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

Optimal. Leaf size=85 \[ \frac {2}{5} a^3 A x^{5/2}+\frac {2}{11} a^2 x^{11/2} (a B+3 A b)+\frac {2}{23} b^2 x^{23/2} (3 a B+A b)+\frac {6}{17} a b x^{17/2} (a B+A b)+\frac {2}{29} b^3 B x^{29/2} \]

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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}{11} a^2 x^{11/2} (a B+3 A b)+\frac {2}{5} a^3 A x^{5/2}+\frac {2}{23} b^2 x^{23/2} (3 a B+A b)+\frac {6}{17} a b x^{17/2} (a B+A b)+\frac {2}{29} b^3 B x^{29/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*b*(A*b + a*B)*x^(17/2))/17 + (2*b^2*(A*b + 3*a*
B)*x^(23/2))/23 + (2*b^3*B*x^(29/2))/29

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^{3/2} \left (a+b x^3\right )^3 \left (A+B x^3\right ) \, dx &=\int \left (a^3 A x^{3/2}+a^2 (3 A b+a B) x^{9/2}+3 a b (A b+a B) x^{15/2}+b^2 (A b+3 a B) x^{21/2}+b^3 B x^{27/2}\right ) \, dx\\ &=\frac {2}{5} a^3 A x^{5/2}+\frac {2}{11} a^2 (3 A b+a B) x^{11/2}+\frac {6}{17} a b (A b+a B) x^{17/2}+\frac {2}{23} b^2 (A b+3 a B) x^{23/2}+\frac {2}{29} b^3 B x^{29/2}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 85, normalized size = 1.00 \begin {gather*} \frac {2}{5} a^3 A x^{5/2}+\frac {2}{11} a^2 x^{11/2} (a B+3 A b)+\frac {2}{23} b^2 x^{23/2} (3 a B+A b)+\frac {6}{17} a b x^{17/2} (a B+A b)+\frac {2}{29} b^3 B x^{29/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*b*(A*b + a*B)*x^(17/2))/17 + (2*b^2*(A*b + 3*a*
B)*x^(23/2))/23 + (2*b^3*B*x^(29/2))/29

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IntegrateAlgebraic [A]  time = 0.04, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (124729 a^3 A x^{5/2}+56695 a^3 B x^{11/2}+170085 a^2 A b x^{11/2}+110055 a^2 b B x^{17/2}+110055 a A b^2 x^{17/2}+81345 a b^2 B x^{23/2}+27115 A b^3 x^{23/2}+21505 b^3 B x^{29/2}\right )}{623645} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(124729*a^3*A*x^(5/2) + 170085*a^2*A*b*x^(11/2) + 56695*a^3*B*x^(11/2) + 110055*a*A*b^2*x^(17/2) + 110055*a
^2*b*B*x^(17/2) + 27115*A*b^3*x^(23/2) + 81345*a*b^2*B*x^(23/2) + 21505*b^3*B*x^(29/2)))/623645

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fricas [A]  time = 0.83, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{623645} \, {\left (21505 \, B b^{3} x^{14} + 27115 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{11} + 110055 \, {\left (B a^{2} b + A a b^{2}\right )} x^{8} + 124729 \, A a^{3} x^{2} + 56695 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/623645*(21505*B*b^3*x^14 + 27115*(3*B*a*b^2 + A*b^3)*x^11 + 110055*(B*a^2*b + A*a*b^2)*x^8 + 124729*A*a^3*x^
2 + 56695*(B*a^3 + 3*A*a^2*b)*x^5)*sqrt(x)

