Optimal. Leaf size=180 \[ -\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 b^{3/2}}+\frac{c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3}-\frac{\sqrt{c+d x^2} (b c-a d) (2 b c-a d)}{2 a^2 b \left (a+b x^2\right )}-\frac{c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.719622, antiderivative size = 180, 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 \[ -\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 b^{3/2}}+\frac{c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3}-\frac{\sqrt{c+d x^2} (b c-a d) (2 b c-a d)}{2 a^2 b \left (a+b x^2\right )}-\frac{c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)^(5/2)/(x^3*(a + b*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 72.7551, size = 155, normalized size = 0.86 \[ - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{2 a b x^{2} \left (a + b x^{2}\right )} + \frac{c \sqrt{c + d x^{2}} \left (a d - 2 b c\right )}{2 a^{2} b x^{2}} - \frac{c^{\frac{3}{2}} \left (5 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 a^{3}} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (a d + 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 a^{3} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**(5/2)/x**3/(b*x**2+a)**2,x)
[Out]
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Mathematica [C] time = 1.18527, size = 349, normalized size = 1.94 \[ -\frac{\frac{(b c-a d)^{3/2} (a d+4 b c) \log \left (\frac{4 a^3 b^{3/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x+i \sqrt{a}\right ) (b c-a d)^{5/2} (a d+4 b c)}\right )}{b^{3/2}}+\frac{(b c-a d)^{3/2} (a d+4 b c) \log \left (\frac{4 a^3 b^{3/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x-i \sqrt{a}\right ) (b c-a d)^{5/2} (a d+4 b c)}\right )}{b^{3/2}}-2 c^{3/2} (4 b c-5 a d) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+2 c^{3/2} \log (x) (4 b c-5 a d)+2 a \sqrt{c+d x^2} \left (\frac{(b c-a d)^2}{b \left (a+b x^2\right )}+\frac{c^2}{x^2}\right )}{4 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)^(5/2)/(x^3*(a + b*x^2)^2),x]
[Out]
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Maple [B] time = 0.031, size = 7590, normalized size = 42.2 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^(5/2)/x^3/(b*x^2+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)^2*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.64854, 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^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((d*x**2+c)**(5/2)/x**3/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.238189, size = 390, normalized size = 2.17 \[ -\frac{1}{2} \, d^{3}{\left (\frac{{\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} d^{3}} + \frac{2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{2} - 2 \, \sqrt{d x^{2} + c} b^{2} c^{3} - 2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d + 3 \, \sqrt{d x^{2} + c} a b c^{2} d +{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{2} - \sqrt{d x^{2} + c} a^{2} c d^{2}}{{\left ({\left (d x^{2} + c\right )}^{2} b - 2 \,{\left (d x^{2} + c\right )} b c + b c^{2} +{\left (d x^{2} + c\right )} a d - a c d\right )} a^{2} b d^{2}} - \frac{{\left (4 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3} b d^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)^2*x^3),x, algorithm="giac")
[Out]