3.393 \(\int x^{7/2} (A+B x) (a+c x^2)^2 \, dx\)

Optimal. Leaf size=77 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{11} a^2 B x^{11/2}+\frac{4}{13} a A c x^{13/2}+\frac{4}{15} a B c x^{15/2}+\frac{2}{17} A c^2 x^{17/2}+\frac{2}{19} B c^2 x^{19/2} \]

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a^2*B*x^(11/2))/11 + (4*a*A*c*x^(13/2))/13 + (4*a*B*c*x^(15/2))/15 + (2*A*c^2*x^(17/2
))/17 + (2*B*c^2*x^(19/2))/19

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Rubi [A]  time = 0.0245772, 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} \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{11} a^2 B x^{11/2}+\frac{4}{13} a A c x^{13/2}+\frac{4}{15} a B c x^{15/2}+\frac{2}{17} A c^2 x^{17/2}+\frac{2}{19} B c^2 x^{19/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a^2*B*x^(11/2))/11 + (4*a*A*c*x^(13/2))/13 + (4*a*B*c*x^(15/2))/15 + (2*A*c^2*x^(17/2
))/17 + (2*B*c^2*x^(19/2))/19

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^{7/2} (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{7/2}+a^2 B x^{9/2}+2 a A c x^{11/2}+2 a B c x^{13/2}+A c^2 x^{15/2}+B c^2 x^{17/2}\right ) \, dx\\ &=\frac{2}{9} a^2 A x^{9/2}+\frac{2}{11} a^2 B x^{11/2}+\frac{4}{13} a A c x^{13/2}+\frac{4}{15} a B c x^{15/2}+\frac{2}{17} A c^2 x^{17/2}+\frac{2}{19} B c^2 x^{19/2}\\ \end{align*}

Mathematica [A]  time = 0.0233011, size = 60, normalized size = 0.78 \[ \frac{2}{99} a^2 x^{9/2} (11 A+9 B x)+\frac{4}{195} a c x^{13/2} (15 A+13 B x)+\frac{2}{323} c^2 x^{17/2} (19 A+17 B x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*x^(9/2)*(11*A + 9*B*x))/99 + (4*a*c*x^(13/2)*(15*A + 13*B*x))/195 + (2*c^2*x^(17/2)*(19*A + 17*B*x))/32
3

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Maple [A]  time = 0.004, size = 54, normalized size = 0.7 \begin{align*}{\frac{218790\,B{c}^{2}{x}^{5}+244530\,A{c}^{2}{x}^{4}+554268\,aBc{x}^{3}+639540\,aAc{x}^{2}+377910\,{a}^{2}Bx+461890\,A{a}^{2}}{2078505}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2078505*x^(9/2)*(109395*B*c^2*x^5+122265*A*c^2*x^4+277134*B*a*c*x^3+319770*A*a*c*x^2+188955*B*a^2*x+230945*A
*a^2)

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Maxima [A]  time = 0.963717, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{4}{15} \, B a c x^{\frac{15}{2}} + \frac{4}{13} \, A a c x^{\frac{13}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*B*c^2*x^(19/2) + 2/17*A*c^2*x^(17/2) + 4/15*B*a*c*x^(15/2) + 4/13*A*a*c*x^(13/2) + 2/11*B*a^2*x^(11/2) +
2/9*A*a^2*x^(9/2)

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Fricas [A]  time = 1.35608, size = 180, normalized size = 2.34 \begin{align*} \frac{2}{2078505} \,{\left (109395 \, B c^{2} x^{9} + 122265 \, A c^{2} x^{8} + 277134 \, B a c x^{7} + 319770 \, A a c x^{6} + 188955 \, B a^{2} x^{5} + 230945 \, A a^{2} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2078505*(109395*B*c^2*x^9 + 122265*A*c^2*x^8 + 277134*B*a*c*x^7 + 319770*A*a*c*x^6 + 188955*B*a^2*x^5 + 2309
45*A*a^2*x^4)*sqrt(x)

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Sympy [A]  time = 21.1513, size = 80, normalized size = 1.04 \begin{align*} \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a c x^{\frac{13}{2}}}{13} + \frac{2 A c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B a^{2} x^{\frac{11}{2}}}{11} + \frac{4 B a c x^{\frac{15}{2}}}{15} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**2*x**(9/2)/9 + 4*A*a*c*x**(13/2)/13 + 2*A*c**2*x**(17/2)/17 + 2*B*a**2*x**(11/2)/11 + 4*B*a*c*x**(15/2)
/15 + 2*B*c**2*x**(19/2)/19

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Giac [A]  time = 1.16654, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{4}{15} \, B a c x^{\frac{15}{2}} + \frac{4}{13} \, A a c x^{\frac{13}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*B*c^2*x^(19/2) + 2/17*A*c^2*x^(17/2) + 4/15*B*a*c*x^(15/2) + 4/13*A*a*c*x^(13/2) + 2/11*B*a^2*x^(11/2) +
2/9*A*a^2*x^(9/2)