3.2.43 \(\int x^{7/2} (a+b x^3)^3 (A+B x^3) \, dx\)

Optimal. Leaf size=85 \[ \frac {2}{9} a^3 A x^{9/2}+\frac {2}{15} a^2 x^{15/2} (a B+3 A b)+\frac {2}{27} b^2 x^{27/2} (3 a B+A b)+\frac {2}{7} a b x^{21/2} (a B+A b)+\frac {2}{33} b^3 B x^{33/2} \]

________________________________________________________________________________________

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}{15} a^2 x^{15/2} (a B+3 A b)+\frac {2}{9} a^3 A x^{9/2}+\frac {2}{27} b^2 x^{27/2} (3 a B+A b)+\frac {2}{7} a b x^{21/2} (a B+A b)+\frac {2}{33} b^3 B x^{33/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x^3)^3*(A + B*x^3),x]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(15/2))/15 + (2*a*b*(A*b + a*B)*x^(21/2))/7 + (2*b^2*(A*b + 3*a*B
)*x^(27/2))/27 + (2*b^3*B*x^(33/2))/33

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^{7/2} \left (a+b x^3\right )^3 \left (A+B x^3\right ) \, dx &=\int \left (a^3 A x^{7/2}+a^2 (3 A b+a B) x^{13/2}+3 a b (A b+a B) x^{19/2}+b^2 (A b+3 a B) x^{25/2}+b^3 B x^{31/2}\right ) \, dx\\ &=\frac {2}{9} a^3 A x^{9/2}+\frac {2}{15} a^2 (3 A b+a B) x^{15/2}+\frac {2}{7} a b (A b+a B) x^{21/2}+\frac {2}{27} b^2 (A b+3 a B) x^{27/2}+\frac {2}{33} b^3 B x^{33/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 71, normalized size = 0.84 \begin {gather*} \frac {2 x^{9/2} \left (1155 a^3 A+693 a^2 x^3 (a B+3 A b)+385 b^2 x^9 (3 a B+A b)+1485 a b x^6 (a B+A b)+315 b^3 B x^{12}\right )}{10395} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x^3)^3*(A + B*x^3),x]

[Out]

(2*x^(9/2)*(1155*a^3*A + 693*a^2*(3*A*b + a*B)*x^3 + 1485*a*b*(A*b + a*B)*x^6 + 385*b^2*(A*b + 3*a*B)*x^9 + 31
5*b^3*B*x^12))/10395

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.04, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (1155 a^3 A x^{9/2}+693 a^3 B x^{15/2}+2079 a^2 A b x^{15/2}+1485 a^2 b B x^{21/2}+1485 a A b^2 x^{21/2}+1155 a b^2 B x^{27/2}+385 A b^3 x^{27/2}+315 b^3 B x^{33/2}\right )}{10395} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(7/2)*(a + b*x^3)^3*(A + B*x^3),x]

[Out]

(2*(1155*a^3*A*x^(9/2) + 2079*a^2*A*b*x^(15/2) + 693*a^3*B*x^(15/2) + 1485*a*A*b^2*x^(21/2) + 1485*a^2*b*B*x^(
21/2) + 385*A*b^3*x^(27/2) + 1155*a*b^2*B*x^(27/2) + 315*b^3*B*x^(33/2)))/10395

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{10395} \, {\left (315 \, B b^{3} x^{16} + 385 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{13} + 1485 \, {\left (B a^{2} b + A a b^{2}\right )} x^{10} + 1155 \, A a^{3} x^{4} + 693 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{7}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^3+a)^3*(B*x^3+A),x, algorithm="fricas")

[Out]

2/10395*(315*B*b^3*x^16 + 385*(3*B*a*b^2 + A*b^3)*x^13 + 1485*(B*a^2*b + A*a*b^2)*x^10 + 1155*A*a^3*x^4 + 693*
(B*a^3 + 3*A*a^2*b)*x^7)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{33} \, B b^{3} x^{\frac {33}{2}} + \frac {2}{9} \, B a b^{2} x^{\frac {27}{2}} + \frac {2}{27} \, A b^{3} x^{\frac {27}{2}} + \frac {2}{7} \, B a^{2} b x^{\frac {21}{2}} + \frac {2}{7} \, A a b^{2} x^{\frac {21}{2}} + \frac {2}{15} \, B a^{3} x^{\frac {15}{2}} + \frac {2}{5} \, A a^{2} b x^{\frac {15}{2}} + \frac {2}{9} \, A a^{3} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^3+a)^3*(B*x^3+A),x, algorithm="giac")

