Optimal. Leaf size=200 \[ -\frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} (b c-a d)^4}+\frac{b^{3/2} (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^4}-\frac{d x (a d+11 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d x}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]
[Out]
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Rubi [A] time = 0.603195, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} (b c-a d)^4}+\frac{b^{3/2} (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^4}-\frac{d x (a d+11 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d x}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x^2)^2*(c + d*x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 117.255, size = 178, normalized size = 0.89 \[ - \frac{3 d x}{4 \left (c + d x^{2}\right )^{2} \left (a d - b c\right )^{2}} + \frac{x}{2 \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} + \frac{d x \left (a d + 11 b c\right )}{8 c \left (c + d x^{2}\right ) \left (a d - b c\right )^{3}} + \frac{\sqrt{d} \left (a^{2} d^{2} - 10 a b c d - 15 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{3}{2}} \left (a d - b c\right )^{4}} + \frac{b^{\frac{3}{2}} \left (5 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a d - b c\right )^{4}} \]
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,x)
[Out]
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Mathematica [A] time = 0.706733, size = 171, normalized size = 0.86 \[ \frac{\frac{\sqrt{d} \left (a^2 d^2-10 a b c d-15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2}}+\frac{4 b^{3/2} (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{4 b^2 x (b c-a d)}{a+b x^2}+\frac{d x (a d-b c) (a d+7 b c)}{c \left (c+d x^2\right )}-\frac{2 d x (b c-a d)^2}{\left (c+d x^2\right )^2}}{8 (b c-a d)^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x^2)^2*(c + d*x^2)^3),x]
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Maple [B] time = 0.022, size = 391, normalized size = 2. \[{\frac{{d}^{4}{x}^{3}{a}^{2}}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}c}}+{\frac{3\,{d}^{3}{x}^{3}ab}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{7\,{d}^{2}{x}^{3}{b}^{2}c}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,abc{d}^{2}x}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{9\,{b}^{2}{c}^{2}dx}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{a}^{2}{d}^{3}x}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{a}^{2}{d}^{3}}{8\, \left ( ad-bc \right ) ^{4}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{5\,ab{d}^{2}}{4\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,{b}^{2}cd}{8\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}xad}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}xc}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,a{b}^{2}d}{2\, \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{3}c}{2\, \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x^2+a)^2/(d*x^2+c)^3,x)
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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(x^2/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="maxima")
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Fricas [A] time = 1.96483, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**2+a)**2/(d*x**2+c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.25718, size = 428, normalized size = 2.14 \[ -\frac{b^{2} x}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (b x^{2} + a\right )}} + \frac{{\left (b^{3} c + 5 \, a b^{2} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{a b}} - \frac{{\left (15 \, b^{2} c^{2} d + 10 \, a b c d^{2} - a^{2} d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )} \sqrt{c d}} - \frac{7 \, b c d^{2} x^{3} + a d^{3} x^{3} + 9 \, b c^{2} d x - a c d^{2} x}{8 \,{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}{\left (d x^{2} + c\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="giac")
[Out]