3.150 \(\int \sqrt{x} (a+b x^3)^3 (A+B x^3) \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{3} a^3 A x^{3/2}+\frac{2}{21} b^2 x^{21/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{27} b^3 B x^{27/2} \]

[Out]

(2*a^3*A*x^(3/2))/3 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (2*a*b*(A*b + a*B)*x^(15/2))/5 + (2*b^2*(A*b + 3*a*B)*
x^(21/2))/21 + (2*b^3*B*x^(27/2))/27

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Rubi [A]  time = 0.0405446, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{3} a^3 A x^{3/2}+\frac{2}{21} b^2 x^{21/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{27} b^3 B x^{27/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x^3)^3*(A + B*x^3),x]

[Out]

(2*a^3*A*x^(3/2))/3 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (2*a*b*(A*b + a*B)*x^(15/2))/5 + (2*b^2*(A*b + 3*a*B)*
x^(21/2))/21 + (2*b^3*B*x^(27/2))/27

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \sqrt{x} \left (a+b x^3\right )^3 \left (A+B x^3\right ) \, dx &=\int \left (a^3 A \sqrt{x}+a^2 (3 A b+a B) x^{7/2}+3 a b (A b+a B) x^{13/2}+b^2 (A b+3 a B) x^{19/2}+b^3 B x^{25/2}\right ) \, dx\\ &=\frac{2}{3} a^3 A x^{3/2}+\frac{2}{9} a^2 (3 A b+a B) x^{9/2}+\frac{2}{5} a b (A b+a B) x^{15/2}+\frac{2}{21} b^2 (A b+3 a B) x^{21/2}+\frac{2}{27} b^3 B x^{27/2}\\ \end{align*}

Mathematica [A]  time = 0.0382633, size = 71, normalized size = 0.84 \[ \frac{2}{945} x^{3/2} \left (105 a^2 x^3 (a B+3 A b)+315 a^3 A+45 b^2 x^9 (3 a B+A b)+189 a b x^6 (a B+A b)+35 b^3 B x^{12}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x^3)^3*(A + B*x^3),x]

[Out]

(2*x^(3/2)*(315*a^3*A + 105*a^2*(3*A*b + a*B)*x^3 + 189*a*b*(A*b + a*B)*x^6 + 45*b^2*(A*b + 3*a*B)*x^9 + 35*b^
3*B*x^12))/945

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Maple [A]  time = 0.007, size = 80, normalized size = 0.9 \begin{align*}{\frac{70\,{b}^{3}B{x}^{12}+90\,{x}^{9}{b}^{3}A+270\,{x}^{9}a{b}^{2}B+378\,{x}^{6}a{b}^{2}A+378\,{x}^{6}{a}^{2}bB+630\,{x}^{3}A{a}^{2}b+210\,{x}^{3}B{a}^{3}+630\,{a}^{3}A}{945}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3*(B*x^3+A)*x^(1/2),x)

[Out]

2/945*x^(3/2)*(35*B*b^3*x^12+45*A*b^3*x^9+135*B*a*b^2*x^9+189*A*a*b^2*x^6+189*B*a^2*b*x^6+315*A*a^2*b*x^3+105*
B*a^3*x^3+315*A*a^3)

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Maxima [A]  time = 0.991515, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{27} \, B b^{3} x^{\frac{27}{2}} + \frac{2}{21} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{21}{2}} + \frac{2}{5} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{15}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} + \frac{2}{9} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3*(B*x^3+A)*x^(1/2),x, algorithm="maxima")

[Out]

2/27*B*b^3*x^(27/2) + 2/21*(3*B*a*b^2 + A*b^3)*x^(21/2) + 2/5*(B*a^2*b + A*a*b^2)*x^(15/2) + 2/3*A*a^3*x^(3/2)
+ 2/9*(B*a^3 + 3*A*a^2*b)*x^(9/2)

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Fricas [A]  time = 1.72516, size = 182, normalized size = 2.14 \begin{align*} \frac{2}{945} \,{\left (35 \, B b^{3} x^{13} + 45 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{10} + 189 \,{\left (B a^{2} b + A a b^{2}\right )} x^{7} + 315 \, A a^{3} x + 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3*(B*x^3+A)*x^(1/2),x, algorithm="fricas")

[Out]

2/945*(35*B*b^3*x^13 + 45*(3*B*a*b^2 + A*b^3)*x^10 + 189*(B*a^2*b + A*a*b^2)*x^7 + 315*A*a^3*x + 105*(B*a^3 +
3*A*a^2*b)*x^4)*sqrt(x)

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Sympy [A]  time = 7.63712, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{2 A a^{2} b x^{\frac{9}{2}}}{3} + \frac{2 A a b^{2} x^{\frac{15}{2}}}{5} + \frac{2 A b^{3} x^{\frac{21}{2}}}{21} + \frac{2 B a^{3} x^{\frac{9}{2}}}{9} + \frac{2 B a^{2} b x^{\frac{15}{2}}}{5} + \frac{2 B a b^{2} x^{\frac{21}{2}}}{7} + \frac{2 B b^{3} x^{\frac{27}{2}}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**3*(B*x**3+A)*x**(1/2),x)

[Out]

2*A*a**3*x**(3/2)/3 + 2*A*a**2*b*x**(9/2)/3 + 2*A*a*b**2*x**(15/2)/5 + 2*A*b**3*x**(21/2)/21 + 2*B*a**3*x**(9/
2)/9 + 2*B*a**2*b*x**(15/2)/5 + 2*B*a*b**2*x**(21/2)/7 + 2*B*b**3*x**(27/2)/27

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Giac [A]  time = 1.12638, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{27} \, B b^{3} x^{\frac{27}{2}} + \frac{2}{7} \, B a b^{2} x^{\frac{21}{2}} + \frac{2}{21} \, A b^{3} x^{\frac{21}{2}} + \frac{2}{5} \, B a^{2} b x^{\frac{15}{2}} + \frac{2}{5} \, A a b^{2} x^{\frac{15}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} b x^{\frac{9}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3*(B*x^3+A)*x^(1/2),x, algorithm="giac")

[Out]

2/27*B*b^3*x^(27/2) + 2/7*B*a*b^2*x^(21/2) + 2/21*A*b^3*x^(21/2) + 2/5*B*a^2*b*x^(15/2) + 2/5*A*a*b^2*x^(15/2)
+ 2/9*B*a^3*x^(9/2) + 2/3*A*a^2*b*x^(9/2) + 2/3*A*a^3*x^(3/2)