[Out]

2/33*B*b^3*x^(33/2) + 2/9*B*a*b^2*x^(27/2) + 2/27*A*b^3*x^(27/2) + 2/7*B*a^2*b*x^(21/2) + 2/7*A*a*b^2*x^(21/2)
+ 2/15*B*a^3*x^(15/2) + 2/5*A*a^2*b*x^(15/2) + 2/9*A*a^3*x^(9/2)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 80, normalized size = 0.94 \begin {gather*} \frac {2 \left (315 B \,b^{3} x^{12}+385 x^{9} A \,b^{3}+1155 x^{9} B a \,b^{2}+1485 x^{6} A a \,b^{2}+1485 x^{6} B \,a^{2} b +2079 x^{3} A \,a^{2} b +693 B \,a^{3} x^{3}+1155 A \,a^{3}\right ) x^{\frac {9}{2}}}{10395} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^3+a)^3*(B*x^3+A),x)

[Out]

2/10395*x^(9/2)*(315*B*b^3*x^12+385*A*b^3*x^9+1155*B*a*b^2*x^9+1485*A*a*b^2*x^6+1485*B*a^2*b*x^6+2079*A*a^2*b*
x^3+693*B*a^3*x^3+1155*A*a^3)

________________________________________________________________________________________

maxima [A]  time = 0.57, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{33} \, B b^{3} x^{\frac {33}{2}} + \frac {2}{27} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {27}{2}} + \frac {2}{7} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {21}{2}} + \frac {2}{9} \, A a^{3} x^{\frac {9}{2}} + \frac {2}{15} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {15}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^3+a)^3*(B*x^3+A),x, algorithm="maxima")

[Out]

2/33*B*b^3*x^(33/2) + 2/27*(3*B*a*b^2 + A*b^3)*x^(27/2) + 2/7*(B*a^2*b + A*a*b^2)*x^(21/2) + 2/9*A*a^3*x^(9/2)
+ 2/15*(B*a^3 + 3*A*a^2*b)*x^(15/2)

________________________________________________________________________________________

mupad [B]  time = 2.52, size = 69, normalized size = 0.81 \begin {gather*} x^{15/2}\,\left (\frac {2\,B\,a^3}{15}+\frac {2\,A\,b\,a^2}{5}\right )+x^{27/2}\,\left (\frac {2\,A\,b^3}{27}+\frac {2\,B\,a\,b^2}{9}\right )+\frac {2\,A\,a^3\,x^{9/2}}{9}+\frac {2\,B\,b^3\,x^{33/2}}{33}+\frac {2\,a\,b\,x^{21/2}\,\left (A\,b+B\,a\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(A + B*x^3)*(a + b*x^3)^3,x)

[Out]

x^(15/2)*((2*B*a^3)/15 + (2*A*a^2*b)/5) + x^(27/2)*((2*A*b^3)/27 + (2*B*a*b^2)/9) + (2*A*a^3*x^(9/2))/9 + (2*B
*b^3*x^(33/2))/33 + (2*a*b*x^(21/2)*(A*b + B*a))/7

________________________________________________________________________________________

sympy [A]  time = 92.99, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A a^{3} x^{\frac {9}{2}}}{9} + \frac {2 A a^{2} b x^{\frac {15}{2}}}{5} + \frac {2 A a b^{2} x^{\frac {21}{2}}}{7} + \frac {2 A b^{3} x^{\frac {27}{2}}}{27} + \frac {2 B a^{3} x^{\frac {15}{2}}}{15} + \frac {2 B a^{2} b x^{\frac {21}{2}}}{7} + \frac {2 B a b^{2} x^{\frac {27}{2}}}{9} + \frac {2 B b^{3} x^{\frac {33}{2}}}{33} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x**3+a)**3*(B*x**3+A),x)

[Out]

2*A*a**3*x**(9/2)/9 + 2*A*a**2*b*x**(15/2)/5 + 2*A*a*b**2*x**(21/2)/7 + 2*A*b**3*x**(27/2)/27 + 2*B*a**3*x**(1
5/2)/15 + 2*B*a**2*b*x**(21/2)/7 + 2*B*a*b**2*x**(27/2)/9 + 2*B*b**3*x**(33/2)/33

________________________________________________________________________________________