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

Optimal. Leaf size=85 \[ \frac {2}{7} a^3 A x^{7/2}+\frac {2}{13} a^2 x^{13/2} (a B+3 A b)+\frac {2}{25} b^2 x^{25/2} (3 a B+A b)+\frac {6}{19} a b x^{19/2} (a B+A b)+\frac {2}{31} b^3 B x^{31/2} \]

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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}{13} a^2 x^{13/2} (a B+3 A b)+\frac {2}{7} a^3 A x^{7/2}+\frac {2}{25} b^2 x^{25/2} (3 a B+A b)+\frac {6}{19} a b x^{19/2} (a B+A b)+\frac {2}{31} b^3 B x^{31/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(19/2))/19 + (2*b^2*(A*b + 3*a*
B)*x^(25/2))/25 + (2*b^3*B*x^(31/2))/31

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 )^3 \left (A+B x^3\right ) \, dx &=\int \left (a^3 A x^{5/2}+a^2 (3 A b+a B) x^{11/2}+3 a b (A b+a B) x^{17/2}+b^2 (A b+3 a B) x^{23/2}+b^3 B x^{29/2}\right ) \, dx\\ &=\frac {2}{7} a^3 A x^{7/2}+\frac {2}{13} a^2 (3 A b+a B) x^{13/2}+\frac {6}{19} a b (A b+a B) x^{19/2}+\frac {2}{25} b^2 (A b+3 a B) x^{25/2}+\frac {2}{31} b^3 B x^{31/2}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 85, normalized size = 1.00 \begin {gather*} \frac {2}{7} a^3 A x^{7/2}+\frac {2}{13} a^2 x^{13/2} (a B+3 A b)+\frac {2}{25} b^2 x^{25/2} (3 a B+A b)+\frac {6}{19} a b x^{19/2} (a B+A b)+\frac {2}{31} b^3 B x^{31/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(19/2))/19 + (2*b^2*(A*b + 3*a*
B)*x^(25/2))/25 + (2*b^3*B*x^(31/2))/31

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IntegrateAlgebraic [A]  time = 0.04, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (191425 a^3 A x^{7/2}+103075 a^3 B x^{13/2}+309225 a^2 A b x^{13/2}+211575 a^2 b B x^{19/2}+211575 a A b^2 x^{19/2}+160797 a b^2 B x^{25/2}+53599 A b^3 x^{25/2}+43225 b^3 B x^{31/2}\right )}{1339975} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(191425*a^3*A*x^(7/2) + 309225*a^2*A*b*x^(13/2) + 103075*a^3*B*x^(13/2) + 211575*a*A*b^2*x^(19/2) + 211575*
a^2*b*B*x^(19/2) + 53599*A*b^3*x^(25/2) + 160797*a*b^2*B*x^(25/2) + 43225*b^3*B*x^(31/2)))/1339975

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fricas [A]  time = 0.82, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{1339975} \, {\left (43225 \, B b^{3} x^{15} + 53599 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{12} + 211575 \, {\left (B a^{2} b + A a b^{2}\right )} x^{9} + 191425 \, A a^{3} x^{3} + 103075 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1339975*(43225*B*b^3*x^15 + 53599*(3*B*a*b^2 + A*b^3)*x^12 + 211575*(B*a^2*b + A*a*b^2)*x^9 + 191425*A*a^3*x
^3 + 103075*(B*a^3 + 3*A*a^2*b)*x^6)*sqrt(x)

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giac [A]  time = 0.15, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{31} \, B b^{3} x^{\frac {31}{2}} + \frac {6}{25} \, B a b^{2} x^{\frac {25}{2}} + \frac {2}{25} \, A b^{3} x^{\frac {25}{2}} + \frac {6}{19} \, B a^{2} b x^{\frac {19}{2}} + \frac {6}{19} \, A a b^{2} x^{\frac {19}{2}} + \frac {2}{13} \, B a^{3} x^{\frac {13}{2}} + \frac {6}{13} \, A a^{2} b x^{\frac {13}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/31*B*b^3*x^(31/2) + 6/25*B*a*b^2*x^(25/2) + 2/25*A*b^3*x^(25/2) + 6/19*B*a^2*b*x^(19/2) + 6/19*A*a*b^2*x^(19
/2) + 2/13*B*a^3*x^(13/2) + 6/13*A*a^2*b*x^(13/2) + 2/7*A*a^3*x^(7/2)

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maple [A]  time = 0.04, size = 80, normalized size = 0.94 \begin {gather*} \frac {2 \left (43225 B \,b^{3} x^{12}+53599 x^{9} A \,b^{3}+160797 x^{9} B a \,b^{2}+211575 x^{6} A a \,b^{2}+211575 x^{6} B \,a^{2} b +309225 x^{3} A \,a^{2} b +103075 B \,a^{3} x^{3}+191425 A \,a^{3}\right ) x^{\frac {7}{2}}}{1339975} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1339975*x^(7/2)*(43225*B*b^3*x^12+53599*A*b^3*x^9+160797*B*a*b^2*x^9+211575*A*a*b^2*x^6+211575*B*a^2*b*x^6+3
09225*A*a^2*b*x^3+103075*B*a^3*x^3+191425*A*a^3)

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maxima [A]  time = 0.50, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{31} \, B b^{3} x^{\frac {31}{2}} + \frac {2}{25} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {25}{2}} + \frac {6}{19} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {19}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} + \frac {2}{13} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {13}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/31*B*b^3*x^(31/2) + 2/25*(3*B*a*b^2 + A*b^3)*x^(25/2) + 6/19*(B*a^2*b + A*a*b^2)*x^(19/2) + 2/7*A*a^3*x^(7/2
) + 2/13*(B*a^3 + 3*A*a^2*b)*x^(13/2)

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mupad [B]  time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{13/2}\,\left (\frac {2\,B\,a^3}{13}+\frac {6\,A\,b\,a^2}{13}\right )+x^{25/2}\,\left (\frac {2\,A\,b^3}{25}+\frac {6\,B\,a\,b^2}{25}\right )+\frac {2\,A\,a^3\,x^{7/2}}{7}+\frac {2\,B\,b^3\,x^{31/2}}{31}+\frac {6\,a\,b\,x^{19/2}\,\left (A\,b+B\,a\right )}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(13/2)*((2*B*a^3)/13 + (6*A*a^2*b)/13) + x^(25/2)*((2*A*b^3)/25 + (6*B*a*b^2)/25) + (2*A*a^3*x^(7/2))/7 + (2
*B*b^3*x^(31/2))/31 + (6*a*b*x^(19/2)*(A*b + B*a))/19

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sympy [A]  time = 58.70, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A a^{3} x^{\frac {7}{2}}}{7} + \frac {6 A a^{2} b x^{\frac {13}{2}}}{13} + \frac {6 A a b^{2} x^{\frac {19}{2}}}{19} + \frac {2 A b^{3} x^{\frac {25}{2}}}{25} + \frac {2 B a^{3} x^{\frac {13}{2}}}{13} + \frac {6 B a^{2} b x^{\frac {19}{2}}}{19} + \frac {6 B a b^{2} x^{\frac {25}{2}}}{25} + \frac {2 B b^{3} x^{\frac {31}{2}}}{31} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**3*x**(7/2)/7 + 6*A*a**2*b*x**(13/2)/13 + 6*A*a*b**2*x**(19/2)/19 + 2*A*b**3*x**(25/2)/25 + 2*B*a**3*x**
(13/2)/13 + 6*B*a**2*b*x**(19/2)/19 + 6*B*a*b**2*x**(25/2)/25 + 2*B*b**3*x**(31/2)/31

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