Optimal. Leaf size=217 \[ \frac{2 d \sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \sqrt{a+b x^2} (b c-a d)^4}-\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (7 b c-4 a d)}{35 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)^2} \]
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Rubi [A] time = 0.648056, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2 d \sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \sqrt{a+b x^2} (b c-a d)^4}-\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (7 b c-4 a d)}{35 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[x^5/((a + b*x^2)^(9/2)*Sqrt[c + d*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 62.3711, size = 204, normalized size = 0.94 \[ \frac{a^{2} \sqrt{c + d x^{2}}}{7 b^{2} \left (a + b x^{2}\right )^{\frac{7}{2}} \left (a d - b c\right )} - \frac{2 a \sqrt{c + d x^{2}} \left (4 a d - 7 b c\right )}{35 b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}} \left (a d - b c\right )^{2}} + \frac{2 d \sqrt{c + d x^{2}} \left (3 a^{2} d^{2} - 14 a b c d + 35 b^{2} c^{2}\right )}{105 b^{2} \sqrt{a + b x^{2}} \left (a d - b c\right )^{4}} + \frac{\sqrt{c + d x^{2}} \left (3 a^{2} d^{2} - 14 a b c d + 35 b^{2} c^{2}\right )}{105 b^{2} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(b*x**2+a)**(9/2)/(d*x**2+c)**(1/2),x)
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Mathematica [A] time = 0.215599, size = 151, normalized size = 0.7 \[ \frac{\sqrt{c+d x^2} \left (7 a^3 d \left (8 c^2-4 c d x^2+3 d^2 x^4\right )+a^2 b \left (-8 c^3+200 c^2 d x^2-101 c d^2 x^4+6 d^3 x^6\right )-7 a b^2 c x^2 \left (4 c^2-37 c d x^2+4 d^2 x^4\right )-35 b^3 c^2 x^4 \left (c-2 d x^2\right )\right )}{105 \left (a+b x^2\right )^{7/2} (b c-a d)^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/((a + b*x^2)^(9/2)*Sqrt[c + d*x^2]),x]
[Out]
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Maple [A] time = 0.016, size = 213, normalized size = 1. \[{\frac{6\,{a}^{2}b{d}^{3}{x}^{6}-28\,a{b}^{2}c{d}^{2}{x}^{6}+70\,{b}^{3}{c}^{2}d{x}^{6}+21\,{a}^{3}{d}^{3}{x}^{4}-101\,{a}^{2}bc{d}^{2}{x}^{4}+259\,a{b}^{2}{c}^{2}d{x}^{4}-35\,{b}^{3}{c}^{3}{x}^{4}-28\,{a}^{3}c{d}^{2}{x}^{2}+200\,{a}^{2}b{c}^{2}d{x}^{2}-28\,a{b}^{2}{c}^{3}{x}^{2}+56\,{a}^{3}{c}^{2}d-8\,{a}^{2}b{c}^{3}}{105\,{a}^{4}{d}^{4}-420\,{a}^{3}bc{d}^{3}+630\,{a}^{2}{c}^{2}{d}^{2}{b}^{2}-420\,a{c}^{3}d{b}^{3}+105\,{c}^{4}{b}^{4}}\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(b*x^2+a)^(9/2)/(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^5/((b*x^2 + a)^(9/2)*sqrt(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.494261, size = 609, normalized size = 2.81 \[ \frac{{\left (2 \,{\left (35 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x^{6} - 8 \, a^{2} b c^{3} + 56 \, a^{3} c^{2} d -{\left (35 \, b^{3} c^{3} - 259 \, a b^{2} c^{2} d + 101 \, a^{2} b c d^{2} - 21 \, a^{3} d^{3}\right )} x^{4} - 4 \,{\left (7 \, a b^{2} c^{3} - 50 \, a^{2} b c^{2} d + 7 \, a^{3} c d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{105 \,{\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} +{\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{8} + 4 \,{\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{6} + 6 \,{\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{4} + 4 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((b*x^2 + a)^(9/2)*sqrt(d*x^2 + c)),x, algorithm="fricas")
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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**5/(b*x**2+a)**(9/2)/(d*x**2+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.31608, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((b*x^2 + a)^(9/2)*sqrt(d*x^2 + c)),x, algorithm="giac")
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