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

Optimal. Leaf size=46 \[ \frac{2 \left (a+b x^3\right )^{5/2} (A b-a B)}{15 b^2}+\frac{2 B \left (a+b x^3\right )^{7/2}}{21 b^2} \]

[Out]

(2*(A*b - a*B)*(a + b*x^3)^(5/2))/(15*b^2) + (2*B*(a + b*x^3)^(7/2))/(21*b^2)

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Rubi [A]  time = 0.141341, antiderivative size = 46, 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 \[ \frac{2 \left (a+b x^3\right )^{5/2} (A b-a B)}{15 b^2}+\frac{2 B \left (a+b x^3\right )^{7/2}}{21 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*(a + b*x^3)^(5/2))/(15*b^2) + (2*B*(a + b*x^3)^(7/2))/(21*b^2)

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Rubi in Sympy [A]  time = 12.0758, size = 41, normalized size = 0.89 \[ \frac{2 B \left (a + b x^{3}\right )^{\frac{7}{2}}}{21 b^{2}} + \frac{2 \left (a + b x^{3}\right )^{\frac{5}{2}} \left (A b - B a\right )}{15 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(a + b*x**3)**(7/2)/(21*b**2) + 2*(a + b*x**3)**(5/2)*(A*b - B*a)/(15*b**2)

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Mathematica [A]  time = 0.0580398, size = 34, normalized size = 0.74 \[ \frac{2 \left (a+b x^3\right )^{5/2} \left (-2 a B+7 A b+5 b B x^3\right )}{105 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 31, normalized size = 0.7 \[{\frac{10\,bB{x}^{3}+14\,Ab-4\,Ba}{105\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.47614, size = 66, normalized size = 1.43 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} A}{15 \, b} + \frac{2}{105} \,{\left (\frac{5 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{b^{2}} - \frac{7 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{b^{2}}\right )} B \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.24338, size = 99, normalized size = 2.15 \[ \frac{2 \,{\left (5 \, B b^{3} x^{9} +{\left (8 \, B a b^{2} + 7 \, A b^{3}\right )} x^{6} - 2 \, B a^{3} + 7 \, A a^{2} b +{\left (B a^{2} b + 14 \, A a b^{2}\right )} x^{3}\right )} \sqrt{b x^{3} + a}}{105 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(5*B*b^3*x^9 + (8*B*a*b^2 + 7*A*b^3)*x^6 - 2*B*a^3 + 7*A*a^2*b + (B*a^2*b
+ 14*A*a*b^2)*x^3)*sqrt(b*x^3 + a)/b^2

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Sympy [A]  time = 5.93264, size = 165, normalized size = 3.59 \[ \begin{cases} \frac{2 A a^{2} \sqrt{a + b x^{3}}}{15 b} + \frac{4 A a x^{3} \sqrt{a + b x^{3}}}{15} + \frac{2 A b x^{6} \sqrt{a + b x^{3}}}{15} - \frac{4 B a^{3} \sqrt{a + b x^{3}}}{105 b^{2}} + \frac{2 B a^{2} x^{3} \sqrt{a + b x^{3}}}{105 b} + \frac{16 B a x^{6} \sqrt{a + b x^{3}}}{105} + \frac{2 B b x^{9} \sqrt{a + b x^{3}}}{21} & \text{for}\: b \neq 0 \\a^{\frac{3}{2}} \left (\frac{A x^{3}}{3} + \frac{B x^{6}}{6}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*A*a**2*sqrt(a + b*x**3)/(15*b) + 4*A*a*x**3*sqrt(a + b*x**3)/15 + 2
*A*b*x**6*sqrt(a + b*x**3)/15 - 4*B*a**3*sqrt(a + b*x**3)/(105*b**2) + 2*B*a**2*
x**3*sqrt(a + b*x**3)/(105*b) + 16*B*a*x**6*sqrt(a + b*x**3)/105 + 2*B*b*x**9*sq
rt(a + b*x**3)/21, Ne(b, 0)), (a**(3/2)*(A*x**3/3 + B*x**6/6), True))

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GIAC/XCAS [A]  time = 0.215665, size = 162, normalized size = 3.52 \[ \frac{2 \,{\left (35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A a + 7 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a\right )} A + \frac{7 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a\right )} B a}{b} + \frac{{\left (15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}\right )} B}{b}\right )}}{315 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(b*x^3 + a)^(3/2)*A*a + 7*(3*(b*x^3 + a)^(5/2) - 5*(b*x^3 + a)^(3/2)*a
)*A + 7*(3*(b*x^3 + a)^(5/2) - 5*(b*x^3 + a)^(3/2)*a)*B*a/b + (15*(b*x^3 + a)^(7
/2) - 42*(b*x^3 + a)^(5/2)*a + 35*(b*x^3 + a)^(3/2)*a^2)*B/b)/b