Optimal. Leaf size=139 \[ \frac{2 B x}{35 a^2 b^2 \sqrt{a+b x^2}}-\frac{2 (4 a C+3 A b)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x (x (4 a C+3 A b)+3 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^3 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
[Out]
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Rubi [A] time = 0.358691, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{2 B x}{35 a^2 b^2 \sqrt{a+b x^2}}-\frac{2 (4 a C+3 A b)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x (x (4 a C+3 A b)+3 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^3 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 29.0823, size = 124, normalized size = 0.89 \[ \frac{2 B x}{35 a^{2} b^{2} \sqrt{a + b x^{2}}} - \frac{x^{3} \left (B a - x \left (A b - C a\right )\right )}{7 a b \left (a + b x^{2}\right )^{\frac{7}{2}}} - \frac{x^{2} \left (3 A b - 3 B b x + 4 C a\right )}{35 a b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}} - \frac{6 A b + 6 B b x + 8 C a}{105 a b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)
[Out]
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Mathematica [A] time = 0.0960272, size = 84, normalized size = 0.6 \[ \frac{-8 a^4 C-2 a^3 b \left (3 A+14 C x^2\right )-7 a^2 b^2 x^2 \left (3 A+5 C x^2\right )+21 a b^3 B x^5+6 b^4 B x^7}{105 a^2 b^3 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]
[Out]
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Maple [A] time = 0.011, size = 85, normalized size = 0.6 \[ -{\frac{-6\,B{b}^{4}{x}^{7}-21\,B{x}^{5}a{b}^{3}+35\,C{x}^{4}{a}^{2}{b}^{2}+21\,A{a}^{2}{b}^{2}{x}^{2}+28\,C{a}^{3}b{x}^{2}+6\,A{a}^{3}b+8\,C{a}^{4}}{105\,{a}^{2}{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x)
[Out]
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Maxima [A] time = 1.38504, size = 242, normalized size = 1.74 \[ -\frac{C x^{4}}{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{B x^{3}}{4 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{4 \, C a x^{2}}{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{A x^{2}}{5 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{3 \, B x}{140 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{2}} + \frac{2 \, B x}{35 \, \sqrt{b x^{2} + a} a^{2} b^{2}} + \frac{B x}{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{2}} - \frac{3 \, B a x}{28 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{8 \, C a^{2}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} - \frac{2 \, A a}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*x^3/(b*x^2 + a)^(9/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285538, size = 177, normalized size = 1.27 \[ \frac{{\left (6 \, B b^{4} x^{7} + 21 \, B a b^{3} x^{5} - 35 \, C a^{2} b^{2} x^{4} - 8 \, C a^{4} - 6 \, A a^{3} b - 7 \,{\left (4 \, C a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a^{2} b^{7} x^{8} + 4 \, a^{3} b^{6} x^{6} + 6 \, a^{4} b^{5} x^{4} + 4 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*x^3/(b*x^2 + a)^(9/2),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(x**3*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)
[Out]
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GIAC/XCAS [A] time = 0.22317, size = 128, normalized size = 0.92 \[ \frac{{\left ({\left (3 \,{\left (\frac{2 \, B b x^{2}}{a^{2}} + \frac{7 \, B}{a}\right )} x - \frac{35 \, C}{b}\right )} x^{2} - \frac{7 \,{\left (4 \, C a^{4} b + 3 \, A a^{3} b^{2}\right )}}{a^{3} b^{3}}\right )} x^{2} - \frac{2 \,{\left (4 \, C a^{5} + 3 \, A a^{4} b\right )}}{a^{3} b^{3}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*x^3/(b*x^2 + a)^(9/2),x, algorithm="giac")
[Out]