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

Optimal. Leaf size=107 \[ \frac{6}{5} a^2 A c x^{5/2}+2 a^3 A \sqrt{x}+\frac{6}{7} a^2 B c x^{7/2}+\frac{2}{3} a^3 B x^{3/2}+\frac{2}{3} a A c^2 x^{9/2}+\frac{6}{11} a B c^2 x^{11/2}+\frac{2}{13} A c^3 x^{13/2}+\frac{2}{15} B c^3 x^{15/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0390424, antiderivative size = 107, 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} \[ \frac{6}{5} a^2 A c x^{5/2}+2 a^3 A \sqrt{x}+\frac{6}{7} a^2 B c x^{7/2}+\frac{2}{3} a^3 B x^{3/2}+\frac{2}{3} a A c^2 x^{9/2}+\frac{6}{11} a B c^2 x^{11/2}+\frac{2}{13} A c^3 x^{13/2}+\frac{2}{15} B c^3 x^{15/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0297251, size = 72, normalized size = 0.67 \[ \frac{2 \sqrt{x} \left (1287 a^2 c x^2 (7 A+5 B x)+5005 a^3 (3 A+B x)+455 a c^2 x^4 (11 A+9 B x)+77 c^3 x^6 (15 A+13 B x)\right )}{15015} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(5005*a^3*(3*A + B*x) + 1287*a^2*c*x^2*(7*A + 5*B*x) + 455*a*c^2*x^4*(11*A + 9*B*x) + 77*c^3*x^6*(1
5*A + 13*B*x)))/15015

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 78, normalized size = 0.7 \begin{align*}{\frac{2002\,B{c}^{3}{x}^{7}+2310\,A{c}^{3}{x}^{6}+8190\,aB{c}^{2}{x}^{5}+10010\,aA{c}^{2}{x}^{4}+12870\,{a}^{2}Bc{x}^{3}+18018\,{a}^{2}Ac{x}^{2}+10010\,{a}^{3}Bx+30030\,A{a}^{3}}{15015}\sqrt{x}} \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]

2/15015*x^(1/2)*(1001*B*c^3*x^7+1155*A*c^3*x^6+4095*B*a*c^2*x^5+5005*A*a*c^2*x^4+6435*B*a^2*c*x^3+9009*A*a^2*c
*x^2+5005*B*a^3*x+15015*A*a^3)

________________________________________________________________________________________

Maxima [A]  time = 1.00935, size = 104, normalized size = 0.97 \begin{align*} \frac{2}{15} \, B c^{3} x^{\frac{15}{2}} + \frac{2}{13} \, A c^{3} x^{\frac{13}{2}} + \frac{6}{11} \, B a c^{2} x^{\frac{11}{2}} + \frac{2}{3} \, A a c^{2} x^{\frac{9}{2}} + \frac{6}{7} \, B a^{2} c x^{\frac{7}{2}} + \frac{6}{5} \, A a^{2} c x^{\frac{5}{2}} + \frac{2}{3} \, B a^{3} x^{\frac{3}{2}} + 2 \, A a^{3} \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="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.32515, size = 211, normalized size = 1.97 \begin{align*} \frac{2}{15015} \,{\left (1001 \, B c^{3} x^{7} + 1155 \, A c^{3} x^{6} + 4095 \, B a c^{2} x^{5} + 5005 \, A a c^{2} x^{4} + 6435 \, B a^{2} c x^{3} + 9009 \, A a^{2} c x^{2} + 5005 \, B a^{3} x + 15015 \, A a^{3}\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]

2/15015*(1001*B*c^3*x^7 + 1155*A*c^3*x^6 + 4095*B*a*c^2*x^5 + 5005*A*a*c^2*x^4 + 6435*B*a^2*c*x^3 + 9009*A*a^2
*c*x^2 + 5005*B*a^3*x + 15015*A*a^3)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 4.43197, size = 112, normalized size = 1.05 \begin{align*} 2 A a^{3} \sqrt{x} + \frac{6 A a^{2} c x^{\frac{5}{2}}}{5} + \frac{2 A a c^{2} x^{\frac{9}{2}}}{3} + \frac{2 A c^{3} x^{\frac{13}{2}}}{13} + \frac{2 B a^{3} x^{\frac{3}{2}}}{3} + \frac{6 B a^{2} c x^{\frac{7}{2}}}{7} + \frac{6 B a c^{2} x^{\frac{11}{2}}}{11} + \frac{2 B c^{3} x^{\frac{15}{2}}}{15} \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*sqrt(x) + 6*A*a**2*c*x**(5/2)/5 + 2*A*a*c**2*x**(9/2)/3 + 2*A*c**3*x**(13/2)/13 + 2*B*a**3*x**(3/2)/3
+ 6*B*a**2*c*x**(7/2)/7 + 6*B*a*c**2*x**(11/2)/11 + 2*B*c**3*x**(15/2)/15

________________________________________________________________________________________

Giac [A]  time = 1.30363, size = 104, normalized size = 0.97 \begin{align*} \frac{2}{15} \, B c^{3} x^{\frac{15}{2}} + \frac{2}{13} \, A c^{3} x^{\frac{13}{2}} + \frac{6}{11} \, B a c^{2} x^{\frac{11}{2}} + \frac{2}{3} \, A a c^{2} x^{\frac{9}{2}} + \frac{6}{7} \, B a^{2} c x^{\frac{7}{2}} + \frac{6}{5} \, A a^{2} c x^{\frac{5}{2}} + \frac{2}{3} \, B a^{3} x^{\frac{3}{2}} + 2 \, A a^{3} \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="giac")

[Out]

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