Optimal. Leaf size=195 \[ \frac{\sqrt{d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^4}+\frac{(b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} b^4}+\frac{d x \sqrt{c+d x^2} (11 b c-12 a d)}{8 b^3}-\frac{x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{3 d x \left (c+d x^2\right )^{3/2}}{4 b^2} \]
[Out]
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Rubi [A] time = 0.634916, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\sqrt{d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^4}+\frac{(b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} b^4}+\frac{d x \sqrt{c+d x^2} (11 b c-12 a d)}{8 b^3}-\frac{x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{3 d x \left (c+d x^2\right )^{3/2}}{4 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 94.1015, size = 182, normalized size = 0.93 \[ - \frac{x \left (c + d x^{2}\right )^{\frac{5}{2}}}{2 b \left (a + b x^{2}\right )} + \frac{3 d x \left (c + d x^{2}\right )^{\frac{3}{2}}}{4 b^{2}} - \frac{d x \sqrt{c + d x^{2}} \left (12 a d - 11 b c\right )}{8 b^{3}} + \frac{\sqrt{d} \left (24 a^{2} d^{2} - 40 a b c d + 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 b^{4}} - \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (6 a d - b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 \sqrt{a} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(d*x**2+c)**(5/2)/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.374174, size = 173, normalized size = 0.89 \[ \frac{\sqrt{d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+b x \sqrt{c+d x^2} \left (-\frac{4 (b c-a d)^2}{a+b x^2}+d (9 b c-8 a d)+2 b d^2 x^2\right )+\frac{4 (b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a}}}{8 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]
[Out]
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Maple [B] time = 0.026, size = 7459, normalized size = 38.3 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(d*x^2+c)^(5/2)/(b*x^2+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}} x^{2}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)*x^2/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.42925, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)*x^2/(b*x^2 + a)^2,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*(d*x**2+c)**(5/2)/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.557498, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)*x^2/(b*x^2 + a)^2,x, algorithm="giac")
[Out]