3.572 \(\int \frac{x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{\sqrt{a+b x^2} (A b-2 a B)}{b^3}+\frac{a (A b-a B)}{b^3 \sqrt{a+b x^2}}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^3} \]

[Out]

(a*(A*b - a*B))/(b^3*Sqrt[a + b*x^2]) + ((A*b - 2*a*B)*Sqrt[a + b*x^2])/b^3 + (B
*(a + b*x^2)^(3/2))/(3*b^3)

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Rubi [A]  time = 0.16596, antiderivative size = 67, 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 \[ \frac{\sqrt{a+b x^2} (A b-2 a B)}{b^3}+\frac{a (A b-a B)}{b^3 \sqrt{a+b x^2}}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(A + B*x^2))/(a + b*x^2)^(3/2),x]

[Out]

(a*(A*b - a*B))/(b^3*Sqrt[a + b*x^2]) + ((A*b - 2*a*B)*Sqrt[a + b*x^2])/b^3 + (B
*(a + b*x^2)^(3/2))/(3*b^3)

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Rubi in Sympy [A]  time = 18.9901, size = 60, normalized size = 0.9 \[ \frac{B \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{3}} + \frac{a \left (A b - B a\right )}{b^{3} \sqrt{a + b x^{2}}} + \frac{\sqrt{a + b x^{2}} \left (A b - 2 B a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x**2+A)/(b*x**2+a)**(3/2),x)

[Out]

B*(a + b*x**2)**(3/2)/(3*b**3) + a*(A*b - B*a)/(b**3*sqrt(a + b*x**2)) + sqrt(a
+ b*x**2)*(A*b - 2*B*a)/b**3

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Mathematica [A]  time = 0.060288, size = 55, normalized size = 0.82 \[ \frac{-8 a^2 B+a \left (6 A b-4 b B x^2\right )+b^2 x^2 \left (3 A+B x^2\right )}{3 b^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(A + B*x^2))/(a + b*x^2)^(3/2),x]

[Out]

(-8*a^2*B + b^2*x^2*(3*A + B*x^2) + a*(6*A*b - 4*b*B*x^2))/(3*b^3*Sqrt[a + b*x^2
])

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Maple [A]  time = 0.008, size = 52, normalized size = 0.8 \[{\frac{{b}^{2}B{x}^{4}+3\,A{b}^{2}{x}^{2}-4\,Bab{x}^{2}+6\,abA-8\,{a}^{2}B}{3\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x^2+A)/(b*x^2+a)^(3/2),x)

[Out]

1/3*(B*b^2*x^4+3*A*b^2*x^2-4*B*a*b*x^2+6*A*a*b-8*B*a^2)/(b*x^2+a)^(1/2)/b^3

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.211825, size = 85, normalized size = 1.27 \[ \frac{{\left (B b^{2} x^{4} - 8 \, B a^{2} + 6 \, A a b -{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^3/(b*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

1/3*(B*b^2*x^4 - 8*B*a^2 + 6*A*a*b - (4*B*a*b - 3*A*b^2)*x^2)*sqrt(b*x^2 + a)/(b
^4*x^2 + a*b^3)

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Sympy [A]  time = 3.43039, size = 117, normalized size = 1.75 \[ \begin{cases} \frac{2 A a}{b^{2} \sqrt{a + b x^{2}}} + \frac{A x^{2}}{b \sqrt{a + b x^{2}}} - \frac{8 B a^{2}}{3 b^{3} \sqrt{a + b x^{2}}} - \frac{4 B a x^{2}}{3 b^{2} \sqrt{a + b x^{2}}} + \frac{B x^{4}}{3 b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{4}}{4} + \frac{B x^{6}}{6}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x**2+A)/(b*x**2+a)**(3/2),x)

[Out]

Piecewise((2*A*a/(b**2*sqrt(a + b*x**2)) + A*x**2/(b*sqrt(a + b*x**2)) - 8*B*a**
2/(3*b**3*sqrt(a + b*x**2)) - 4*B*a*x**2/(3*b**2*sqrt(a + b*x**2)) + B*x**4/(3*b
*sqrt(a + b*x**2)), Ne(b, 0)), ((A*x**4/4 + B*x**6/6)/a**(3/2), True))

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GIAC/XCAS [A]  time = 0.228403, size = 88, normalized size = 1.31 \[ \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} B - 6 \, \sqrt{b x^{2} + a} B a + 3 \, \sqrt{b x^{2} + a} A b - \frac{3 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x^{2} + a}}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^3/(b*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

1/3*((b*x^2 + a)^(3/2)*B - 6*sqrt(b*x^2 + a)*B*a + 3*sqrt(b*x^2 + a)*A*b - 3*(B*
a^2 - A*a*b)/sqrt(b*x^2 + a))/b^3