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

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \frac {\left (a+b x^3\right )^3 \left (A+B x^3\right )}{x^{5/2}} \, dx &=\int \left (\frac {a^3 A}{x^{5/2}}+a^2 (3 A b+a B) \sqrt {x}+3 a b (A b+a B) x^{7/2}+b^2 (A b+3 a B) x^{13/2}+b^3 B x^{19/2}\right ) \, dx\\ &=-\frac {2 a^3 A}{3 x^{3/2}}+\frac {2}{3} a^2 (3 A b+a B) x^{3/2}+\frac {2}{3} a b (A b+a B) x^{9/2}+\frac {2}{15} b^2 (A b+3 a B) x^{15/2}+\frac {2}{21} b^3 B x^{21/2}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 77, normalized size = 0.91 \begin {gather*} \frac {2 \left (-35 a^3 \left (A-B x^3\right )+35 a^2 b x^3 \left (3 A+B x^3\right )+7 a b^2 x^6 \left (5 A+3 B x^3\right )+b^3 x^9 \left (7 A+5 B x^3\right )\right )}{105 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-35*a^3*(A - B*x^3) + 35*a^2*b*x^3*(3*A + B*x^3) + 7*a*b^2*x^6*(5*A + 3*B*x^3) + b^3*x^9*(7*A + 5*B*x^3)))
/(105*x^(3/2))

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IntegrateAlgebraic [A]  time = 0.04, size = 83, normalized size = 0.98 \begin {gather*} \frac {2 \left (-35 a^3 A+35 a^3 B x^3+105 a^2 A b x^3+35 a^2 b B x^6+35 a A b^2 x^6+21 a b^2 B x^9+7 A b^3 x^9+5 b^3 B x^{12}\right )}{105 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-35*a^3*A + 105*a^2*A*b*x^3 + 35*a^3*B*x^3 + 35*a*A*b^2*x^6 + 35*a^2*b*B*x^6 + 7*A*b^3*x^9 + 21*a*b^2*B*x^
9 + 5*b^3*B*x^12))/(105*x^(3/2))

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fricas [A]  time = 0.79, size = 75, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (5 \, B b^{3} x^{12} + 7 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 35 \, {\left (B a^{2} b + A a b^{2}\right )} x^{6} - 35 \, A a^{3} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )}}{105 \, 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^(5/2),x, algorithm="fricas")

[Out]

2/105*(5*B*b^3*x^12 + 7*(3*B*a*b^2 + A*b^3)*x^9 + 35*(B*a^2*b + A*a*b^2)*x^6 - 35*A*a^3 + 35*(B*a^3 + 3*A*a^2*
b)*x^3)/x^(3/2)

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giac [A]  time = 0.16, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{21} \, B b^{3} x^{\frac {21}{2}} + \frac {2}{5} \, B a b^{2} x^{\frac {15}{2}} + \frac {2}{15} \, A b^{3} x^{\frac {15}{2}} + \frac {2}{3} \, B a^{2} b x^{\frac {9}{2}} + \frac {2}{3} \, A a b^{2} x^{\frac {9}{2}} + \frac {2}{3} \, B a^{3} x^{\frac {3}{2}} + 2 \, A a^{2} b x^{\frac {3}{2}} - \frac {2 \, A a^{3}}{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^(5/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 80, normalized size = 0.94 \begin {gather*} -\frac {2 \left (-5 B \,b^{3} x^{12}-7 x^{9} A \,b^{3}-21 x^{9} B a \,b^{2}-35 x^{6} A a \,b^{2}-35 x^{6} B \,a^{2} b -105 x^{3} A \,a^{2} b -35 B \,a^{3} x^{3}+35 A \,a^{3}\right )}{105 x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(-5*B*b^3*x^12-7*A*b^3*x^9-21*B*a*b^2*x^9-35*A*a*b^2*x^6-35*B*a^2*b*x^6-105*A*a^2*b*x^3-35*B*a^3*x^3+35
*A*a^3)/x^(3/2)

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maxima [A]  time = 0.64, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{21} \, B b^{3} x^{\frac {21}{2}} + \frac {2}{15} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {15}{2}} + \frac {2}{3} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {9}{2}} - \frac {2 \, A a^{3}}{3 \, x^{\frac {3}{2}}} + \frac {2}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} 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^(5/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 28.37, size = 112, normalized size = 1.32 \begin {gather*} - \frac {2 A a^{3}}{3 x^{\frac {3}{2}}} + 2 A a^{2} b x^{\frac {3}{2}} + \frac {2 A a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 A b^{3} x^{\frac {15}{2}}}{15} + \frac {2 B a^{3} x^{\frac {3}{2}}}{3} + \frac {2 B a^{2} b x^{\frac {9}{2}}}{3} + \frac {2 B a b^{2} x^{\frac {15}{2}}}{5} + \frac {2 B b^{3} x^{\frac {21}{2}}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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