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

Optimal. Leaf size=105 \[ -\frac{x (b c-a d) (2 a d+3 b c)}{3 c^2 d^2 \sqrt{c+d x^2}}-\frac{x \left (a+b x^2\right ) (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{d^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.136059, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{x (b c-a d) (2 a d+3 b c)}{3 c^2 d^2 \sqrt{c+d x^2}}-\frac{x \left (a+b x^2\right ) (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.4313, size = 94, normalized size = 0.9 \[ \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{d^{\frac{5}{2}}} + \frac{x \left (a + b x^{2}\right ) \left (a d - b c\right )}{3 c d \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{x \left (a d - b c\right ) \left (2 a d + 3 b c\right )}{3 c^{2} d^{2} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.222373, size = 101, normalized size = 0.96 \[ \frac{x \left (a^2 d^2 \left (3 c+2 d x^2\right )+2 a b c d^2 x^2-b^2 c^2 \left (3 c+4 d x^2\right )\right )}{3 c^2 d^2 \left (c+d x^2\right )^{3/2}}+\frac{b^2 \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 136, normalized size = 1.3 \[{\frac{{a}^{2}x}{3\,c} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,{a}^{2}x}{3\,{c}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{{b}^{2}{x}^{3}}{3\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{{b}^{2}x}{{d}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{{b}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}-{\frac{2\,abx}{3\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,abx}{3\,cd}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235764, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (2 \,{\left (2 \, b^{2} c^{2} d - a b c d^{2} - a^{2} d^{3}\right )} x^{3} + 3 \,{\left (b^{2} c^{3} - a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} - 3 \,{\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{6 \,{\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )} \sqrt{d}}, -\frac{{\left (2 \,{\left (2 \, b^{2} c^{2} d - a b c d^{2} - a^{2} d^{3}\right )} x^{3} + 3 \,{\left (b^{2} c^{3} - a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} - 3 \,{\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{3 \,{\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.237555, size = 142, normalized size = 1.35 \[ -\frac{x{\left (\frac{2 \,{\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3} - a^{2} d^{4}\right )} x^{2}}{c^{2} d^{3}} + \frac{3 \,{\left (b^{2} c^{3} d - a^{2} c d^{3}\right )}}{c^{2} d^{3}}\right )}}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} - \frac{b^{2}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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