3.405 \(\int \sqrt{x} (A+B x) (a+c x^2)^3 \, dx\)

Optimal. Leaf size=109 \[ \frac{6}{7} a^2 A c x^{7/2}+\frac{2}{3} a^3 A x^{3/2}+\frac{2}{3} a^2 B c x^{9/2}+\frac{2}{5} a^3 B x^{5/2}+\frac{6}{11} a A c^2 x^{11/2}+\frac{6}{13} a B c^2 x^{13/2}+\frac{2}{15} A c^3 x^{15/2}+\frac{2}{17} B c^3 x^{17/2} \]

[Out]

(2*a^3*A*x^(3/2))/3 + (2*a^3*B*x^(5/2))/5 + (6*a^2*A*c*x^(7/2))/7 + (2*a^2*B*c*x^(9/2))/3 + (6*a*A*c^2*x^(11/2
))/11 + (6*a*B*c^2*x^(13/2))/13 + (2*A*c^3*x^(15/2))/15 + (2*B*c^3*x^(17/2))/17

________________________________________________________________________________________

Rubi [A]  time = 0.0372586, antiderivative size = 109, 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} \[ \frac{6}{7} a^2 A c x^{7/2}+\frac{2}{3} a^3 A x^{3/2}+\frac{2}{3} a^2 B c x^{9/2}+\frac{2}{5} a^3 B x^{5/2}+\frac{6}{11} a A c^2 x^{11/2}+\frac{6}{13} a B c^2 x^{13/2}+\frac{2}{15} A c^3 x^{15/2}+\frac{2}{17} B c^3 x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(A + B*x)*(a + c*x^2)^3,x]

[Out]

(2*a^3*A*x^(3/2))/3 + (2*a^3*B*x^(5/2))/5 + (6*a^2*A*c*x^(7/2))/7 + (2*a^2*B*c*x^(9/2))/3 + (6*a*A*c^2*x^(11/2
))/11 + (6*a*B*c^2*x^(13/2))/13 + (2*A*c^3*x^(15/2))/15 + (2*B*c^3*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 \sqrt{x} (A+B x) \left (a+c x^2\right )^3 \, dx &=\int \left (a^3 A \sqrt{x}+a^3 B x^{3/2}+3 a^2 A c x^{5/2}+3 a^2 B c x^{7/2}+3 a A c^2 x^{9/2}+3 a B c^2 x^{11/2}+A c^3 x^{13/2}+B c^3 x^{15/2}\right ) \, dx\\ &=\frac{2}{3} a^3 A x^{3/2}+\frac{2}{5} a^3 B x^{5/2}+\frac{6}{7} a^2 A c x^{7/2}+\frac{2}{3} a^2 B c x^{9/2}+\frac{6}{11} a A c^2 x^{11/2}+\frac{6}{13} a B c^2 x^{13/2}+\frac{2}{15} A c^3 x^{15/2}+\frac{2}{17} B c^3 x^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0466382, size = 83, normalized size = 0.76 \[ \frac{2}{21} a^2 c x^{7/2} (9 A+7 B x)+\frac{2}{15} a^3 x^{3/2} (5 A+3 B x)+\frac{6}{143} a c^2 x^{11/2} (13 A+11 B x)+\frac{2}{255} c^3 x^{15/2} (17 A+15 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(A + B*x)*(a + c*x^2)^3,x]

[Out]

(2*a^3*x^(3/2)*(5*A + 3*B*x))/15 + (2*a^2*c*x^(7/2)*(9*A + 7*B*x))/21 + (6*a*c^2*x^(11/2)*(13*A + 11*B*x))/143
+ (2*c^3*x^(15/2)*(17*A + 15*B*x))/255

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 78, normalized size = 0.7 \begin{align*}{\frac{30030\,B{c}^{3}{x}^{7}+34034\,A{c}^{3}{x}^{6}+117810\,aB{c}^{2}{x}^{5}+139230\,aA{c}^{2}{x}^{4}+170170\,{a}^{2}Bc{x}^{3}+218790\,{a}^{2}Ac{x}^{2}+102102\,{a}^{3}Bx+170170\,A{a}^{3}}{255255}{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)^3*x^(1/2),x)

[Out]

