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

Optimal. Leaf size=121 \[ -\frac{b (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 d^{7/2}}+\frac{2 b x (b c-a d)}{d^3 \sqrt{c+d x^2}}+\frac{x^3 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{b^2 x \sqrt{c+d x^2}}{2 d^3} \]

[Out]

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

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Rubi [A]  time = 0.32199, antiderivative size = 121, 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 \[ -\frac{b (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 d^{7/2}}+\frac{2 b x (b c-a d)}{d^3 \sqrt{c+d x^2}}+\frac{x^3 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac{b^2 x \sqrt{c+d x^2}}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 58.8152, size = 110, normalized size = 0.91 \[ \frac{b^{2} x \sqrt{c + d x^{2}}}{2 d^{3}} - \frac{2 b x \left (a d - b c\right )}{d^{3} \sqrt{c + d x^{2}}} + \frac{b \left (4 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 d^{\frac{7}{2}}} + \frac{x^{3} \left (a d - b c\right )^{2}}{3 c d^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.188548, size = 118, normalized size = 0.98 \[ \frac{x \left (2 a^2 d^3 x^2-4 a b c d \left (3 c+4 d x^2\right )+b^2 c \left (15 c^2+20 c d x^2+3 d^2 x^4\right )\right )}{6 c d^3 \left (c+d x^2\right )^{3/2}}+\frac{b (4 a d-5 b c) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{2 d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(2*a^2*d^3*x^2 - 4*a*b*c*d*(3*c + 4*d*x^2) + b^2*c*(15*c^2 + 20*c*d*x^2 + 3*d
^2*x^4)))/(6*c*d^3*(c + d*x^2)^(3/2)) + (b*(-5*b*c + 4*a*d)*Log[d*x + Sqrt[d]*Sq
rt[c + d*x^2]])/(2*d^(7/2))

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Maple [A]  time = 0.015, size = 185, normalized size = 1.5 \[ -{\frac{{a}^{2}x}{3\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{{a}^{2}x}{3\,cd}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{{b}^{2}{x}^{5}}{2\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,{b}^{2}c{x}^{3}}{6\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,{b}^{2}cx}{2\,{d}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{5\,{b}^{2}c}{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}}-{\frac{2\,ab{x}^{3}}{3\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{abx}{{d}^{2}\sqrt{d{x}^{2}+c}}}+2\,{\frac{ab\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }{{d}^{5/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25625, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \, b^{2} c d^{2} x^{5} + 2 \,{\left (10 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 3 \,{\left (5 \, b^{2} c^{3} - 4 \, a b c^{2} d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} - 3 \,{\left (5 \, b^{2} c^{4} - 4 \, a b c^{3} d +{\left (5 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3}\right )} x^{4} + 2 \,{\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{12 \,{\left (c d^{5} x^{4} + 2 \, c^{2} d^{4} x^{2} + c^{3} d^{3}\right )} \sqrt{d}}, \frac{{\left (3 \, b^{2} c d^{2} x^{5} + 2 \,{\left (10 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 3 \,{\left (5 \, b^{2} c^{3} - 4 \, a b c^{2} d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} - 3 \,{\left (5 \, b^{2} c^{4} - 4 \, a b c^{3} d +{\left (5 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3}\right )} x^{4} + 2 \,{\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{6 \,{\left (c d^{5} x^{4} + 2 \, c^{2} d^{4} x^{2} + c^{3} d^{3}\right )} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(2*(3*b^2*c*d^2*x^5 + 2*(10*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3)*x^3 + 3*(5*
b^2*c^3 - 4*a*b*c^2*d)*x)*sqrt(d*x^2 + c)*sqrt(d) - 3*(5*b^2*c^4 - 4*a*b*c^3*d +
(5*b^2*c^2*d^2 - 4*a*b*c*d^3)*x^4 + 2*(5*b^2*c^3*d - 4*a*b*c^2*d^2)*x^2)*log(-2
*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/((c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3
*d^3)*sqrt(d)), 1/6*((3*b^2*c*d^2*x^5 + 2*(10*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3)
*x^3 + 3*(5*b^2*c^3 - 4*a*b*c^2*d)*x)*sqrt(d*x^2 + c)*sqrt(-d) - 3*(5*b^2*c^4 -
4*a*b*c^3*d + (5*b^2*c^2*d^2 - 4*a*b*c*d^3)*x^4 + 2*(5*b^2*c^3*d - 4*a*b*c^2*d^2
)*x^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/((c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3
)*sqrt(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*(a + b*x**2)**2/(c + d*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.242582, size = 176, normalized size = 1.45 \[ \frac{{\left ({\left (\frac{3 \, b^{2} x^{2}}{d} + \frac{2 \,{\left (10 \, b^{2} c^{2} d^{3} - 8 \, a b c d^{4} + a^{2} d^{5}\right )}}{c d^{5}}\right )} x^{2} + \frac{3 \,{\left (5 \, b^{2} c^{3} d^{2} - 4 \, a b c^{2} d^{3}\right )}}{c d^{5}}\right )} x}{6 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} + \frac{{\left (5 \, b^{2} c - 4 \, a b d\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{2 \, d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*((3*b^2*x^2/d + 2*(10*b^2*c^2*d^3 - 8*a*b*c*d^4 + a^2*d^5)/(c*d^5))*x^2 + 3*
(5*b^2*c^3*d^2 - 4*a*b*c^2*d^3)/(c*d^5))*x/(d*x^2 + c)^(3/2) + 1/2*(5*b^2*c - 4*
a*b*d)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(7/2)