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

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

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Rubi [A]  time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.07, size = 71, normalized size = 0.84 \begin {gather*} \frac {2}{945} x^{3/2} \left (315 a^3 A+105 a^2 x^3 (a B+3 A b)+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 ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.05, size = 97, normalized size = 1.14 \begin {gather*} \frac {2}{945} \left (315 a^3 A x^{3/2}+105 a^3 B x^{9/2}+315 a^2 A b x^{9/2}+189 a^2 b B x^{15/2}+189 a A b^2 x^{15/2}+135 a b^2 B x^{21/2}+45 A b^3 x^{21/2}+35 b^3 B x^{27/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(315*a^3*A*x^(3/2) + 315*a^2*A*b*x^(9/2) + 105*a^3*B*x^(9/2) + 189*a*A*b^2*x^(15/2) + 189*a^2*b*B*x^(15/2)
+ 45*A*b^3*x^(21/2) + 135*a*b^2*B*x^(21/2) + 35*b^3*B*x^(27/2)))/945

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fricas [A]  time = 0.80, size = 76, normalized size = 0.89 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.16, size = 77, normalized size = 0.91 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 80, normalized size = 0.94 \begin {gather*} \frac {2 \left (35 B \,b^{3} x^{12}+45 x^{9} A \,b^{3}+135 x^{9} B a \,b^{2}+189 x^{6} A a \,b^{2}+189 x^{6} B \,a^{2} b +315 x^{3} A \,a^{2} b +105 B \,a^{3} x^{3}+315 A \,a^{3}\right ) x^{\frac {3}{2}}}{945} \end {gather*}

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 = 0.67, size = 73, normalized size = 0.86 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{9/2}\,\left (\frac {2\,B\,a^3}{9}+\frac {2\,A\,b\,a^2}{3}\right )+x^{21/2}\,\left (\frac {2\,A\,b^3}{21}+\frac {2\,B\,a\,b^2}{7}\right )+\frac {2\,A\,a^3\,x^{3/2}}{3}+\frac {2\,B\,b^3\,x^{27/2}}{27}+\frac {2\,a\,b\,x^{15/2}\,\left (A\,b+B\,a\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 6.34, size = 114, normalized size = 1.34 \begin {gather*} \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 {gather*}

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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