3.140 \(\int x^{5/2} (a+b x^3)^2 (A+B x^3) \, dx\)

Optimal. Leaf size=63 \[ \frac {2}{7} a^2 A x^{7/2}+\frac {2}{19} b x^{19/2} (2 a B+A b)+\frac {2}{13} a x^{13/2} (a B+2 A b)+\frac {2}{25} b^2 B x^{25/2} \]

[Out]

2/7*a^2*A*x^(7/2)+2/13*a*(2*A*b+B*a)*x^(13/2)+2/19*b*(A*b+2*B*a)*x^(19/2)+2/25*b^2*B*x^(25/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 63, 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} \[ \frac {2}{7} a^2 A x^{7/2}+\frac {2}{19} b x^{19/2} (2 a B+A b)+\frac {2}{13} a x^{13/2} (a B+2 A b)+\frac {2}{25} b^2 B x^{25/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x^3)^2*(A + B*x^3),x]

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(13/2))/13 + (2*b*(A*b + 2*a*B)*x^(19/2))/19 + (2*b^2*B*x^(25/2))/2
5

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 x^{5/2} \left (a+b x^3\right )^2 \left (A+B x^3\right ) \, dx &=\int \left (a^2 A x^{5/2}+a (2 A b+a B) x^{11/2}+b (A b+2 a B) x^{17/2}+b^2 B x^{23/2}\right ) \, dx\\ &=\frac {2}{7} a^2 A x^{7/2}+\frac {2}{13} a (2 A b+a B) x^{13/2}+\frac {2}{19} b (A b+2 a B) x^{19/2}+\frac {2}{25} b^2 B x^{25/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 63, normalized size = 1.00 \[ \frac {2}{7} a^2 A x^{7/2}+\frac {2}{19} b x^{19/2} (2 a B+A b)+\frac {2}{13} a x^{13/2} (a B+2 A b)+\frac {2}{25} b^2 B x^{25/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x^3)^2*(A + B*x^3),x]

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(13/2))/13 + (2*b*(A*b + 2*a*B)*x^(19/2))/19 + (2*b^2*B*x^(25/2))/2
5

________________________________________________________________________________________

fricas [A]  time = 0.86, size = 56, normalized size = 0.89 \[ \frac {2}{43225} \, {\left (1729 \, B b^{2} x^{12} + 2275 \, {\left (2 \, B a b + A b^{2}\right )} x^{9} + 3325 \, {\left (B a^{2} + 2 \, A a b\right )} x^{6} + 6175 \, A a^{2} x^{3}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^3+a)^2*(B*x^3+A),x, algorithm="fricas")

[Out]

2/43225*(1729*B*b^2*x^12 + 2275*(2*B*a*b + A*b^2)*x^9 + 3325*(B*a^2 + 2*A*a*b)*x^6 + 6175*A*a^2*x^3)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 53, normalized size = 0.84 \[ \frac {2}{25} \, B b^{2} x^{\frac {25}{2}} + \frac {4}{19} \, B a b x^{\frac {19}{2}} + \frac {2}{19} \, A b^{2} x^{\frac {19}{2}} + \frac {2}{13} \, B a^{2} x^{\frac {13}{2}} + \frac {4}{13} \, A a b x^{\frac {13}{2}} + \frac {2}{7} \, A a^{2} x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^3+a)^2*(B*x^3+A),x, algorithm="giac")

[Out]

2/25*B*b^2*x^(25/2) + 4/19*B*a*b*x^(19/2) + 2/19*A*b^2*x^(19/2) + 2/13*B*a^2*x^(13/2) + 4/13*A*a*b*x^(13/2) +
2/7*A*a^2*x^(7/2)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 56, normalized size = 0.89 \[ \frac {2 \left (1729 b^{2} B \,x^{9}+2275 A \,b^{2} x^{6}+4550 B a b \,x^{6}+6650 A a b \,x^{3}+3325 B \,a^{2} x^{3}+6175 a^{2} A \right ) x^{\frac {7}{2}}}{43225} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x^3+a)^2*(B*x^3+A),x)

[Out]

2/43225*x^(7/2)*(1729*B*b^2*x^9+2275*A*b^2*x^6+4550*B*a*b*x^6+6650*A*a*b*x^3+3325*B*a^2*x^3+6175*A*a^2)

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 51, normalized size = 0.81 \[ \frac {2}{25} \, B b^{2} x^{\frac {25}{2}} + \frac {2}{19} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {19}{2}} + \frac {2}{13} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {13}{2}} + \frac {2}{7} \, A a^{2} x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^3+a)^2*(B*x^3+A),x, algorithm="maxima")

[Out]

2/25*B*b^2*x^(25/2) + 2/19*(2*B*a*b + A*b^2)*x^(19/2) + 2/13*(B*a^2 + 2*A*a*b)*x^(13/2) + 2/7*A*a^2*x^(7/2)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 51, normalized size = 0.81 \[ x^{13/2}\,\left (\frac {2\,B\,a^2}{13}+\frac {4\,A\,b\,a}{13}\right )+x^{19/2}\,\left (\frac {2\,A\,b^2}{19}+\frac {4\,B\,a\,b}{19}\right )+\frac {2\,A\,a^2\,x^{7/2}}{7}+\frac {2\,B\,b^2\,x^{25/2}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(A + B*x^3)*(a + b*x^3)^2,x)

[Out]

x^(13/2)*((2*B*a^2)/13 + (4*A*a*b)/13) + x^(19/2)*((2*A*b^2)/19 + (4*B*a*b)/19) + (2*A*a^2*x^(7/2))/7 + (2*B*b
^2*x^(25/2))/25

________________________________________________________________________________________

sympy [A]  time = 29.39, size = 80, normalized size = 1.27 \[ \frac {2 A a^{2} x^{\frac {7}{2}}}{7} + \frac {4 A a b x^{\frac {13}{2}}}{13} + \frac {2 A b^{2} x^{\frac {19}{2}}}{19} + \frac {2 B a^{2} x^{\frac {13}{2}}}{13} + \frac {4 B a b x^{\frac {19}{2}}}{19} + \frac {2 B b^{2} x^{\frac {25}{2}}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(b*x**3+a)**2*(B*x**3+A),x)

[Out]

2*A*a**2*x**(7/2)/7 + 4*A*a*b*x**(13/2)/13 + 2*A*b**2*x**(19/2)/19 + 2*B*a**2*x**(13/2)/13 + 4*B*a*b*x**(19/2)
/19 + 2*B*b**2*x**(25/2)/25

________________________________________________________________________________________