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

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

[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

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Rubi [A]  time = 0.040558, antiderivative size = 83, normalized size of antiderivative = 1., 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} \[ 2 a^2 \sqrt{x} (a B+3 A b)-\frac{2 a^3 A}{5 x^{5/2}}+\frac{2}{13} b^2 x^{13/2} (3 a B+A b)+\frac{6}{7} a b x^{7/2} (a B+A b)+\frac{2}{19} b^3 B x^{19/2} \]

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*}

Mathematica [A]  time = 0.0237802, size = 78, normalized size = 0.94 \[ \frac{7410 a^2 b x^3 \left (7 A+B x^3\right )-3458 a^3 \left (A-5 B x^3\right )+570 a b^2 x^6 \left (13 A+7 B x^3\right )+70 b^3 x^9 \left (19 A+13 B x^3\right )}{8645 x^{5/2}} \]

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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Maple [A]  time = 0.006, size = 80, normalized size = 1. \begin{align*} -{\frac{-910\,{b}^{3}B{x}^{12}-1330\,{x}^{9}{b}^{3}A-3990\,{x}^{9}a{b}^{2}B-7410\,{x}^{6}a{b}^{2}A-7410\,{x}^{6}{a}^{2}bB-51870\,A{a}^{2}b{x}^{3}-17290\,B{a}^{3}{x}^{3}+3458\,{a}^{3}A}{8645}{x}^{-{\frac{5}{2}}}} \end{align*}

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 = 0.936169, size = 99, normalized size = 1.19 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.67592, size = 186, normalized size = 2.24 \begin{align*} \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{align*}

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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Sympy [A]  time = 39.641, size = 110, normalized size = 1.33 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.11192, size = 104, normalized size = 1.25 \begin{align*} \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{align*}

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)