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

Optimal. Leaf size=83 \[ \frac{2}{5} a^2 x^{5/2} (a B+3 A b)-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2} \]

[Out]

(-2*a^3*A)/Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(5/2))/5 + (6*a*b*(A*b + a*B)*x^(11/2))/11 + (2*b^2*(A*b + 3*a*B)*
x^(17/2))/17 + (2*b^3*B*x^(23/2))/23

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Rubi [A]  time = 0.040886, 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} \[ \frac{2}{5} a^2 x^{5/2} (a B+3 A b)-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*A)/Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(5/2))/5 + (6*a*b*(A*b + a*B)*x^(11/2))/11 + (2*b^2*(A*b + 3*a*B)*
x^(17/2))/17 + (2*b^3*B*x^(23/2))/23

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

Mathematica [A]  time = 0.0239988, size = 81, normalized size = 0.98 \[ \frac{2346 a^2 b x^3 \left (11 A+5 B x^3\right )-8602 a^3 \left (5 A-B x^3\right )+690 a b^2 x^6 \left (17 A+11 B x^3\right )+110 b^3 x^9 \left (23 A+17 B x^3\right )}{21505 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8602*a^3*(5*A - B*x^3) + 2346*a^2*b*x^3*(11*A + 5*B*x^3) + 690*a*b^2*x^6*(17*A + 11*B*x^3) + 110*b^3*x^9*(23
*A + 17*B*x^3))/(21505*Sqrt[x])

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Maple [A]  time = 0.006, size = 80, normalized size = 1. \begin{align*} -{\frac{-1870\,{b}^{3}B{x}^{12}-2530\,{x}^{9}{b}^{3}A-7590\,{x}^{9}a{b}^{2}B-11730\,{x}^{6}a{b}^{2}A-11730\,{x}^{6}{a}^{2}bB-25806\,{x}^{3}A{a}^{2}b-8602\,{x}^{3}B{a}^{3}+43010\,{a}^{3}A}{21505}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21505*(-935*B*b^3*x^12-1265*A*b^3*x^9-3795*B*a*b^2*x^9-5865*A*a*b^2*x^6-5865*B*a^2*b*x^6-12903*A*a^2*b*x^3-
4301*B*a^3*x^3+21505*A*a^3)/x^(1/2)

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Maxima [A]  time = 0.942053, size = 99, normalized size = 1.19 \begin{align*} \frac{2}{23} \, B b^{3} x^{\frac{23}{2}} + \frac{2}{17} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{17}{2}} + \frac{6}{11} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{11}{2}} - \frac{2 \, A a^{3}}{\sqrt{x}} + \frac{2}{5} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} 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^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.72498, size = 190, normalized size = 2.29 \begin{align*} \frac{2 \,{\left (935 \, B b^{3} x^{12} + 1265 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 5865 \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} - 21505 \, A a^{3} + 4301 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )}}{21505 \, \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^(3/2),x, algorithm="fricas")

[Out]

2/21505*(935*B*b^3*x^12 + 1265*(3*B*a*b^2 + A*b^3)*x^9 + 5865*(B*a^2*b + A*a*b^2)*x^6 - 21505*A*a^3 + 4301*(B*
a^3 + 3*A*a^2*b)*x^3)/sqrt(x)

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Sympy [A]  time = 26.7273, size = 112, normalized size = 1.35 \begin{align*} - \frac{2 A a^{3}}{\sqrt{x}} + \frac{6 A a^{2} b x^{\frac{5}{2}}}{5} + \frac{6 A a b^{2} x^{\frac{11}{2}}}{11} + \frac{2 A b^{3} x^{\frac{17}{2}}}{17} + \frac{2 B a^{3} x^{\frac{5}{2}}}{5} + \frac{6 B a^{2} b x^{\frac{11}{2}}}{11} + \frac{6 B a b^{2} x^{\frac{17}{2}}}{17} + \frac{2 B b^{3} x^{\frac{23}{2}}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12058, size = 104, normalized size = 1.25 \begin{align*} \frac{2}{23} \, B b^{3} x^{\frac{23}{2}} + \frac{6}{17} \, B a b^{2} x^{\frac{17}{2}} + \frac{2}{17} \, A b^{3} x^{\frac{17}{2}} + \frac{6}{11} \, B a^{2} b x^{\frac{11}{2}} + \frac{6}{11} \, A a b^{2} x^{\frac{11}{2}} + \frac{2}{5} \, B a^{3} x^{\frac{5}{2}} + \frac{6}{5} \, A a^{2} b x^{\frac{5}{2}} - \frac{2 \, A a^{3}}{\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^(3/2),x, algorithm="giac")

[Out]

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