Optimal. Leaf size=174 \[ \frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{3/2} b^3}+\frac{d^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^3}-\frac{d x \sqrt{c+d x^2} (b c-2 a d)}{2 a b^2}+\frac{x \left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.527996, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{3/2} b^3}+\frac{d^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^3}-\frac{d x \sqrt{c+d x^2} (b c-2 a d)}{2 a b^2}+\frac{x \left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)^(5/2)/(a + b*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 74.451, size = 155, normalized size = 0.89 \[ - \frac{d^{\frac{3}{2}} \left (4 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 b^{3}} - \frac{x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{2 a b \left (a + b x^{2}\right )} + \frac{d x \sqrt{c + d x^{2}} \left (2 a d - b c\right )}{2 a b^{2}} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (4 a d + b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{3}{2}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**(5/2)/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.325655, size = 143, normalized size = 0.82 \[ \frac{\frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2}}+d^{3/2} (5 b c-4 a d) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+b x \sqrt{c+d x^2} \left (\frac{(b c-a d)^2}{a \left (a+b x^2\right )}+d^2\right )}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)^(5/2)/(a + b*x^2)^2,x]
[Out]
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Maple [B] time = 0.025, size = 7451, normalized size = 42.8 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.985829, 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)/(b*x^2 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x^{2}\right )^{\frac{5}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**(5/2)/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.558972, 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)/(b*x^2 + a)^2,x, algorithm="giac")
[Out]