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

Optimal. Leaf size=83 \[ -\frac {2 a^3 A}{5 x^{5/2}}+2 a^2 \sqrt {x} (a B+3 A b)+\frac {2}{13} b^2 x^{13/2} (3 a B+A b)+\frac {6}{7} a b x^{7/2} (a B+A b)+\frac {2}{19} b^3 B x^{19/2} \]

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Rubi [A]  time = 0.04, antiderivative size = 83, 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*} 2 a^2 \sqrt {x} (a B+3 A b)-\frac {2 a^3 A}{5 x^{5/2}}+\frac {2}{13} b^2 x^{13/2} (3 a B+A b)+\frac {6}{7} a b x^{7/2} (a B+A b)+\frac {2}{19} b^3 B x^{19/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*A)/(5*x^(5/2)) + 2*a^2*(3*A*b + a*B)*Sqrt[x] + (6*a*b*(A*b + a*B)*x^(7/2))/7 + (2*b^2*(A*b + 3*a*B)*x^
(13/2))/13 + (2*b^3*B*x^(19/2))/19

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

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Mathematica [A]  time = 0.09, size = 78, normalized size = 0.94 \begin {gather*} \frac {-3458 a^3 \left (A-5 B x^3\right )+7410 a^2 b x^3 \left (7 A+B x^3\right )+570 a b^2 x^6 \left (13 A+7 B x^3\right )+70 b^3 x^9 \left (19 A+13 B x^3\right )}{8645 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3458*a^3*(A - 5*B*x^3) + 7410*a^2*b*x^3*(7*A + B*x^3) + 570*a*b^2*x^6*(13*A + 7*B*x^3) + 70*b^3*x^9*(19*A +
13*B*x^3))/(8645*x^(5/2))

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IntegrateAlgebraic [A]  time = 0.05, size = 83, normalized size = 1.00 \begin {gather*} \frac {2 \left (-1729 a^3 A+8645 a^3 B x^3+25935 a^2 A b x^3+3705 a^2 b B x^6+3705 a A b^2 x^6+1995 a b^2 B x^9+665 A b^3 x^9+455 b^3 B x^{12}\right )}{8645 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1729*a^3*A + 25935*a^2*A*b*x^3 + 8645*a^3*B*x^3 + 3705*a*A*b^2*x^6 + 3705*a^2*b*B*x^6 + 665*A*b^3*x^9 + 1
995*a*b^2*B*x^9 + 455*b^3*B*x^12))/(8645*x^(5/2))

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fricas [A]  time = 0.68, size = 75, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (455 \, B b^{3} x^{12} + 665 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 3705 \, {\left (B a^{2} b + A a b^{2}\right )} x^{6} - 1729 \, A a^{3} + 8645 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )}}{8645 \, x^{\frac {5}{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^(7/2),x, algorithm="fricas")

[Out]

2/8645*(455*B*b^3*x^12 + 665*(3*B*a*b^2 + A*b^3)*x^9 + 3705*(B*a^2*b + A*a*b^2)*x^6 - 1729*A*a^3 + 8645*(B*a^3
+ 3*A*a^2*b)*x^3)/x^(5/2)

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giac [A]  time = 0.16, size = 77, normalized size = 0.93 \begin {gather*} \frac {2}{19} \, B b^{3} x^{\frac {19}{2}} + \frac {6}{13} \, B a b^{2} x^{\frac {13}{2}} + \frac {2}{13} \, A b^{3} x^{\frac {13}{2}} + \frac {6}{7} \, B a^{2} b x^{\frac {7}{2}} + \frac {6}{7} \, A a b^{2} x^{\frac {7}{2}} + 2 \, B a^{3} \sqrt {x} + 6 \, A a^{2} b \sqrt {x} - \frac {2 \, A a^{3}}{5 \, x^{\frac {5}{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^(7/2),x, algorithm="giac")

[Out]

2/19*B*b^3*x^(19/2) + 6/13*B*a*b^2*x^(13/2) + 2/13*A*b^3*x^(13/2) + 6/7*B*a^2*b*x^(7/2) + 6/7*A*a*b^2*x^(7/2)
+ 2*B*a^3*sqrt(x) + 6*A*a^2*b*sqrt(x) - 2/5*A*a^3/x^(5/2)

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maple [A]  time = 0.04, size = 80, normalized size = 0.96 \begin {gather*} -\frac {2 \left (-455 B \,b^{3} x^{12}-665 x^{9} A \,b^{3}-1995 x^{9} B a \,b^{2}-3705 x^{6} A a \,b^{2}-3705 x^{6} B \,a^{2} b -25935 x^{3} A \,a^{2} b -8645 B \,a^{3} x^{3}+1729 A \,a^{3}\right )}{8645 x^{\frac {5}{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^(7/2),x)

[Out]

-2/8645*(-455*B*b^3*x^12-665*A*b^3*x^9-1995*B*a*b^2*x^9-3705*A*a*b^2*x^6-3705*B*a^2*b*x^6-25935*A*a^2*b*x^3-86
45*B*a^3*x^3+1729*A*a^3)/x^(5/2)

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maxima [A]  time = 0.55, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{19} \, B b^{3} x^{\frac {19}{2}} + \frac {2}{13} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {13}{2}} + \frac {6}{7} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {7}{2}} - \frac {2 \, A a^{3}}{5 \, x^{\frac {5}{2}}} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*B*b^3*x^(19/2) + 2/13*(3*B*a*b^2 + A*b^3)*x^(13/2) + 6/7*(B*a^2*b + A*a*b^2)*x^(7/2) - 2/5*A*a^3/x^(5/2)
+ 2*(B*a^3 + 3*A*a^2*b)*sqrt(x)

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mupad [B]  time = 0.03, size = 69, normalized size = 0.83 \begin {gather*} \sqrt {x}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+x^{13/2}\,\left (\frac {2\,A\,b^3}{13}+\frac {6\,B\,a\,b^2}{13}\right )-\frac {2\,A\,a^3}{5\,x^{5/2}}+\frac {2\,B\,b^3\,x^{19/2}}{19}+\frac {6\,a\,b\,x^{7/2}\,\left (A\,b+B\,a\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1/2)*(2*B*a^3 + 6*A*a^2*b) + x^(13/2)*((2*A*b^3)/13 + (6*B*a*b^2)/13) - (2*A*a^3)/(5*x^(5/2)) + (2*B*b^3*x^
(19/2))/19 + (6*a*b*x^(7/2)*(A*b + B*a))/7

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sympy [A]  time = 29.28, size = 110, normalized size = 1.33 \begin {gather*} - \frac {2 A a^{3}}{5 x^{\frac {5}{2}}} + 6 A a^{2} b \sqrt {x} + \frac {6 A a b^{2} x^{\frac {7}{2}}}{7} + \frac {2 A b^{3} x^{\frac {13}{2}}}{13} + 2 B a^{3} \sqrt {x} + \frac {6 B a^{2} b x^{\frac {7}{2}}}{7} + \frac {6 B a b^{2} x^{\frac {13}{2}}}{13} + \frac {2 B b^{3} x^{\frac {19}{2}}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**3/(5*x**(5/2)) + 6*A*a**2*b*sqrt(x) + 6*A*a*b**2*x**(7/2)/7 + 2*A*b**3*x**(13/2)/13 + 2*B*a**3*sqrt(x)
+ 6*B*a**2*b*x**(7/2)/7 + 6*B*a*b**2*x**(13/2)/13 + 2*B*b**3*x**(19/2)/19

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