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

Optimal. Leaf size=103 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{5 c x^5}-\frac{2 a \left (c+d x^2\right )^{3/2} (5 b c-a d)}{15 c^2 x^3}-\frac{b^2 \sqrt{c+d x^2}}{x}+b^2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right ) \]

[Out]

-((b^2*Sqrt[c + d*x^2])/x) - (a^2*(c + d*x^2)^(3/2))/(5*c*x^5) - (2*a*(5*b*c - a
*d)*(c + d*x^2)^(3/2))/(15*c^2*x^3) + b^2*Sqrt[d]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d
*x^2]]

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Rubi [A]  time = 0.187503, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{5 c x^5}-\frac{2 a \left (c+d x^2\right )^{3/2} (5 b c-a d)}{15 c^2 x^3}-\frac{b^2 \sqrt{c+d x^2}}{x}+b^2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((b^2*Sqrt[c + d*x^2])/x) - (a^2*(c + d*x^2)^(3/2))/(5*c*x^5) - (2*a*(5*b*c - a
*d)*(c + d*x^2)^(3/2))/(15*c^2*x^3) + b^2*Sqrt[d]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d
*x^2]]

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Rubi in Sympy [A]  time = 24.6507, size = 92, normalized size = 0.89 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{5 c x^{5}} + \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - 5 b c\right )}{15 c^{2} x^{3}} + b^{2} \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )} - \frac{b^{2} \sqrt{c + d x^{2}}}{x} \]

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**6,x)

[Out]

-a**2*(c + d*x**2)**(3/2)/(5*c*x**5) + 2*a*(c + d*x**2)**(3/2)*(a*d - 5*b*c)/(15
*c**2*x**3) + b**2*sqrt(d)*atanh(sqrt(d)*x/sqrt(c + d*x**2)) - b**2*sqrt(c + d*x
**2)/x

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Mathematica [A]  time = 0.131193, size = 104, normalized size = 1.01 \[ b^2 \sqrt{d} \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )-\frac{\sqrt{c+d x^2} \left (a^2 \left (3 c^2+c d x^2-2 d^2 x^4\right )+10 a b c x^2 \left (c+d x^2\right )+15 b^2 c^2 x^4\right )}{15 c^2 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d*x^2]*(15*b^2*c^2*x^4 + 10*a*b*c*x^2*(c + d*x^2) + a^2*(3*c^2 + c*d*
x^2 - 2*d^2*x^4)))/(15*c^2*x^5) + b^2*Sqrt[d]*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]]

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Maple [A]  time = 0.017, size = 123, normalized size = 1.2 \[ -{\frac{{a}^{2}}{5\,c{x}^{5}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{2\,{a}^{2}d}{15\,{c}^{2}{x}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}}{cx} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}dx}{c}\sqrt{d{x}^{2}+c}}+{b}^{2}\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) -{\frac{2\,ab}{3\,c{x}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^6,x)

[Out]

-1/5*a^2*(d*x^2+c)^(3/2)/c/x^5+2/15*a^2*d/c^2/x^3*(d*x^2+c)^(3/2)-b^2/c/x*(d*x^2
+c)^(3/2)+b^2*d/c*x*(d*x^2+c)^(1/2)+b^2*d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-2/
3*a*b/c/x^3*(d*x^2+c)^(3/2)