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giac [A]  time = 0.17, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{29} \, B b^{3} x^{\frac {29}{2}} + \frac {6}{23} \, B a b^{2} x^{\frac {23}{2}} + \frac {2}{23} \, A b^{3} x^{\frac {23}{2}} + \frac {6}{17} \, B a^{2} b x^{\frac {17}{2}} + \frac {6}{17} \, A a b^{2} x^{\frac {17}{2}} + \frac {2}{11} \, B a^{3} x^{\frac {11}{2}} + \frac {6}{11} \, A a^{2} b x^{\frac {11}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/29*B*b^3*x^(29/2) + 6/23*B*a*b^2*x^(23/2) + 2/23*A*b^3*x^(23/2) + 6/17*B*a^2*b*x^(17/2) + 6/17*A*a*b^2*x^(17
/2) + 2/11*B*a^3*x^(11/2) + 6/11*A*a^2*b*x^(11/2) + 2/5*A*a^3*x^(5/2)

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maple [A]  time = 0.05, size = 80, normalized size = 0.94 \begin {gather*} \frac {2 \left (21505 B \,b^{3} x^{12}+27115 x^{9} A \,b^{3}+81345 x^{9} B a \,b^{2}+110055 x^{6} A a \,b^{2}+110055 x^{6} B \,a^{2} b +170085 x^{3} A \,a^{2} b +56695 B \,a^{3} x^{3}+124729 A \,a^{3}\right ) x^{\frac {5}{2}}}{623645} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/623645*x^(5/2)*(21505*B*b^3*x^12+27115*A*b^3*x^9+81345*B*a*b^2*x^9+110055*A*a*b^2*x^6+110055*B*a^2*b*x^6+170
085*A*a^2*b*x^3+56695*B*a^3*x^3+124729*A*a^3)

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maxima [A]  time = 0.48, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{29} \, B b^{3} x^{\frac {29}{2}} + \frac {2}{23} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {23}{2}} + \frac {6}{17} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {17}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} + \frac {2}{11} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {11}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/29*B*b^3*x^(29/2) + 2/23*(3*B*a*b^2 + A*b^3)*x^(23/2) + 6/17*(B*a^2*b + A*a*b^2)*x^(17/2) + 2/5*A*a^3*x^(5/2
) + 2/11*(B*a^3 + 3*A*a^2*b)*x^(11/2)

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mupad [B]  time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{11/2}\,\left (\frac {2\,B\,a^3}{11}+\frac {6\,A\,b\,a^2}{11}\right )+x^{23/2}\,\left (\frac {2\,A\,b^3}{23}+\frac {6\,B\,a\,b^2}{23}\right )+\frac {2\,A\,a^3\,x^{5/2}}{5}+\frac {2\,B\,b^3\,x^{29/2}}{29}+\frac {6\,a\,b\,x^{17/2}\,\left (A\,b+B\,a\right )}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(11/2)*((2*B*a^3)/11 + (6*A*a^2*b)/11) + x^(23/2)*((2*A*b^3)/23 + (6*B*a*b^2)/23) + (2*A*a^3*x^(5/2))/5 + (2
*B*b^3*x^(29/2))/29 + (6*a*b*x^(17/2)*(A*b + B*a))/17

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sympy [A]  time = 39.60, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A a^{3} x^{\frac {5}{2}}}{5} + \frac {6 A a^{2} b x^{\frac {11}{2}}}{11} + \frac {6 A a b^{2} x^{\frac {17}{2}}}{17} + \frac {2 A b^{3} x^{\frac {23}{2}}}{23} + \frac {2 B a^{3} x^{\frac {11}{2}}}{11} + \frac {6 B a^{2} b x^{\frac {17}{2}}}{17} + \frac {6 B a b^{2} x^{\frac {23}{2}}}{23} + \frac {2 B b^{3} x^{\frac {29}{2}}}{29} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**3*x**(5/2)/5 + 6*A*a**2*b*x**(11/2)/11 + 6*A*a*b**2*x**(17/2)/17 + 2*A*b**3*x**(23/2)/23 + 2*B*a**3*x**
(11/2)/11 + 6*B*a**2*b*x**(17/2)/17 + 6*B*a*b**2*x**(23/2)/23 + 2*B*b**3*x**(29/2)/29

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