3.623 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=147 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac{b \left (c+d x^2\right )^{3/2} (4 a d+3 b c)}{3 c x}+\frac{b d x \sqrt{c+d x^2} (4 a d+3 b c)}{2 c}+\frac{1}{2} b \sqrt{d} (4 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )-\frac{2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3} \]

[Out]

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

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Rubi [A]  time = 0.252298, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac{b \left (c+d x^2\right )^{3/2} (4 a d+3 b c)}{3 c x}+\frac{b d x \sqrt{c+d x^2} (4 a d+3 b c)}{2 c}+\frac{1}{2} b \sqrt{d} (4 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )-\frac{2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 26.9589, size = 134, normalized size = 0.91 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 c x^{5}} - \frac{2 a b \left (c + d x^{2}\right )^{\frac{5}{2}}}{3 c x^{3}} + \frac{b \sqrt{d} \left (4 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2} + \frac{b d x \sqrt{c + d x^{2}} \left (4 a d + 3 b c\right )}{2 c} - \frac{b \left (c + d x^{2}\right )^{\frac{3}{2}} \left (4 a d + 3 b c\right )}{3 c x} \]

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**6,x)

[Out]

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

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Mathematica [A]  time = 0.186524, size = 113, normalized size = 0.77 \[ \frac{1}{2} b \sqrt{d} (4 a d+3 b c) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )-\frac{\sqrt{c+d x^2} \left (6 a^2 \left (c+d x^2\right )^2+20 a b c x^2 \left (c+4 d x^2\right )+15 b^2 c x^4 \left (2 c-d x^2\right )\right )}{30 c x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^6,x]

[Out]

-(Sqrt[c + d*x^2]*(15*b^2*c*x^4*(2*c - d*x^2) + 6*a^2*(c + d*x^2)^2 + 20*a*b*c*x
^2*(c + 4*d*x^2)))/(30*c*x^5) + (b*Sqrt[d]*(3*b*c + 4*a*d)*Log[d*x + Sqrt[d]*Sqr
t[c + d*x^2]])/2

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Maple [A]  time = 0.018, size = 203, normalized size = 1.4 \[ -{\frac{{a}^{2}}{5\,c{x}^{5}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}}{cx} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}dx}{c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{2}dx}{2}\sqrt{d{x}^{2}+c}}+{\frac{3\,{b}^{2}c}{2}\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }-{\frac{2\,ab}{3\,c{x}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{4\,dab}{3\,{c}^{2}x} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{4\,ab{d}^{2}x}{3\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{ab{d}^{2}x\sqrt{d{x}^{2}+c}}{c}}+2\,ab{d}^{3/2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^6,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282195, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt{d} x^{5} \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) + 2 \,{\left (15 \, b^{2} c d x^{6} - 2 \,{\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 4 \,{\left (5 \, a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{60 \, c x^{5}}, \frac{15 \,{\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt{-d} x^{5} \arctan \left (\frac{d x}{\sqrt{d x^{2} + c} \sqrt{-d}}\right ) +{\left (15 \, b^{2} c d x^{6} - 2 \,{\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 4 \,{\left (5 \, a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{30 \, c x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x^6,x, algorithm="fricas")

[Out]

[1/60*(15*(3*b^2*c^2 + 4*a*b*c*d)*sqrt(d)*x^5*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*s
qrt(d)*x - c) + 2*(15*b^2*c*d*x^6 - 2*(15*b^2*c^2 + 40*a*b*c*d + 3*a^2*d^2)*x^4
- 6*a^2*c^2 - 4*(5*a*b*c^2 + 3*a^2*c*d)*x^2)*sqrt(d*x^2 + c))/(c*x^5), 1/30*(15*
(3*b^2*c^2 + 4*a*b*c*d)*sqrt(-d)*x^5*arctan(d*x/(sqrt(d*x^2 + c)*sqrt(-d))) + (1
5*b^2*c*d*x^6 - 2*(15*b^2*c^2 + 40*a*b*c*d + 3*a^2*d^2)*x^4 - 6*a^2*c^2 - 4*(5*a
*b*c^2 + 3*a^2*c*d)*x^2)*sqrt(d*x^2 + c))/(c*x^5)]

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Sympy [A]  time = 22.9441, size = 304, normalized size = 2.07 \[ - \frac{a^{2} c \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{5 x^{4}} - \frac{2 a^{2} d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{5 x^{2}} - \frac{a^{2} d^{\frac{5}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{5 c} - \frac{2 a b \sqrt{c} d}{x \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{2 a b c \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 x^{2}} - \frac{2 a b d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3} + 2 a b d^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )} - \frac{2 a b d^{2} x}{\sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{3}{2}}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{b^{2} \sqrt{c} d x \sqrt{1 + \frac{d x^{2}}{c}}}{2} - \frac{b^{2} \sqrt{c} d x}{\sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} c \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x**6,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.254954, size = 549, normalized size = 3.73 \[ \frac{1}{2} \, \sqrt{d x^{2} + c} b^{2} d x - \frac{1}{4} \,{\left (3 \, b^{2} c \sqrt{d} + 4 \, a b d^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right ) + \frac{2 \,{\left (15 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} b^{2} c^{2} \sqrt{d} + 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a b c d^{\frac{3}{2}} + 15 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a^{2} d^{\frac{5}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b^{2} c^{3} \sqrt{d} - 180 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a b c^{2} d^{\frac{3}{2}} + 90 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{2} c^{4} \sqrt{d} + 220 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{3} d^{\frac{3}{2}} + 30 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac{5}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c^{5} \sqrt{d} - 140 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{4} d^{\frac{3}{2}} + 15 \, b^{2} c^{6} \sqrt{d} + 40 \, a b c^{5} d^{\frac{3}{2}} + 3 \, a^{2} c^{4} d^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)/x^6,x, algorithm="giac")

[Out]

1/2*sqrt(d*x^2 + c)*b^2*d*x - 1/4*(3*b^2*c*sqrt(d) + 4*a*b*d^(3/2))*ln((sqrt(d)*
x - sqrt(d*x^2 + c))^2) + 2/15*(15*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^2*sqrt(
d) + 60*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*c*d^(3/2) + 15*(sqrt(d)*x - sqrt(d*x
^2 + c))^8*a^2*d^(5/2) - 60*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^3*sqrt(d) - 18
0*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a*b*c^2*d^(3/2) + 90*(sqrt(d)*x - sqrt(d*x^2 +
c))^4*b^2*c^4*sqrt(d) + 220*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^3*d^(3/2) + 3
0*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*c^2*d^(5/2) - 60*(sqrt(d)*x - sqrt(d*x^2 +
c))^2*b^2*c^5*sqrt(d) - 140*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^4*d^(3/2) + 1
5*b^2*c^6*sqrt(d) + 40*a*b*c^5*d^(3/2) + 3*a^2*c^4*d^(5/2))/((sqrt(d)*x - sqrt(d
*x^2 + c))^2 - c)^5