0.594885, antiderivative size = 219, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{4 c x^4}+\frac{1}{40} \left (c+d x^2\right )^{5/2} \left (\frac{5 a d (3 a d+8 b c)}{c^2}+8 b^2\right )+\frac{\left (c+d x^2\right )^{3/2} \left (5 a d (3 a d+8 b c)+8 b^2 c^2\right )}{24 c}+\frac{1}{8} \sqrt{c+d x^2} \left (5 a d (3 a d+8 b c)+8 b^2 c^2\right )-\frac{1}{8} \sqrt{c} \left (5 a d (3 a d+8 b c)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )-\frac{a \left (c+d x^2\right )^{7/2} (3 a d+8 b c)}{8 c^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

((8*b^2*c^2 + 5*a*d*(8*b*c + 3*a*d))*Sqrt[c + d*x^2])/8 + ((8*b^2*c^2 + 5*a*d*(8
*b*c + 3*a*d))*(c + d*x^2)^(3/2))/(24*c) + ((8*b^2 + (5*a*d*(8*b*c + 3*a*d))/c^2
)*(c + d*x^2)^(5/2))/40 - (a^2*(c + d*x^2)^(7/2))/(4*c*x^4) - (a*(8*b*c + 3*a*d)
*(c + d*x^2)^(7/2))/(8*c^2*x^2) - (Sqrt[c]*(8*b^2*c^2 + 5*a*d*(8*b*c + 3*a*d))*A
rcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/8

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Rubi in Sympy [A]  time = 38.0435, size = 207, normalized size = 0.93 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{4 c x^{4}} - \frac{a \left (c + d x^{2}\right )^{\frac{7}{2}} \left (3 a d + 8 b c\right )}{8 c^{2} x^{2}} - \frac{\sqrt{c} \left (5 a d \left (3 a d + 8 b c\right ) + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{8} + \sqrt{c + d x^{2}} \left (\frac{5 a d \left (3 a d + 8 b c\right )}{8} + b^{2} c^{2}\right ) + \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (5 a d \left (3 a d + 8 b c\right ) + 8 b^{2} c^{2}\right )}{24 c} + \frac{\left (c + d x^{2}\right )^{\frac{5}{2}} \left (5 a d \left (3 a d + 8 b c\right ) + 8 b^{2} c^{2}\right )}{40 c^{2}} \]

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**5,x)

[Out]

-a**2*(c + d*x**2)**(7/2)/(4*c*x**4) - a*(c + d*x**2)**(7/2)*(3*a*d + 8*b*c)/(8*
c**2*x**2) - sqrt(c)*(5*a*d*(3*a*d + 8*b*c) + 8*b**2*c**2)*atanh(sqrt(c + d*x**2
)/sqrt(c))/8 + sqrt(c + d*x**2)*(5*a*d*(3*a*d + 8*b*c)/8 + b**2*c**2) + (c + d*x
**2)**(3/2)*(5*a*d*(3*a*d + 8*b*c) + 8*b**2*c**2)/(24*c) + (c + d*x**2)**(5/2)*(
5*a*d*(3*a*d + 8*b*c) + 8*b**2*c**2)/(40*c**2)

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Mathematica [A]  time = 0.443782, size = 186, normalized size = 0.84 \[ -\frac{1}{8} \sqrt{c} \left (15 a^2 d^2+40 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+\sqrt{c+d x^2} \left (\frac{a^2 \left (-2 c^2-9 c d x^2+8 d^2 x^4\right )}{8 x^4}+\frac{1}{3} a b \left (-\frac{3 c^2}{x^2}+14 c d+2 d^2 x^2\right )+\frac{1}{15} b^2 \left (23 c^2+11 c d x^2+3 d^2 x^4\right )\right )+\frac{1}{8} \sqrt{c} \log (x) \left (15 a^2 d^2+40 a b c d+8 b^2 c^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x^2]*((a*b*(14*c*d - (3*c^2)/x^2 + 2*d^2*x^2))/3 + (b^2*(23*c^2 + 11*
c*d*x^2 + 3*d^2*x^4))/15 + (a^2*(-2*c^2 - 9*c*d*x^2 + 8*d^2*x^4))/(8*x^4)) + (Sq
rt[c]*(8*b^2*c^2 + 40*a*b*c*d + 15*a^2*d^2)*Log[x])/8 - (Sqrt[c]*(8*b^2*c^2 + 40
*a*b*c*d + 15*a^2*d^2)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/8

