3.408 \(\int \frac{(A+B x) (a+c x^2)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=103 \[ 6 a^2 A c \sqrt{x}-\frac{2 a^3 A}{3 x^{3/2}}+2 a^2 B c x^{3/2}-\frac{2 a^3 B}{\sqrt{x}}+\frac{6}{5} a A c^2 x^{5/2}+\frac{6}{7} a B c^2 x^{7/2}+\frac{2}{9} A c^3 x^{9/2}+\frac{2}{11} B c^3 x^{11/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0372254, antiderivative size = 103, 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} \[ 6 a^2 A c \sqrt{x}-\frac{2 a^3 A}{3 x^{3/2}}+2 a^2 B c x^{3/2}-\frac{2 a^3 B}{\sqrt{x}}+\frac{6}{5} a A c^2 x^{5/2}+\frac{6}{7} a B c^2 x^{7/2}+\frac{2}{9} A c^3 x^{9/2}+\frac{2}{11} B c^3 x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/x^(5/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0330627, size = 78, normalized size = 0.76 \[ 2 a^2 c \sqrt{x} (3 A+B x)-\frac{2 a^3 (A+3 B x)}{3 x^{3/2}}+\frac{6}{35} a c^2 x^{5/2} (7 A+5 B x)+\frac{2}{99} c^3 x^{9/2} (11 A+9 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/x^(5/2),x]

[Out]

2*a^2*c*Sqrt[x]*(3*A + B*x) - (2*a^3*(A + 3*B*x))/(3*x^(3/2)) + (6*a*c^2*x^(5/2)*(7*A + 5*B*x))/35 + (2*c^3*x^
(9/2)*(11*A + 9*B*x))/99

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 78, normalized size = 0.8 \begin{align*} -{\frac{-630\,B{c}^{3}{x}^{7}-770\,A{c}^{3}{x}^{6}-2970\,aB{c}^{2}{x}^{5}-4158\,aA{c}^{2}{x}^{4}-6930\,{a}^{2}Bc{x}^{3}-20790\,{a}^{2}Ac{x}^{2}+6930\,{a}^{3}Bx+2310\,A{a}^{3}}{3465}{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^(5/2),x)

[Out]

-2/3465*(-315*B*c^3*x^7-385*A*c^3*x^6-1485*B*a*c^2*x^5-2079*A*a*c^2*x^4-3465*B*a^2*c*x^3-10395*A*a^2*c*x^2+346
5*B*a^3*x+1155*A*a^3)/x^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 1.01428, size = 104, normalized size = 1.01 \begin{align*} \frac{2}{11} \, B c^{3} x^{\frac{11}{2}} + \frac{2}{9} \, A c^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B a c^{2} x^{\frac{7}{2}} + \frac{6}{5} \, A a c^{2} x^{\frac{5}{2}} + 2 \, B a^{2} c x^{\frac{3}{2}} + 6 \, A a^{2} c \sqrt{x} - \frac{2 \,{\left (3 \, B a^{3} x + A a^{3}\right )}}{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^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.4349, size = 207, normalized size = 2.01 \begin{align*} \frac{2 \,{\left (315 \, B c^{3} x^{7} + 385 \, A c^{3} x^{6} + 1485 \, B a c^{2} x^{5} + 2079 \, A a c^{2} x^{4} + 3465 \, B a^{2} c x^{3} + 10395 \, A a^{2} c x^{2} - 3465 \, B a^{3} x - 1155 \, A a^{3}\right )}}{3465 \, 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^(5/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*c^3*x^7 + 385*A*c^3*x^6 + 1485*B*a*c^2*x^5 + 2079*A*a*c^2*x^4 + 3465*B*a^2*c*x^3 + 10395*A*a^2*c
*x^2 - 3465*B*a^3*x - 1155*A*a^3)/x^(3/2)

________________________________________________________________________________________

Sympy [A]  time = 5.57855, size = 109, normalized size = 1.06 \begin{align*} - \frac{2 A a^{3}}{3 x^{\frac{3}{2}}} + 6 A a^{2} c \sqrt{x} + \frac{6 A a c^{2} x^{\frac{5}{2}}}{5} + \frac{2 A c^{3} x^{\frac{9}{2}}}{9} - \frac{2 B a^{3}}{\sqrt{x}} + 2 B a^{2} c x^{\frac{3}{2}} + \frac{6 B a c^{2} x^{\frac{7}{2}}}{7} + \frac{2 B c^{3} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/x**(5/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.24446, size = 104, normalized size = 1.01 \begin{align*} \frac{2}{11} \, B c^{3} x^{\frac{11}{2}} + \frac{2}{9} \, A c^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B a c^{2} x^{\frac{7}{2}} + \frac{6}{5} \, A a c^{2} x^{\frac{5}{2}} + 2 \, B a^{2} c x^{\frac{3}{2}} + 6 \, A a^{2} c \sqrt{x} - \frac{2 \,{\left (3 \, B a^{3} x + A a^{3}\right )}}{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^(5/2),x, algorithm="giac")

[Out]

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