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

Optimal. Leaf size=146 \[ -\frac{c \left (8 a^2 d^2+b c (5 b c-12 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 d^{7/2}}+\frac{x \sqrt{c+d x^2} \left (8 a^2 d^2+b c (5 b c-12 a d)\right )}{16 d^3}-\frac{b x^3 \sqrt{c+d x^2} (5 b c-12 a d)}{24 d^2}+\frac{b^2 x^5 \sqrt{c+d x^2}}{6 d} \]

[Out]

((8*a^2*d^2 + b*c*(5*b*c - 12*a*d))*x*Sqrt[c + d*x^2])/(16*d^3) - (b*(5*b*c - 12
*a*d)*x^3*Sqrt[c + d*x^2])/(24*d^2) + (b^2*x^5*Sqrt[c + d*x^2])/(6*d) - (c*(8*a^
2*d^2 + b*c*(5*b*c - 12*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d^(7/2))

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Rubi [A]  time = 0.389686, antiderivative size = 146, 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 \[ \frac{x \sqrt{c+d x^2} \left (8 a^2+\frac{b c (5 b c-12 a d)}{d^2}\right )}{16 d}-\frac{c \left (8 a^2 d^2+b c (5 b c-12 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 d^{7/2}}-\frac{b x^3 \sqrt{c+d x^2} (5 b c-12 a d)}{24 d^2}+\frac{b^2 x^5 \sqrt{c+d x^2}}{6 d} \]

Antiderivative was successfully verified.

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

[Out]

((8*a^2 + (b*c*(5*b*c - 12*a*d))/d^2)*x*Sqrt[c + d*x^2])/(16*d) - (b*(5*b*c - 12
*a*d)*x^3*Sqrt[c + d*x^2])/(24*d^2) + (b^2*x^5*Sqrt[c + d*x^2])/(6*d) - (c*(8*a^
2*d^2 + b*c*(5*b*c - 12*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d^(7/2))

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Rubi in Sympy [A]  time = 34.0838, size = 138, normalized size = 0.95 \[ \frac{b^{2} x^{5} \sqrt{c + d x^{2}}}{6 d} + \frac{b x^{3} \sqrt{c + d x^{2}} \left (12 a d - 5 b c\right )}{24 d^{2}} - \frac{c \left (8 a^{2} d^{2} - b c \left (12 a d - 5 b c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{16 d^{\frac{7}{2}}} + \frac{x \sqrt{c + d x^{2}} \left (8 a^{2} d^{2} - b c \left (12 a d - 5 b c\right )\right )}{16 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**5*sqrt(c + d*x**2)/(6*d) + b*x**3*sqrt(c + d*x**2)*(12*a*d - 5*b*c)/(24*
d**2) - c*(8*a**2*d**2 - b*c*(12*a*d - 5*b*c))*atanh(sqrt(d)*x/sqrt(c + d*x**2))
/(16*d**(7/2)) + x*sqrt(c + d*x**2)*(8*a**2*d**2 - b*c*(12*a*d - 5*b*c))/(16*d**
3)

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Mathematica [A]  time = 0.138043, size = 125, normalized size = 0.86 \[ \frac{\sqrt{d} x \sqrt{c+d x^2} \left (24 a^2 d^2+12 a b d \left (2 d x^2-3 c\right )+b^2 \left (15 c^2-10 c d x^2+8 d^2 x^4\right )\right )-3 c \left (8 a^2 d^2-12 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{48 d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^2]*(24*a^2*d^2 + 12*a*b*d*(-3*c + 2*d*x^2) + b^2*(15*c^2
- 10*c*d*x^2 + 8*d^2*x^4)) - 3*c*(5*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*Log[d*x +
Sqrt[d]*Sqrt[c + d*x^2]])/(48*d^(7/2))

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Maple [A]  time = 0.014, size = 197, normalized size = 1.4 \[{\frac{{a}^{2}x}{2\,d}\sqrt{d{x}^{2}+c}}-{\frac{{a}^{2}c}{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}+{\frac{{b}^{2}{x}^{5}}{6\,d}\sqrt{d{x}^{2}+c}}-{\frac{5\,{b}^{2}c{x}^{3}}{24\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{5\,{b}^{2}{c}^{2}x}{16\,{d}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{5\,{b}^{2}{c}^{3}}{16}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}}+{\frac{ab{x}^{3}}{2\,d}\sqrt{d{x}^{2}+c}}-{\frac{3\,abcx}{4\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,ab{c}^{2}}{4}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29354, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, b^{2} d^{2} x^{5} - 2 \,{\left (5 \, b^{2} c d - 12 \, a b d^{2}\right )} x^{3} + 3 \,{\left (5 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 3 \,{\left (5 \, b^{2} c^{3} - 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{96 \, d^{\frac{7}{2}}}, \frac{{\left (8 \, b^{2} d^{2} x^{5} - 2 \,{\left (5 \, b^{2} c d - 12 \, a b d^{2}\right )} x^{3} + 3 \,{\left (5 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} - 3 \,{\left (5 \, b^{2} c^{3} - 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{48 \, \sqrt{-d} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(2*(8*b^2*d^2*x^5 - 2*(5*b^2*c*d - 12*a*b*d^2)*x^3 + 3*(5*b^2*c^2 - 12*a*b
*c*d + 8*a^2*d^2)*x)*sqrt(d*x^2 + c)*sqrt(d) + 3*(5*b^2*c^3 - 12*a*b*c^2*d + 8*a
^2*c*d^2)*log(2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/d^(7/2), 1/48*((8*
b^2*d^2*x^5 - 2*(5*b^2*c*d - 12*a*b*d^2)*x^3 + 3*(5*b^2*c^2 - 12*a*b*c*d + 8*a^2
*d^2)*x)*sqrt(d*x^2 + c)*sqrt(-d) - 3*(5*b^2*c^3 - 12*a*b*c^2*d + 8*a^2*c*d^2)*a
rctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/(sqrt(-d)*d^3)]

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Sympy [A]  time = 39.8398, size = 301, normalized size = 2.06 \[ \frac{a^{2} \sqrt{c} x \sqrt{1 + \frac{d x^{2}}{c}}}{2 d} - \frac{a^{2} c \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 d^{\frac{3}{2}}} - \frac{3 a b c^{\frac{3}{2}} x}{4 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a b \sqrt{c} x^{3}}{4 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{4 d^{\frac{5}{2}}} + \frac{a b x^{5}}{2 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} c^{\frac{5}{2}} x}{16 d^{3} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} c^{\frac{3}{2}} x^{3}}{48 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} \sqrt{c} x^{5}}{24 d \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{5 b^{2} c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{16 d^{\frac{7}{2}}} + \frac{b^{2} x^{7}}{6 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235995, size = 182, normalized size = 1.25 \[ \frac{1}{48} \,{\left (2 \,{\left (\frac{4 \, b^{2} x^{2}}{d} - \frac{5 \, b^{2} c d^{3} - 12 \, a b d^{4}}{d^{5}}\right )} x^{2} + \frac{3 \,{\left (5 \, b^{2} c^{2} d^{2} - 12 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )}}{d^{5}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (5 \, b^{2} c^{3} - 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{16 \, d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*(4*b^2*x^2/d - (5*b^2*c*d^3 - 12*a*b*d^4)/d^5)*x^2 + 3*(5*b^2*c^2*d^2 -
12*a*b*c*d^3 + 8*a^2*d^4)/d^5)*sqrt(d*x^2 + c)*x + 1/16*(5*b^2*c^3 - 12*a*b*c^2*
d + 8*a^2*c*d^2)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(7/2)