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Maple [A]  time = 0.018, size = 305, normalized size = 1.4 \[ -{\frac{{a}^{2}}{4\,c{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{a}^{2}d}{8\,{c}^{2}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{a}^{2}{d}^{2}}{8\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{2}{d}^{2}}{8\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{a}^{2}{d}^{2}}{8}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) }+{\frac{15\,{a}^{2}{d}^{2}}{8}\sqrt{d{x}^{2}+c}}+{\frac{{b}^{2}}{5} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}c}{3} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{b}^{2}{c}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) +{b}^{2}\sqrt{d{x}^{2}+c}{c}^{2}-{\frac{ab}{c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{abd}{c} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,abd}{3} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-5\,abd{c}^{3/2}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) +5\,abdc\sqrt{d{x}^{2}+c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*a^2*(d*x^2+c)^(7/2)/c/x^4-3/8*a^2*d/c^2/x^2*(d*x^2+c)^(7/2)+3/8*a^2*d^2/c^2
*(d*x^2+c)^(5/2)+5/8*a^2*d^2/c*(d*x^2+c)^(3/2)-15/8*a^2*d^2*c^(1/2)*ln((2*c+2*c^
(1/2)*(d*x^2+c)^(1/2))/x)+15/8*a^2*d^2*(d*x^2+c)^(1/2)+1/5*b^2*(d*x^2+c)^(5/2)+1
/3*b^2*c*(d*x^2+c)^(3/2)-b^2*c^(5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+b^2*(
d*x^2+c)^(1/2)*c^2-a*b/c/x^2*(d*x^2+c)^(7/2)+a*b*d/c*(d*x^2+c)^(5/2)+5/3*a*b*d*(
d*x^2+c)^(3/2)-5*a*b*d*c^(3/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+5*a*b*d*c*(
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)^(5/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250508, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (8 \, b^{2} c^{2} + 40 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt{c} x^{4} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (24 \, b^{2} d^{2} x^{8} + 8 \,{\left (11 \, b^{2} c d + 10 \, a b d^{2}\right )} x^{6} + 8 \,{\left (23 \, b^{2} c^{2} + 70 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{4} - 30 \, a^{2} c^{2} - 15 \,{\left (8 \, a b c^{2} + 9 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{240 \, x^{4}}, -\frac{15 \,{\left (8 \, b^{2} c^{2} + 40 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt{-c} x^{4} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (24 \, b^{2} d^{2} x^{8} + 8 \,{\left (11 \, b^{2} c d + 10 \, a b d^{2}\right )} x^{6} + 8 \,{\left (23 \, b^{2} c^{2} + 70 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{4} - 30 \, a^{2} c^{2} - 15 \,{\left (8 \, a b c^{2} + 9 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{120 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/240*(15*(8*b^2*c^2 + 40*a*b*c*d + 15*a^2*d^2)*sqrt(c)*x^4*log(-(d*x^2 - 2*sqr
t(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 2*(24*b^2*d^2*x^8 + 8*(11*b^2*c*d + 10*a*b*d^
2)*x^6 + 8*(23*b^2*c^2 + 70*a*b*c*d + 15*a^2*d^2)*x^4 - 30*a^2*c^2 - 15*(8*a*b*c
^2 + 9*a^2*c*d)*x^2)*sqrt(d*x^2 + c))/x^4, -1/120*(15*(8*b^2*c^2 + 40*a*b*c*d +
15*a^2*d^2)*sqrt(-c)*x^4*arctan(c/(sqrt(d*x^2 + c)*sqrt(-c))) - (24*b^2*d^2*x^8
+ 8*(11*b^2*c*d + 10*a*b*d^2)*x^6 + 8*(23*b^2*c^2 + 70*a*b*c*d + 15*a^2*d^2)*x^4
- 30*a^2*c^2 - 15*(8*a*b*c^2 + 9*a^2*c*d)*x^2)*sqrt(d*x^2 + c))/x^4]

