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

Optimal. Leaf size=46 \[ \frac{\left (a+b x^2\right )^{5/2} (A b-a B)}{5 b^2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b^2} \]

[Out]

((A*b - a*B)*(a + b*x^2)^(5/2))/(5*b^2) + (B*(a + b*x^2)^(7/2))/(7*b^2)

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Rubi [A]  time = 0.102413, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\left (a+b x^2\right )^{5/2} (A b-a B)}{5 b^2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*(a + b*x^2)^(5/2))/(5*b^2) + (B*(a + b*x^2)^(7/2))/(7*b^2)

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Rubi in Sympy [A]  time = 13.0507, size = 37, normalized size = 0.8 \[ \frac{B \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x**2)**(7/2)/(7*b**2) + (a + b*x**2)**(5/2)*(A*b - B*a)/(5*b**2)

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Mathematica [A]  time = 0.0502828, size = 34, normalized size = 0.74 \[ \frac{\left (a+b x^2\right )^{5/2} \left (-2 a B+7 A b+5 b B x^2\right )}{35 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^2)^(5/2)*(7*A*b - 2*a*B + 5*b*B*x^2))/(35*b^2)

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Maple [A]  time = 0.006, size = 31, normalized size = 0.7 \[{\frac{5\,bB{x}^{2}+7\,Ab-2\,Ba}{35\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35*(b*x^2+a)^(5/2)*(5*B*b*x^2+7*A*b-2*B*a)/b^2

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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)*(b*x^2 + a)^(3/2)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.212391, size = 99, normalized size = 2.15 \[ \frac{{\left (5 \, B b^{3} x^{6} +{\left (8 \, B a b^{2} + 7 \, A b^{3}\right )} x^{4} - 2 \, B a^{3} + 7 \, A a^{2} b +{\left (B a^{2} b + 14 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{35 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35*(5*B*b^3*x^6 + (8*B*a*b^2 + 7*A*b^3)*x^4 - 2*B*a^3 + 7*A*a^2*b + (B*a^2*b +
14*A*a*b^2)*x^2)*sqrt(b*x^2 + a)/b^2

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Sympy [A]  time = 3.69978, size = 158, normalized size = 3.43 \[ \begin{cases} \frac{A a^{2} \sqrt{a + b x^{2}}}{5 b} + \frac{2 A a x^{2} \sqrt{a + b x^{2}}}{5} + \frac{A b x^{4} \sqrt{a + b x^{2}}}{5} - \frac{2 B a^{3} \sqrt{a + b x^{2}}}{35 b^{2}} + \frac{B a^{2} x^{2} \sqrt{a + b x^{2}}}{35 b} + \frac{8 B a x^{4} \sqrt{a + b x^{2}}}{35} + \frac{B b x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\a^{\frac{3}{2}} \left (\frac{A x^{2}}{2} + \frac{B x^{4}}{4}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((A*a**2*sqrt(a + b*x**2)/(5*b) + 2*A*a*x**2*sqrt(a + b*x**2)/5 + A*b*x
**4*sqrt(a + b*x**2)/5 - 2*B*a**3*sqrt(a + b*x**2)/(35*b**2) + B*a**2*x**2*sqrt(
a + b*x**2)/(35*b) + 8*B*a*x**4*sqrt(a + b*x**2)/35 + B*b*x**6*sqrt(a + b*x**2)/
7, Ne(b, 0)), (a**(3/2)*(A*x**2/2 + B*x**4/4), True))

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GIAC/XCAS [A]  time = 0.245832, size = 162, normalized size = 3.52 \[ \frac{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a + 7 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} A + \frac{7 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} B a}{b} + \frac{{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} B}{b}}{105 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(35*(b*x^2 + a)^(3/2)*A*a + 7*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a
)*A + 7*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*B*a/b + (15*(b*x^2 + a)^(7
/2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)*B/b)/b