3.396 \(\int \sqrt{x} (A+B x) (a+c x^2)^2 \, dx\)

Optimal. Leaf size=77 \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{5} a^2 B x^{5/2}+\frac{4}{7} a A c x^{7/2}+\frac{4}{9} a B c x^{9/2}+\frac{2}{11} A c^2 x^{11/2}+\frac{2}{13} B c^2 x^{13/2} \]

[Out]

(2*a^2*A*x^(3/2))/3 + (2*a^2*B*x^(5/2))/5 + (4*a*A*c*x^(7/2))/7 + (4*a*B*c*x^(9/2))/9 + (2*A*c^2*x^(11/2))/11
+ (2*B*c^2*x^(13/2))/13

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Rubi [A]  time = 0.0240956, 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} \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{5} a^2 B x^{5/2}+\frac{4}{7} a A c x^{7/2}+\frac{4}{9} a B c x^{9/2}+\frac{2}{11} A c^2 x^{11/2}+\frac{2}{13} B c^2 x^{13/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*A*x^(3/2))/3 + (2*a^2*B*x^(5/2))/5 + (4*a*A*c*x^(7/2))/7 + (4*a*B*c*x^(9/2))/9 + (2*A*c^2*x^(11/2))/11
+ (2*B*c^2*x^(13/2))/13

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 \sqrt{x} (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A \sqrt{x}+a^2 B x^{3/2}+2 a A c x^{5/2}+2 a B c x^{7/2}+A c^2 x^{9/2}+B c^2 x^{11/2}\right ) \, dx\\ &=\frac{2}{3} a^2 A x^{3/2}+\frac{2}{5} a^2 B x^{5/2}+\frac{4}{7} a A c x^{7/2}+\frac{4}{9} a B c x^{9/2}+\frac{2}{11} A c^2 x^{11/2}+\frac{2}{13} B c^2 x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0212842, size = 60, normalized size = 0.78 \[ \frac{2}{15} a^2 x^{3/2} (5 A+3 B x)+\frac{4}{63} a c x^{7/2} (9 A+7 B x)+\frac{2}{143} c^2 x^{11/2} (13 A+11 B x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*x^(3/2)*(5*A + 3*B*x))/15 + (4*a*c*x^(7/2)*(9*A + 7*B*x))/63 + (2*c^2*x^(11/2)*(13*A + 11*B*x))/143

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Maple [A]  time = 0.007, size = 54, normalized size = 0.7 \begin{align*}{\frac{6930\,B{c}^{2}{x}^{5}+8190\,A{c}^{2}{x}^{4}+20020\,aBc{x}^{3}+25740\,aAc{x}^{2}+18018\,{a}^{2}Bx+30030\,A{a}^{2}}{45045}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2*x^(1/2),x)

[Out]

2/45045*x^(3/2)*(3465*B*c^2*x^5+4095*A*c^2*x^4+10010*B*a*c*x^3+12870*A*a*c*x^2+9009*B*a^2*x+15015*A*a^2)

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Maxima [A]  time = 1.01344, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{13} \, B c^{2} x^{\frac{13}{2}} + \frac{2}{11} \, A c^{2} x^{\frac{11}{2}} + \frac{4}{9} \, B a c x^{\frac{9}{2}} + \frac{4}{7} \, A a c x^{\frac{7}{2}} + \frac{2}{5} \, B a^{2} x^{\frac{5}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2*x^(1/2),x, algorithm="maxima")

[Out]

2/13*B*c^2*x^(13/2) + 2/11*A*c^2*x^(11/2) + 4/9*B*a*c*x^(9/2) + 4/7*A*a*c*x^(7/2) + 2/5*B*a^2*x^(5/2) + 2/3*A*
a^2*x^(3/2)

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Fricas [A]  time = 1.41547, size = 162, normalized size = 2.1 \begin{align*} \frac{2}{45045} \,{\left (3465 \, B c^{2} x^{6} + 4095 \, A c^{2} x^{5} + 10010 \, B a c x^{4} + 12870 \, A a c x^{3} + 9009 \, B a^{2} x^{2} + 15015 \, A a^{2} x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2*x^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3465*B*c^2*x^6 + 4095*A*c^2*x^5 + 10010*B*a*c*x^4 + 12870*A*a*c*x^3 + 9009*B*a^2*x^2 + 15015*A*a^2*x)
*sqrt(x)

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Sympy [A]  time = 2.61674, size = 80, normalized size = 1.04 \begin{align*} \frac{2 A a^{2} x^{\frac{3}{2}}}{3} + \frac{4 A a c x^{\frac{7}{2}}}{7} + \frac{2 A c^{2} x^{\frac{11}{2}}}{11} + \frac{2 B a^{2} x^{\frac{5}{2}}}{5} + \frac{4 B a c x^{\frac{9}{2}}}{9} + \frac{2 B c^{2} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**2*x**(1/2),x)

[Out]

2*A*a**2*x**(3/2)/3 + 4*A*a*c*x**(7/2)/7 + 2*A*c**2*x**(11/2)/11 + 2*B*a**2*x**(5/2)/5 + 4*B*a*c*x**(9/2)/9 +
2*B*c**2*x**(13/2)/13

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Giac [A]  time = 1.21368, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{13} \, B c^{2} x^{\frac{13}{2}} + \frac{2}{11} \, A c^{2} x^{\frac{11}{2}} + \frac{4}{9} \, B a c x^{\frac{9}{2}} + \frac{4}{7} \, A a c x^{\frac{7}{2}} + \frac{2}{5} \, B a^{2} x^{\frac{5}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2*x^(1/2),x, algorithm="giac")

[Out]

2/13*B*c^2*x^(13/2) + 2/11*A*c^2*x^(11/2) + 4/9*B*a*c*x^(9/2) + 4/7*A*a*c*x^(7/2) + 2/5*B*a^2*x^(5/2) + 2/3*A*
a^2*x^(3/2)