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

Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{17} b^2 B x^{17/2} \]

[Out]

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

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Rubi [A]  time = 0.0303508, antiderivative size = 61, 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} \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{17} b^2 B x^{17/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0189261, size = 60, normalized size = 0.98 \[ \frac{-374 a^2 \left (5 A-B x^3\right )+68 a b x^3 \left (11 A+5 B x^3\right )+10 b^2 x^6 \left (17 A+11 B x^3\right )}{935 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-374*a^2*(5*A - B*x^3) + 68*a*b*x^3*(11*A + 5*B*x^3) + 10*b^2*x^6*(17*A + 11*B*x^3))/(935*Sqrt[x])

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Maple [A]  time = 0.006, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-110\,B{b}^{2}{x}^{9}-170\,A{b}^{2}{x}^{6}-340\,B{x}^{6}ab-748\,aAb{x}^{3}-374\,B{x}^{3}{a}^{2}+1870\,{a}^{2}A}{935}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/935*(-55*B*b^2*x^9-85*A*b^2*x^6-170*B*a*b*x^6-374*A*a*b*x^3-187*B*a^2*x^3+935*A*a^2)/x^(1/2)

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Maxima [A]  time = 0.940555, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{17} \, B b^{2} x^{\frac{17}{2}} + \frac{2}{11} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{5} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{5}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.68896, size = 131, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (55 \, B b^{2} x^{9} + 85 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + 187 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} - 935 \, A a^{2}\right )}}{935 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/935*(55*B*b^2*x^9 + 85*(2*B*a*b + A*b^2)*x^6 + 187*(B*a^2 + 2*A*a*b)*x^3 - 935*A*a^2)/sqrt(x)

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Sympy [A]  time = 7.95219, size = 78, normalized size = 1.28 \begin{align*} - \frac{2 A a^{2}}{\sqrt{x}} + \frac{4 A a b x^{\frac{5}{2}}}{5} + \frac{2 A b^{2} x^{\frac{11}{2}}}{11} + \frac{2 B a^{2} x^{\frac{5}{2}}}{5} + \frac{4 B a b x^{\frac{11}{2}}}{11} + \frac{2 B b^{2} x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**2/sqrt(x) + 4*A*a*b*x**(5/2)/5 + 2*A*b**2*x**(11/2)/11 + 2*B*a**2*x**(5/2)/5 + 4*B*a*b*x**(11/2)/11 +
2*B*b**2*x**(17/2)/17

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Giac [A]  time = 1.1013, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{17} \, B b^{2} x^{\frac{17}{2}} + \frac{4}{11} \, B a b x^{\frac{11}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} + \frac{2}{5} \, B a^{2} x^{\frac{5}{2}} + \frac{4}{5} \, A a b x^{\frac{5}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*b^2*x^(17/2) + 4/11*B*a*b*x^(11/2) + 2/11*A*b^2*x^(11/2) + 2/5*B*a^2*x^(5/2) + 4/5*A*a*b*x^(5/2) - 2*A*
a^2/sqrt(x)