Optimal. Leaf size=95 \[ \frac{2}{3} x^{3/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{5 x^{5/2}}+\frac{4}{7} b d x^{7/2} (a d+b c)-\frac{4 a c (a d+b c)}{\sqrt{x}}+\frac{2}{11} b^2 d^2 x^{11/2} \]
[Out]
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Rubi [A] time = 0.134953, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{2}{3} x^{3/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{5 x^{5/2}}+\frac{4}{7} b d x^{7/2} (a d+b c)-\frac{4 a c (a d+b c)}{\sqrt{x}}+\frac{2}{11} b^2 d^2 x^{11/2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(c + d*x^2)^2)/x^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 22.2182, size = 100, normalized size = 1.05 \[ - \frac{2 a^{2} c^{2}}{5 x^{\frac{5}{2}}} - \frac{4 a c \left (a d + b c\right )}{\sqrt{x}} + \frac{2 b^{2} d^{2} x^{\frac{11}{2}}}{11} + \frac{4 b d x^{\frac{7}{2}} \left (a d + b c\right )}{7} + x^{\frac{3}{2}} \left (\frac{2 a^{2} d^{2}}{3} + \frac{8 a b c d}{3} + \frac{2 b^{2} c^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**2/x**(7/2),x)
[Out]
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Mathematica [A] time = 0.0616044, size = 83, normalized size = 0.87 \[ \frac{2 \left (385 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-231 a^2 c^2+330 b d x^6 (a d+b c)-2310 a c x^2 (a d+b c)+105 b^2 d^2 x^8\right )}{1155 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^(7/2),x]
[Out]
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Maple [A] time = 0.01, size = 97, normalized size = 1. \[ -{\frac{-210\,{b}^{2}{d}^{2}{x}^{8}-660\,{x}^{6}ab{d}^{2}-660\,{x}^{6}{b}^{2}cd-770\,{x}^{4}{a}^{2}{d}^{2}-3080\,{x}^{4}abcd-770\,{x}^{4}{b}^{2}{c}^{2}+4620\,{x}^{2}{a}^{2}cd+4620\,a{c}^{2}b{x}^{2}+462\,{a}^{2}{c}^{2}}{1155}{x}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^2/x^(7/2),x)
[Out]
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Maxima [A] time = 1.34768, size = 117, normalized size = 1.23 \[ \frac{2}{11} \, b^{2} d^{2} x^{\frac{11}{2}} + \frac{4}{7} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{7}{2}} + \frac{2}{3} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (a^{2} c^{2} + 10 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{5 \, x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^2/x^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222071, size = 117, normalized size = 1.23 \[ \frac{2 \,{\left (105 \, b^{2} d^{2} x^{8} + 330 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 385 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 231 \, a^{2} c^{2} - 2310 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{1155 \, x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^2/x^(7/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 33.7085, size = 133, normalized size = 1.4 \[ - \frac{2 a^{2} c^{2}}{5 x^{\frac{5}{2}}} - \frac{4 a^{2} c d}{\sqrt{x}} + \frac{2 a^{2} d^{2} x^{\frac{3}{2}}}{3} - \frac{4 a b c^{2}}{\sqrt{x}} + \frac{8 a b c d x^{\frac{3}{2}}}{3} + \frac{4 a b d^{2} x^{\frac{7}{2}}}{7} + \frac{2 b^{2} c^{2} x^{\frac{3}{2}}}{3} + \frac{4 b^{2} c d x^{\frac{7}{2}}}{7} + \frac{2 b^{2} d^{2} x^{\frac{11}{2}}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)**2/x**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.249829, size = 130, normalized size = 1.37 \[ \frac{2}{11} \, b^{2} d^{2} x^{\frac{11}{2}} + \frac{4}{7} \, b^{2} c d x^{\frac{7}{2}} + \frac{4}{7} \, a b d^{2} x^{\frac{7}{2}} + \frac{2}{3} \, b^{2} c^{2} x^{\frac{3}{2}} + \frac{8}{3} \, a b c d x^{\frac{3}{2}} + \frac{2}{3} \, a^{2} d^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b c^{2} x^{2} + 10 \, a^{2} c d x^{2} + a^{2} c^{2}\right )}}{5 \, x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^2/x^(7/2),x, algorithm="giac")
[Out]