3.394 \(\int x^{5/2} (A+B x) (a+c x^2)^2 \, dx\)

Optimal. Leaf size=77 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{9} a^2 B x^{9/2}+\frac{4}{11} a A c x^{11/2}+\frac{4}{13} a B c x^{13/2}+\frac{2}{15} A c^2 x^{15/2}+\frac{2}{17} B c^2 x^{17/2} \]

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a^2*B*x^(9/2))/9 + (4*a*A*c*x^(11/2))/11 + (4*a*B*c*x^(13/2))/13 + (2*A*c^2*x^(15/2))
/15 + (2*B*c^2*x^(17/2))/17

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Rubi [A]  time = 0.0257879, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {766} \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{9} a^2 B x^{9/2}+\frac{4}{11} a A c x^{11/2}+\frac{4}{13} a B c x^{13/2}+\frac{2}{15} A c^2 x^{15/2}+\frac{2}{17} B c^2 x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(A + B*x)*(a + c*x^2)^2,x]

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a^2*B*x^(9/2))/9 + (4*a*A*c*x^(11/2))/11 + (4*a*B*c*x^(13/2))/13 + (2*A*c^2*x^(15/2))
/15 + (2*B*c^2*x^(17/2))/17

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^{5/2} (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{5/2}+a^2 B x^{7/2}+2 a A c x^{9/2}+2 a B c x^{11/2}+A c^2 x^{13/2}+B c^2 x^{15/2}\right ) \, dx\\ &=\frac{2}{7} a^2 A x^{7/2}+\frac{2}{9} a^2 B x^{9/2}+\frac{4}{11} a A c x^{11/2}+\frac{4}{13} a B c x^{13/2}+\frac{2}{15} A c^2 x^{15/2}+\frac{2}{17} B c^2 x^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.023185, size = 60, normalized size = 0.78 \[ \frac{2}{63} a^2 x^{7/2} (9 A+7 B x)+\frac{4}{143} a c x^{11/2} (13 A+11 B x)+\frac{2}{255} c^2 x^{15/2} (17 A+15 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(A + B*x)*(a + c*x^2)^2,x]

[Out]

(2*a^2*x^(7/2)*(9*A + 7*B*x))/63 + (4*a*c*x^(11/2)*(13*A + 11*B*x))/143 + (2*c^2*x^(15/2)*(17*A + 15*B*x))/255

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Maple [A]  time = 0.004, size = 54, normalized size = 0.7 \begin{align*}{\frac{90090\,B{c}^{2}{x}^{5}+102102\,A{c}^{2}{x}^{4}+235620\,aBc{x}^{3}+278460\,aAc{x}^{2}+170170\,{a}^{2}Bx+218790\,A{a}^{2}}{765765}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x+A)*(c*x^2+a)^2,x)

[Out]

2/765765*x^(7/2)*(45045*B*c^2*x^5+51051*A*c^2*x^4+117810*B*a*c*x^3+139230*A*a*c*x^2+85085*B*a^2*x+109395*A*a^2
)

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Maxima [A]  time = 0.994453, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{2}{15} \, A c^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B a c x^{\frac{13}{2}} + \frac{4}{11} \, A a c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{2} x^{\frac{9}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+a)^2,x, algorithm="maxima")

[Out]

2/17*B*c^2*x^(17/2) + 2/15*A*c^2*x^(15/2) + 4/13*B*a*c*x^(13/2) + 4/11*A*a*c*x^(11/2) + 2/9*B*a^2*x^(9/2) + 2/
7*A*a^2*x^(7/2)

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Fricas [A]  time = 1.27833, size = 174, normalized size = 2.26 \begin{align*} \frac{2}{765765} \,{\left (45045 \, B c^{2} x^{8} + 51051 \, A c^{2} x^{7} + 117810 \, B a c x^{6} + 139230 \, A a c x^{5} + 85085 \, B a^{2} x^{4} + 109395 \, A a^{2} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+a)^2,x, algorithm="fricas")

[Out]

2/765765*(45045*B*c^2*x^8 + 51051*A*c^2*x^7 + 117810*B*a*c*x^6 + 139230*A*a*c*x^5 + 85085*B*a^2*x^4 + 109395*A
*a^2*x^3)*sqrt(x)

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Sympy [A]  time = 10.5877, size = 80, normalized size = 1.04 \begin{align*} \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{4 A a c x^{\frac{11}{2}}}{11} + \frac{2 A c^{2} x^{\frac{15}{2}}}{15} + \frac{2 B a^{2} x^{\frac{9}{2}}}{9} + \frac{4 B a c x^{\frac{13}{2}}}{13} + \frac{2 B c^{2} x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x+A)*(c*x**2+a)**2,x)

[Out]

2*A*a**2*x**(7/2)/7 + 4*A*a*c*x**(11/2)/11 + 2*A*c**2*x**(15/2)/15 + 2*B*a**2*x**(9/2)/9 + 4*B*a*c*x**(13/2)/1
3 + 2*B*c**2*x**(17/2)/17

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Giac [A]  time = 1.20202, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{2}{15} \, A c^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B a c x^{\frac{13}{2}} + \frac{4}{11} \, A a c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{2} x^{\frac{9}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+a)^2,x, algorithm="giac")

[Out]

2/17*B*c^2*x^(17/2) + 2/15*A*c^2*x^(15/2) + 4/13*B*a*c*x^(13/2) + 4/11*A*a*c*x^(11/2) + 2/9*B*a^2*x^(9/2) + 2/
7*A*a^2*x^(7/2)