Optimal. Leaf size=120 \[ \frac{1}{3} d x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{3 x^3}-\frac{a c^2 (3 a d+2 b c)}{x}+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{7} b^2 d^3 x^7 \]
[Out]
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Rubi [A] time = 0.166695, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{1}{3} d x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{3 x^3}-\frac{a c^2 (3 a d+2 b c)}{x}+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{7} b^2 d^3 x^7 \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(c + d*x^2)^3)/x^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} c^{3}}{3 x^{3}} - \frac{a c^{2} \left (3 a d + 2 b c\right )}{x} + \frac{b^{2} d^{3} x^{7}}{7} + \frac{b d^{2} x^{5} \left (2 a d + 3 b c\right )}{5} + \frac{d x^{3} \left (a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{3} + \frac{c \left (3 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right ) \int b^{2}\, dx}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**3/x**4,x)
[Out]
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Mathematica [A] time = 0.0808203, size = 120, normalized size = 1. \[ \frac{1}{3} d x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{3 x^3}-\frac{a c^2 (3 a d+2 b c)}{x}+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{7} b^2 d^3 x^7 \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^4,x]
[Out]
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Maple [A] time = 0.008, size = 124, normalized size = 1. \[{\frac{{b}^{2}{d}^{3}{x}^{7}}{7}}+{\frac{2\,{x}^{5}ab{d}^{3}}{5}}+{\frac{3\,{x}^{5}{b}^{2}c{d}^{2}}{5}}+{\frac{{x}^{3}{a}^{2}{d}^{3}}{3}}+2\,{x}^{3}abc{d}^{2}+{x}^{3}{b}^{2}{c}^{2}d+3\,x{a}^{2}c{d}^{2}+6\,xab{c}^{2}d+x{b}^{2}{c}^{3}-{\frac{{a}^{2}{c}^{3}}{3\,{x}^{3}}}-{\frac{a{c}^{2} \left ( 3\,ad+2\,bc \right ) }{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^3/x^4,x)
[Out]
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Maxima [A] time = 1.33104, size = 170, normalized size = 1.42 \[ \frac{1}{7} \, b^{2} d^{3} x^{7} + \frac{1}{5} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{5} + \frac{1}{3} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x - \frac{a^{2} c^{3} + 3 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229272, size = 174, normalized size = 1.45 \[ \frac{15 \, b^{2} d^{3} x^{10} + 21 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 35 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 35 \, a^{2} c^{3} + 105 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} - 105 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{105 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3/x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.04625, size = 129, normalized size = 1.08 \[ \frac{b^{2} d^{3} x^{7}}{7} + x^{5} \left (\frac{2 a b d^{3}}{5} + \frac{3 b^{2} c d^{2}}{5}\right ) + x^{3} \left (\frac{a^{2} d^{3}}{3} + 2 a b c d^{2} + b^{2} c^{2} d\right ) + x \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3}\right ) - \frac{a^{2} c^{3} + x^{2} \left (9 a^{2} c^{2} d + 6 a b c^{3}\right )}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)**3/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.221275, size = 174, normalized size = 1.45 \[ \frac{1}{7} \, b^{2} d^{3} x^{7} + \frac{3}{5} \, b^{2} c d^{2} x^{5} + \frac{2}{5} \, a b d^{3} x^{5} + b^{2} c^{2} d x^{3} + 2 \, a b c d^{2} x^{3} + \frac{1}{3} \, a^{2} d^{3} x^{3} + b^{2} c^{3} x + 6 \, a b c^{2} d x + 3 \, a^{2} c d^{2} x - \frac{6 \, a b c^{3} x^{2} + 9 \, a^{2} c^{2} d x^{2} + a^{2} c^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3/x^4,x, algorithm="giac")
[Out]