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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*sqrt(d*x^2 + c)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251361, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{2} c^{2} \sqrt{d} x^{5} \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) - 2 \,{\left ({\left (15 \, b^{2} c^{2} + 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} +{\left (10 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{30 \, c^{2} x^{5}}, \frac{15 \, b^{2} c^{2} \sqrt{-d} x^{5} \arctan \left (\frac{d x}{\sqrt{d x^{2} + c} \sqrt{-d}}\right ) -{\left ({\left (15 \, b^{2} c^{2} + 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} +{\left (10 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \, c^{2} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(15*b^2*c^2*sqrt(d)*x^5*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) -
2*((15*b^2*c^2 + 10*a*b*c*d - 2*a^2*d^2)*x^4 + 3*a^2*c^2 + (10*a*b*c^2 + a^2*c*d
)*x^2)*sqrt(d*x^2 + c))/(c^2*x^5), 1/15*(15*b^2*c^2*sqrt(-d)*x^5*arctan(d*x/(sqr
t(d*x^2 + c)*sqrt(-d))) - ((15*b^2*c^2 + 10*a*b*c*d - 2*a^2*d^2)*x^4 + 3*a^2*c^2
+ (10*a*b*c^2 + a^2*c*d)*x^2)*sqrt(d*x^2 + c))/(c^2*x^5)]

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Sympy [A]  time = 10.4667, size = 199, normalized size = 1.93 \[ - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{5 x^{4}} - \frac{a^{2} d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c x^{2}} + \frac{2 a^{2} d^{\frac{5}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c^{2}} - \frac{2 a b \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 x^{2}} - \frac{2 a b d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3 c} - \frac{b^{2} \sqrt{c}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + b^{2} \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )} - \frac{b^{2} d x}{\sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**6,x)

[Out]

-a**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(5*x**4) - a**2*d**(3/2)*sqrt(c/(d*x**2) + 1)
/(15*c*x**2) + 2*a**2*d**(5/2)*sqrt(c/(d*x**2) + 1)/(15*c**2) - 2*a*b*sqrt(d)*sq
rt(c/(d*x**2) + 1)/(3*x**2) - 2*a*b*d**(3/2)*sqrt(c/(d*x**2) + 1)/(3*c) - b**2*s
qrt(c)/(x*sqrt(1 + d*x**2/c)) + b**2*sqrt(d)*asinh(sqrt(d)*x/sqrt(c)) - b**2*d*x
/(sqrt(c)*sqrt(1 + d*x**2/c))

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GIAC/XCAS [A]  time = 0.260052, size = 544, normalized size = 5.28 \[ -\frac{1}{2} \, b^{2} \sqrt{d}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right ) + \frac{2 \,{\left (15 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} b^{2} c \sqrt{d} + 30 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a b d^{\frac{3}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b^{2} c^{2} \sqrt{d} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a b c d^{\frac{3}{2}} + 30 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a^{2} d^{\frac{5}{2}} + 90 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{2} c^{3} \sqrt{d} + 40 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{2} d^{\frac{3}{2}} + 10 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c d^{\frac{5}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c^{4} \sqrt{d} - 20 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{3} d^{\frac{3}{2}} + 10 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c^{2} d^{\frac{5}{2}} + 15 \, b^{2} c^{5} \sqrt{d} + 10 \, a b c^{4} d^{\frac{3}{2}} - 2 \, a^{2} c^{3} d^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b^2*sqrt(d)*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2) + 2/15*(15*(sqrt(d)*x - sqr
t(d*x^2 + c))^8*b^2*c*sqrt(d) + 30*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*d^(3/2) -
60*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^2*sqrt(d) - 60*(sqrt(d)*x - sqrt(d*x^2
+ c))^6*a*b*c*d^(3/2) + 30*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*d^(5/2) + 90*(sq
rt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^3*sqrt(d) + 40*(sqrt(d)*x - sqrt(d*x^2 + c))^
4*a*b*c^2*d^(3/2) + 10*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*c*d^(5/2) - 60*(sqrt(
d)*x - sqrt(d*x^2 + c))^2*b^2*c^4*sqrt(d) - 20*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a
*b*c^3*d^(3/2) + 10*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c^2*d^(5/2) + 15*b^2*c^5
*sqrt(d) + 10*a*b*c^4*d^(3/2) - 2*a^2*c^3*d^(5/2))/((sqrt(d)*x - sqrt(d*x^2 + c)
)^2 - c)^5