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

Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]

[Out]

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

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Rubi [A]  time = 0.0310831, 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} \[ -\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0161413, size = 57, normalized size = 0.93 \[ \frac{2 \left (-91 a^2 \left (A-5 B x^3\right )+130 a b x^3 \left (7 A+B x^3\right )+5 b^2 x^6 \left (13 A+7 B x^3\right )\right )}{455 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-91*a^2*(A - 5*B*x^3) + 130*a*b*x^3*(7*A + B*x^3) + 5*b^2*x^6*(13*A + 7*B*x^3)))/(455*x^(5/2))

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-70\,B{b}^{2}{x}^{9}-130\,A{b}^{2}{x}^{6}-260\,B{x}^{6}ab-1820\,aAb{x}^{3}-910\,B{x}^{3}{a}^{2}+182\,{a}^{2}A}{455}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/455*(-35*B*b^2*x^9-65*A*b^2*x^6-130*B*a*b*x^6-910*A*a*b*x^3-455*B*a^2*x^3+91*A*a^2)/x^(5/2)

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Maxima [A]  time = 0.943316, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{2}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{7}{2}} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} \sqrt{x} - \frac{2 \, A a^{2}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.72558, size = 130, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (35 \, B b^{2} x^{9} + 65 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + 455 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} - 91 \, A a^{2}\right )}}{455 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/455*(35*B*b^2*x^9 + 65*(2*B*a*b + A*b^2)*x^6 + 455*(B*a^2 + 2*A*a*b)*x^3 - 91*A*a^2)/x^(5/2)

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Sympy [A]  time = 12.0103, size = 76, normalized size = 1.25 \begin{align*} - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} + 4 A a b \sqrt{x} + \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + 2 B a^{2} \sqrt{x} + \frac{4 B a b x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**2/(5*x**(5/2)) + 4*A*a*b*sqrt(x) + 2*A*b**2*x**(7/2)/7 + 2*B*a**2*sqrt(x) + 4*B*a*b*x**(7/2)/7 + 2*B*b
**2*x**(13/2)/13

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Giac [A]  time = 1.16006, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{7} \, B a b x^{\frac{7}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + 2 \, B a^{2} \sqrt{x} + 4 \, A a b \sqrt{x} - \frac{2 \, A a^{2}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*B*b^2*x^(13/2) + 4/7*B*a*b*x^(7/2) + 2/7*A*b^2*x^(7/2) + 2*B*a^2*sqrt(x) + 4*A*a*b*sqrt(x) - 2/5*A*a^2/x^
(5/2)