2/255255*x^(3/2)*(15015*B*c^3*x^7+17017*A*c^3*x^6+58905*B*a*c^2*x^5+69615*A*a*c^2*x^4+85085*B*a^2*c*x^3+109395
*A*a^2*c*x^2+51051*B*a^3*x+85085*A*a^3)

________________________________________________________________________________________

Maxima [A]  time = 1.00981, size = 104, normalized size = 0.95 \begin{align*} \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{15} \, A c^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B a c^{2} x^{\frac{13}{2}} + \frac{6}{11} \, A a c^{2} x^{\frac{11}{2}} + \frac{2}{3} \, B a^{2} c x^{\frac{9}{2}} + \frac{6}{7} \, A a^{2} c x^{\frac{7}{2}} + \frac{2}{5} \, B a^{3} x^{\frac{5}{2}} + \frac{2}{3} \, A a^{3} 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)^3*x^(1/2),x, algorithm="maxima")

[Out]

2/17*B*c^3*x^(17/2) + 2/15*A*c^3*x^(15/2) + 6/13*B*a*c^2*x^(13/2) + 6/11*A*a*c^2*x^(11/2) + 2/3*B*a^2*c*x^(9/2
) + 6/7*A*a^2*c*x^(7/2) + 2/5*B*a^3*x^(5/2) + 2/3*A*a^3*x^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.38318, size = 228, normalized size = 2.09 \begin{align*} \frac{2}{255255} \,{\left (15015 \, B c^{3} x^{8} + 17017 \, A c^{3} x^{7} + 58905 \, B a c^{2} x^{6} + 69615 \, A a c^{2} x^{5} + 85085 \, B a^{2} c x^{4} + 109395 \, A a^{2} c x^{3} + 51051 \, B a^{3} x^{2} + 85085 \, A a^{3} 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)^3*x^(1/2),x, algorithm="fricas")

[Out]

2/255255*(15015*B*c^3*x^8 + 17017*A*c^3*x^7 + 58905*B*a*c^2*x^6 + 69615*A*a*c^2*x^5 + 85085*B*a^2*c*x^4 + 1093
95*A*a^2*c*x^3 + 51051*B*a^3*x^2 + 85085*A*a^3*x)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 4.2439, size = 114, normalized size = 1.05 \begin{align*} \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{6 A a^{2} c x^{\frac{7}{2}}}{7} + \frac{6 A a c^{2} x^{\frac{11}{2}}}{11} + \frac{2 A c^{3} x^{\frac{15}{2}}}{15} + \frac{2 B a^{3} x^{\frac{5}{2}}}{5} + \frac{2 B a^{2} c x^{\frac{9}{2}}}{3} + \frac{6 B a c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B c^{3} x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3*x**(1/2),x)

[Out]

2*A*a**3*x**(3/2)/3 + 6*A*a**2*c*x**(7/2)/7 + 6*A*a*c**2*x**(11/2)/11 + 2*A*c**3*x**(15/2)/15 + 2*B*a**3*x**(5
/2)/5 + 2*B*a**2*c*x**(9/2)/3 + 6*B*a*c**2*x**(13/2)/13 + 2*B*c**3*x**(17/2)/17

________________________________________________________________________________________

Giac [A]  time = 1.23229, size = 104, normalized size = 0.95 \begin{align*} \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{15} \, A c^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B a c^{2} x^{\frac{13}{2}} + \frac{6}{11} \, A a c^{2} x^{\frac{11}{2}} + \frac{2}{3} \, B a^{2} c x^{\frac{9}{2}} + \frac{6}{7} \, A a^{2} c x^{\frac{7}{2}} + \frac{2}{5} \, B a^{3} x^{\frac{5}{2}} + \frac{2}{3} \, A a^{3} 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)^3*x^(1/2),x, algorithm="giac")

[Out]

2/17*B*c^3*x^(17/2) + 2/15*A*c^3*x^(15/2) + 6/13*B*a*c^2*x^(13/2) + 6/11*A*a*c^2*x^(11/2) + 2/3*B*a^2*c*x^(9/2
) + 6/7*A*a^2*c*x^(7/2) + 2/5*B*a^3*x^(5/2) + 2/3*A*a^3*x^(3/2)