Optimal. Leaf size=258 \[ \frac{x \sqrt{c+d x^2} \left (32 a^2 d^2-52 a b c d+19 b^2 c^2\right )}{16 b^4}+\frac{\left (-64 a^3 d^3+120 a^2 b c d^2-60 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^5 \sqrt{d}}-\frac{\sqrt{a} (3 b c-8 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^5}+\frac{d x^3 \sqrt{c+d x^2} (7 b c-8 a d)}{8 b^3}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2} \]
[Out]
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Rubi [A] time = 1.18009, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x \sqrt{c+d x^2} \left (32 a^2 d^2-52 a b c d+19 b^2 c^2\right )}{16 b^4}+\frac{\left (-64 a^3 d^3+120 a^2 b c d^2-60 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^5 \sqrt{d}}-\frac{\sqrt{a} (3 b c-8 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^5}+\frac{d x^3 \sqrt{c+d x^2} (7 b c-8 a d)}{8 b^3}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 148.015, size = 246, normalized size = 0.95 \[ \frac{\sqrt{a} \left (a d - b c\right )^{\frac{3}{2}} \left (8 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 b^{5}} - \frac{x^{3} \left (c + d x^{2}\right )^{\frac{5}{2}}}{2 b \left (a + b x^{2}\right )} + \frac{2 d x^{3} \left (c + d x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} - \frac{d x^{3} \sqrt{c + d x^{2}} \left (8 a d - 7 b c\right )}{8 b^{3}} + \frac{x \sqrt{c + d x^{2}} \left (32 a^{2} d^{2} - 52 a b c d + 19 b^{2} c^{2}\right )}{16 b^{4}} - \frac{\left (64 a^{3} d^{3} - 120 a^{2} b c d^{2} + 60 a b^{2} c^{2} d - 5 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{16 b^{5} \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(d*x**2+c)**(5/2)/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.576895, size = 219, normalized size = 0.85 \[ \frac{b x \sqrt{c+d x^2} \left (72 a^2 d^2+2 b d x^2 (13 b c-12 a d)+\frac{24 a (b c-a d)^2}{a+b x^2}-108 a b c d+33 b^2 c^2+8 b^2 d^2 x^4\right )+\frac{3 \left (-64 a^3 d^3+120 a^2 b c d^2-60 a b^2 c^2 d+5 b^3 c^3\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}+24 \sqrt{a} (8 a d-3 b c) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{48 b^5} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]
[Out]
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Maple [B] time = 0.041, size = 7611, normalized size = 29.5 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(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. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}} x^{4}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)*x^4/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 3.03263, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)*x^4/(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**4*(d*x**2+c)**(5/2)/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.581435, 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^4/(b*x^2 + a)^2,x, algorithm="giac")
[Out]