Optimal. Leaf size=130 \[ \frac{x (a d+2 b c)}{2 b \sqrt{c+d x^2} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}-\frac{3 \sqrt{a} c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 (b c-a d)^{5/2}} \]
[Out]
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Rubi [A] time = 0.28252, antiderivative size = 130, 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 \[ \frac{x (a d+2 b c)}{2 b \sqrt{c+d x^2} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}-\frac{3 \sqrt{a} c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 (b c-a d)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[x^4/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 52.0721, size = 110, normalized size = 0.85 \[ - \frac{3 \sqrt{a} c \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 \left (a d - b c\right )^{\frac{5}{2}}} - \frac{a x}{2 b \left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{x \left (a d + 2 b c\right )}{2 b \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.435764, size = 106, normalized size = 0.82 \[ \frac{1}{2} \left (\frac{3 a c x+a d x^3+2 b c x^3}{\left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)^2}-\frac{3 \sqrt{a} c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{5/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]
[Out]
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Maple [B] time = 0.032, size = 1498, normalized size = 11.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^2+a)^2/(d*x^2+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.50435, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b c d x^{4} + a c^{2} +{\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c x\right )} \sqrt{d x^{2} + c}}{8 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}, -\frac{3 \,{\left (b c d x^{4} + a c^{2} +{\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} x \sqrt{\frac{a}{b c - a d}}}\right ) - 2 \,{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c x\right )} \sqrt{d x^{2} + c}}{4 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 3.34081, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^(3/2)),x, algorithm="giac")
[Out]