Optimal. Leaf size=74 \[ \frac{2 d \sqrt{c+d x^2}}{3 \sqrt{a+b x^2} (b c-a d)^2}-\frac{\sqrt{c+d x^2}}{3 \left (a+b x^2\right )^{3/2} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.147639, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 d \sqrt{c+d x^2}}{3 \sqrt{a+b x^2} (b c-a d)^2}-\frac{\sqrt{c+d x^2}}{3 \left (a+b x^2\right )^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 15.3092, size = 61, normalized size = 0.82 \[ \frac{2 d \sqrt{c + d x^{2}}}{3 \sqrt{a + b x^{2}} \left (a d - b c\right )^{2}} + \frac{\sqrt{c + d x^{2}}}{3 \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0628408, size = 52, normalized size = 0.7 \[ \frac{\sqrt{c+d x^2} \left (3 a d-b c+2 b d x^2\right )}{3 \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]
[Out]
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Maple [A] time = 0.007, size = 60, normalized size = 0.8 \[{\frac{2\,bd{x}^{2}+3\,ad-bc}{3\,{a}^{2}{d}^{2}-6\,cabd+3\,{b}^{2}{c}^{2}}\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270516, size = 170, normalized size = 2.3 \[ \frac{{\left (2 \, b d x^{2} - b c + 3 \, a d\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (a + b x^{2}\right )^{\frac{5}{2}} \sqrt{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.239711, size = 174, normalized size = 2.35 \[ \frac{4 \,{\left (b^{2} c - a b d - 3 \,{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt{b d} b^{2} d}{3 \,{\left (b^{2} c - a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{3}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)),x, algorithm="giac")
[Out]