Optimal. Leaf size=295 \[ \frac{2 b^5 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^7 \left (a+b x^2\right )}-\frac{2 a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac{10 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt{d x} \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.229594, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 b^5 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^7 \left (a+b x^2\right )}-\frac{2 a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac{10 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt{d x} \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(d*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 28.897, size = 238, normalized size = 0.81 \[ \frac{16384 a^{5} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{1155 d \left (d x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )} - \frac{4096 a^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{231 d \left (d x\right )^{\frac{5}{2}}} + \frac{512 a^{3} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{231 d \left (d x\right )^{\frac{5}{2}}} + \frac{128 a^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{231 d \left (d x\right )^{\frac{5}{2}}} + \frac{8 a \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{33 d \left (d x\right )^{\frac{5}{2}}} + \frac{2 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{15 d \left (d x\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/(d*x)**(7/2),x)
[Out]
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Mathematica [A] time = 0.0504978, size = 88, normalized size = 0.3 \[ \frac{2 x \sqrt{\left (a+b x^2\right )^2} \left (-231 a^5-5775 a^4 b x^2+3850 a^3 b^2 x^4+1650 a^2 b^3 x^6+525 a b^4 x^8+77 b^5 x^{10}\right )}{1155 (d x)^{7/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(d*x)^(7/2),x]
[Out]
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Maple [A] time = 0.009, size = 83, normalized size = 0.3 \[ -{\frac{2\, \left ( -77\,{b}^{5}{x}^{10}-525\,a{b}^{4}{x}^{8}-1650\,{a}^{2}{b}^{3}{x}^{6}-3850\,{a}^{3}{b}^{2}{x}^{4}+5775\,{a}^{4}b{x}^{2}+231\,{a}^{5} \right ) x}{1155\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( dx \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/(d*x)^(7/2),x)
[Out]
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Maxima [A] time = 0.724518, size = 203, normalized size = 0.69 \[ \frac{2 \,{\left (7 \,{\left (11 \, b^{5} \sqrt{d} x^{3} + 15 \, a b^{4} \sqrt{d} x\right )} x^{\frac{9}{2}} + 60 \,{\left (7 \, a b^{4} \sqrt{d} x^{3} + 11 \, a^{2} b^{3} \sqrt{d} x\right )} x^{\frac{5}{2}} + 330 \,{\left (3 \, a^{2} b^{3} \sqrt{d} x^{3} + 7 \, a^{3} b^{2} \sqrt{d} x\right )} \sqrt{x} + \frac{1540 \,{\left (a^{3} b^{2} \sqrt{d} x^{3} - 3 \, a^{4} b \sqrt{d} x\right )}}{x^{\frac{3}{2}}} - \frac{231 \,{\left (5 \, a^{4} b \sqrt{d} x^{3} + a^{5} \sqrt{d} x\right )}}{x^{\frac{7}{2}}}\right )}}{1155 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/(d*x)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273514, size = 90, normalized size = 0.31 \[ \frac{2 \,{\left (77 \, b^{5} x^{10} + 525 \, a b^{4} x^{8} + 1650 \, a^{2} b^{3} x^{6} + 3850 \, a^{3} b^{2} x^{4} - 5775 \, a^{4} b x^{2} - 231 \, a^{5}\right )}}{1155 \, \sqrt{d x} d^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/(d*x)^(7/2),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((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/(d*x)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.271414, size = 219, normalized size = 0.74 \[ -\frac{2 \,{\left (\frac{231 \,{\left (25 \, a^{4} b d^{3} x^{2}{\rm sign}\left (b x^{2} + a\right ) + a^{5} d^{3}{\rm sign}\left (b x^{2} + a\right )\right )}}{\sqrt{d x} d^{2} x^{2}} - \frac{77 \, \sqrt{d x} b^{5} d^{105} x^{7}{\rm sign}\left (b x^{2} + a\right ) + 525 \, \sqrt{d x} a b^{4} d^{105} x^{5}{\rm sign}\left (b x^{2} + a\right ) + 1650 \, \sqrt{d x} a^{2} b^{3} d^{105} x^{3}{\rm sign}\left (b x^{2} + a\right ) + 3850 \, \sqrt{d x} a^{3} b^{2} d^{105} x{\rm sign}\left (b x^{2} + a\right )}{d^{105}}\right )}}{1155 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/(d*x)^(7/2),x, algorithm="giac")
[Out]