Optimal. Leaf size=68 \[ -\frac{A b-2 a B}{b^3 \sqrt{a+b x^2}}+\frac{a (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{B \sqrt{a+b x^2}}{b^3} \]
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Rubi [A] time = 0.171386, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{A b-2 a B}{b^3 \sqrt{a+b x^2}}+\frac{a (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{B \sqrt{a+b x^2}}{b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(A + B*x^2))/(a + b*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 19.6958, size = 60, normalized size = 0.88 \[ \frac{B \sqrt{a + b x^{2}}}{b^{3}} + \frac{a \left (A b - B a\right )}{3 b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{A b - 2 B a}{b^{3} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x**2+A)/(b*x**2+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0640779, size = 54, normalized size = 0.79 \[ \frac{8 a^2 B-2 a b \left (A-6 B x^2\right )+3 b^2 x^2 \left (B x^2-A\right )}{3 b^3 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(A + B*x^2))/(a + b*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.008, size = 53, normalized size = 0.8 \[ -{\frac{-3\,{b}^{2}B{x}^{4}+3\,A{b}^{2}{x}^{2}-12\,Bab{x}^{2}+2\,abA-8\,{a}^{2}B}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x^2+A)/(b*x^2+a)^(5/2),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((B*x^2 + A)*x^3/(b*x^2 + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226029, size = 101, normalized size = 1.49 \[ \frac{{\left (3 \, B b^{2} x^{4} + 8 \, B a^{2} - 2 \, A a b + 3 \,{\left (4 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(b*x^2 + a)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.12173, size = 240, normalized size = 3.53 \[ \begin{cases} - \frac{2 A a b}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} - \frac{3 A b^{2} x^{2}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} + \frac{8 B a^{2}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} + \frac{12 B a b x^{2}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} + \frac{3 B b^{2} x^{4}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{4}}{4} + \frac{B x^{6}}{6}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x**2+A)/(b*x**2+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.231109, size = 82, normalized size = 1.21 \[ \frac{3 \, \sqrt{b x^{2} + a} B + \frac{6 \,{\left (b x^{2} + a\right )} B a - B a^{2} - 3 \,{\left (b x^{2} + a\right )} A b + A a b}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(b*x^2 + a)^(5/2),x, algorithm="giac")
[Out]