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

Optimal. Leaf size=152 \[ \frac{\left (8 a^2 d^2-24 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{7/2}}-\frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-24 a b c d+15 b^2 c^2\right )}{8 c d^3}+\frac{x^3 (b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 x^3 \sqrt{c+d x^2}}{4 d^2} \]

[Out]

((b*c - a*d)^2*x^3)/(c*d^2*Sqrt[c + d*x^2]) - ((15*b^2*c^2 - 24*a*b*c*d + 8*a^2*
d^2)*x*Sqrt[c + d*x^2])/(8*c*d^3) + (b^2*x^3*Sqrt[c + d*x^2])/(4*d^2) + ((15*b^2
*c^2 - 24*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(7/2))

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Rubi [A]  time = 0.336539, antiderivative size = 152, 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 \[ \frac{\left (8 a^2 d^2-24 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{7/2}}-\frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-24 a b c d+15 b^2 c^2\right )}{8 c d^3}+\frac{x^3 (b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 x^3 \sqrt{c+d x^2}}{4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*x^3)/(c*d^2*Sqrt[c + d*x^2]) - ((15*b^2*c^2 - 24*a*b*c*d + 8*a^2*
d^2)*x*Sqrt[c + d*x^2])/(8*c*d^3) + (b^2*x^3*Sqrt[c + d*x^2])/(4*d^2) + ((15*b^2
*c^2 - 24*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(7/2))

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Rubi in Sympy [A]  time = 44.444, size = 143, normalized size = 0.94 \[ \frac{b^{2} x^{3} \sqrt{c + d x^{2}}}{4 d^{2}} + \frac{\left (8 a^{2} d^{2} - 24 a b c d + 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 d^{\frac{7}{2}}} + \frac{x^{3} \left (a d - b c\right )^{2}}{c d^{2} \sqrt{c + d x^{2}}} - \frac{x \sqrt{c + d x^{2}} \left (8 a^{2} d^{2} - 24 a b c d + 15 b^{2} c^{2}\right )}{8 c d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**3*sqrt(c + d*x**2)/(4*d**2) + (8*a**2*d**2 - 24*a*b*c*d + 15*b**2*c**2)*
atanh(sqrt(d)*x/sqrt(c + d*x**2))/(8*d**(7/2)) + x**3*(a*d - b*c)**2/(c*d**2*sqr
t(c + d*x**2)) - x*sqrt(c + d*x**2)*(8*a**2*d**2 - 24*a*b*c*d + 15*b**2*c**2)/(8
*c*d**3)

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Mathematica [A]  time = 0.18136, size = 124, normalized size = 0.82 \[ \frac{\left (8 a^2 d^2-24 a b c d+15 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{8 d^{7/2}}+\sqrt{c+d x^2} \left (-\frac{x (a d-b c)^2}{d^3 \left (c+d x^2\right )}-\frac{b x (7 b c-8 a d)}{8 d^3}+\frac{b^2 x^3}{4 d^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x^2]*(-(b*(7*b*c - 8*a*d)*x)/(8*d^3) + (b^2*x^3)/(4*d^2) - ((-(b*c) +
a*d)^2*x)/(d^3*(c + d*x^2))) + ((15*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*Log[d*x +
Sqrt[d]*Sqrt[c + d*x^2]])/(8*d^(7/2))

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Maple [A]  time = 0.015, size = 192, normalized size = 1.3 \[ -{\frac{{a}^{2}x}{d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{{a}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}+{\frac{{b}^{2}{x}^{5}}{4\,d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{5\,{b}^{2}c{x}^{3}}{8\,{d}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{15\,{b}^{2}{c}^{2}x}{8\,{d}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{15\,{b}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}}+{\frac{ab{x}^{3}}{d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+3\,{\frac{abcx}{{d}^{2}\sqrt{d{x}^{2}+c}}}-3\,{\frac{abc\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }{{d}^{5/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250528, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, b^{2} d^{2} x^{5} -{\left (5 \, b^{2} c d - 8 \, a b d^{2}\right )} x^{3} -{\left (15 \, b^{2} c^{2} - 24 \, a b c d + 8 \, a^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} +{\left (15 \, b^{2} c^{3} - 24 \, a b c^{2} d + 8 \, a^{2} c d^{2} +{\left (15 \, b^{2} c^{2} d - 24 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{16 \,{\left (d^{4} x^{2} + c d^{3}\right )} \sqrt{d}}, \frac{{\left (2 \, b^{2} d^{2} x^{5} -{\left (5 \, b^{2} c d - 8 \, a b d^{2}\right )} x^{3} -{\left (15 \, b^{2} c^{2} - 24 \, a b c d + 8 \, a^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} +{\left (15 \, b^{2} c^{3} - 24 \, a b c^{2} d + 8 \, a^{2} c d^{2} +{\left (15 \, b^{2} c^{2} d - 24 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{8 \,{\left (d^{4} x^{2} + c d^{3}\right )} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(2*(2*b^2*d^2*x^5 - (5*b^2*c*d - 8*a*b*d^2)*x^3 - (15*b^2*c^2 - 24*a*b*c*d
+ 8*a^2*d^2)*x)*sqrt(d*x^2 + c)*sqrt(d) + (15*b^2*c^3 - 24*a*b*c^2*d + 8*a^2*c*
d^2 + (15*b^2*c^2*d - 24*a*b*c*d^2 + 8*a^2*d^3)*x^2)*log(-2*sqrt(d*x^2 + c)*d*x
- (2*d*x^2 + c)*sqrt(d)))/((d^4*x^2 + c*d^3)*sqrt(d)), 1/8*((2*b^2*d^2*x^5 - (5*
b^2*c*d - 8*a*b*d^2)*x^3 - (15*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*x)*sqrt(d*x^2 +
c)*sqrt(-d) + (15*b^2*c^3 - 24*a*b*c^2*d + 8*a^2*c*d^2 + (15*b^2*c^2*d - 24*a*b
*c*d^2 + 8*a^2*d^3)*x^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/((d^4*x^2 + c*d^3)*
sqrt(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*(a + b*x**2)**2/(c + d*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.238108, size = 177, normalized size = 1.16 \[ \frac{{\left ({\left (\frac{2 \, b^{2} x^{2}}{d} - \frac{5 \, b^{2} c d^{3} - 8 \, a b d^{4}}{d^{5}}\right )} x^{2} - \frac{15 \, b^{2} c^{2} d^{2} - 24 \, a b c d^{3} + 8 \, a^{2} d^{4}}{d^{5}}\right )} x}{8 \, \sqrt{d x^{2} + c}} - \frac{{\left (15 \, b^{2} c^{2} - 24 \, a b c d + 8 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{8 \, d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((2*b^2*x^2/d - (5*b^2*c*d^3 - 8*a*b*d^4)/d^5)*x^2 - (15*b^2*c^2*d^2 - 24*a*
b*c*d^3 + 8*a^2*d^4)/d^5)*x/sqrt(d*x^2 + c) - 1/8*(15*b^2*c^2 - 24*a*b*c*d + 8*a
^2*d^2)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(7/2)