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Sympy [A]  time = 116.474, size = 473, normalized size = 2.13 \[ - \frac{15 a^{2} \sqrt{c} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{8} - \frac{a^{2} c^{3}}{4 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a^{2} c^{2} \sqrt{d}}{8 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a^{2} c d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{x} + \frac{7 a^{2} c d^{\frac{3}{2}}}{8 x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} d^{\frac{5}{2}} x}{\sqrt{\frac{c}{d x^{2}} + 1}} - 5 a b c^{\frac{3}{2}} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )} - \frac{a b c^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{x} + \frac{4 a b c^{2} \sqrt{d}}{x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{4 a b c d^{\frac{3}{2}} x}{\sqrt{\frac{c}{d x^{2}} + 1}} + 2 a b d^{2} \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) - b^{2} c^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )} + \frac{b^{2} c^{3}}{\sqrt{d} x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{b^{2} c^{2} \sqrt{d} x}{\sqrt{\frac{c}{d x^{2}} + 1}} + 2 b^{2} c d \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) + b^{2} d^{2} \left (\begin{cases} - \frac{2 c^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{c x^{2} \sqrt{c + d x^{2}}}{15 d} + \frac{x^{4} \sqrt{c + d x^{2}}}{5} & \text{for}\: d \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*a**2*sqrt(c)*d**2*asinh(sqrt(c)/(sqrt(d)*x))/8 - a**2*c**3/(4*sqrt(d)*x**5*s
qrt(c/(d*x**2) + 1)) - 3*a**2*c**2*sqrt(d)/(8*x**3*sqrt(c/(d*x**2) + 1)) - a**2*
c*d**(3/2)*sqrt(c/(d*x**2) + 1)/x + 7*a**2*c*d**(3/2)/(8*x*sqrt(c/(d*x**2) + 1))
+ a**2*d**(5/2)*x/sqrt(c/(d*x**2) + 1) - 5*a*b*c**(3/2)*d*asinh(sqrt(c)/(sqrt(d
)*x)) - a*b*c**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/x + 4*a*b*c**2*sqrt(d)/(x*sqrt(c/(
d*x**2) + 1)) + 4*a*b*c*d**(3/2)*x/sqrt(c/(d*x**2) + 1) + 2*a*b*d**2*Piecewise((
sqrt(c)*x**2/2, Eq(d, 0)), ((c + d*x**2)**(3/2)/(3*d), True)) - b**2*c**(5/2)*as
inh(sqrt(c)/(sqrt(d)*x)) + b**2*c**3/(sqrt(d)*x*sqrt(c/(d*x**2) + 1)) + b**2*c**
2*sqrt(d)*x/sqrt(c/(d*x**2) + 1) + 2*b**2*c*d*Piecewise((sqrt(c)*x**2/2, Eq(d, 0
)), ((c + d*x**2)**(3/2)/(3*d), True)) + b**2*d**2*Piecewise((-2*c**2*sqrt(c + d
*x**2)/(15*d**2) + c*x**2*sqrt(c + d*x**2)/(15*d) + x**4*sqrt(c + d*x**2)/5, Ne(
d, 0)), (sqrt(c)*x**4/4, True))

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GIAC/XCAS [A]  time = 0.245732, size = 327, normalized size = 1.47 \[ \frac{24 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} d + 40 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c d + 120 \, \sqrt{d x^{2} + c} b^{2} c^{2} d + 80 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b d^{2} + 480 \, \sqrt{d x^{2} + c} a b c d^{2} + 120 \, \sqrt{d x^{2} + c} a^{2} d^{3} + \frac{15 \,{\left (8 \, b^{2} c^{3} d + 40 \, a b c^{2} d^{2} + 15 \, a^{2} c d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{15 \,{\left (8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c^{2} d^{2} - 8 \, \sqrt{d x^{2} + c} a b c^{3} d^{2} + 9 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} c d^{3} - 7 \, \sqrt{d x^{2} + c} a^{2} c^{2} d^{3}\right )}}{d^{2} x^{4}}}{120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/120*(24*(d*x^2 + c)^(5/2)*b^2*d + 40*(d*x^2 + c)^(3/2)*b^2*c*d + 120*sqrt(d*x^
2 + c)*b^2*c^2*d + 80*(d*x^2 + c)^(3/2)*a*b*d^2 + 480*sqrt(d*x^2 + c)*a*b*c*d^2
+ 120*sqrt(d*x^2 + c)*a^2*d^3 + 15*(8*b^2*c^3*d + 40*a*b*c^2*d^2 + 15*a^2*c*d^3)
*arctan(sqrt(d*x^2 + c)/sqrt(-c))/sqrt(-c) - 15*(8*(d*x^2 + c)^(3/2)*a*b*c^2*d^2
- 8*sqrt(d*x^2 + c)*a*b*c^3*d^2 + 9*(d*x^2 + c)^(3/2)*a^2*c*d^3 - 7*sqrt(d*x^2
+ c)*a^2*c^2*d^3)/(d^2*x^4))/d