3.411 \(\int \frac{(A+B x) (a+c x^2)^3}{x^{11/2}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0390749, 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} \[ -\frac{6 a^2 A c}{5 x^{5/2}}-\frac{2 a^3 A}{9 x^{9/2}}-\frac{2 a^2 B c}{x^{3/2}}-\frac{2 a^3 B}{7 x^{7/2}}-\frac{6 a A c^2}{\sqrt{x}}+6 a B c^2 \sqrt{x}+\frac{2}{3} A c^3 x^{3/2}+\frac{2}{5} B c^3 x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/x^(11/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0274516, size = 71, normalized size = 0.69 \[ -\frac{2 \left (63 a^2 c x^2 (3 A+5 B x)+5 a^3 (7 A+9 B x)+945 a c^2 x^4 (A-B x)-21 c^3 x^6 (5 A+3 B x)\right )}{315 x^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/x^(11/2),x]

[Out]

(-2*(945*a*c^2*x^4*(A - B*x) - 21*c^3*x^6*(5*A + 3*B*x) + 63*a^2*c*x^2*(3*A + 5*B*x) + 5*a^3*(7*A + 9*B*x)))/(
315*x^(9/2))

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 78, normalized size = 0.8 \begin{align*} -{\frac{-126\,B{c}^{3}{x}^{7}-210\,A{c}^{3}{x}^{6}-1890\,aB{c}^{2}{x}^{5}+1890\,aA{c}^{2}{x}^{4}+630\,{a}^{2}Bc{x}^{3}+378\,{a}^{2}Ac{x}^{2}+90\,{a}^{3}Bx+70\,A{a}^{3}}{315}{x}^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/x^(11/2),x)

[Out]

-2/315*(-63*B*c^3*x^7-105*A*c^3*x^6-945*B*a*c^2*x^5+945*A*a*c^2*x^4+315*B*a^2*c*x^3+189*A*a^2*c*x^2+45*B*a^3*x
+35*A*a^3)/x^(9/2)

________________________________________________________________________________________

Maxima [A]  time = 1.00417, size = 105, normalized size = 1.02 \begin{align*} \frac{2}{5} \, B c^{3} x^{\frac{5}{2}} + \frac{2}{3} \, A c^{3} x^{\frac{3}{2}} + 6 \, B a c^{2} \sqrt{x} - \frac{2 \,{\left (945 \, A a c^{2} x^{4} + 315 \, B a^{2} c x^{3} + 189 \, A a^{2} c x^{2} + 45 \, B a^{3} x + 35 \, A a^{3}\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(11/2),x, algorithm="maxima")

[Out]

2/5*B*c^3*x^(5/2) + 2/3*A*c^3*x^(3/2) + 6*B*a*c^2*sqrt(x) - 2/315*(945*A*a*c^2*x^4 + 315*B*a^2*c*x^3 + 189*A*a
^2*c*x^2 + 45*B*a^3*x + 35*A*a^3)/x^(9/2)

________________________________________________________________________________________

Fricas [A]  time = 1.18719, size = 192, normalized size = 1.86 \begin{align*} \frac{2 \,{\left (63 \, B c^{3} x^{7} + 105 \, A c^{3} x^{6} + 945 \, B a c^{2} x^{5} - 945 \, A a c^{2} x^{4} - 315 \, B a^{2} c x^{3} - 189 \, A a^{2} c x^{2} - 45 \, B a^{3} x - 35 \, A a^{3}\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(11/2),x, algorithm="fricas")

[Out]

2/315*(63*B*c^3*x^7 + 105*A*c^3*x^6 + 945*B*a*c^2*x^5 - 945*A*a*c^2*x^4 - 315*B*a^2*c*x^3 - 189*A*a^2*c*x^2 -
45*B*a^3*x - 35*A*a^3)/x^(9/2)

________________________________________________________________________________________

Sympy [A]  time = 13.6632, size = 109, normalized size = 1.06 \begin{align*} - \frac{2 A a^{3}}{9 x^{\frac{9}{2}}} - \frac{6 A a^{2} c}{5 x^{\frac{5}{2}}} - \frac{6 A a c^{2}}{\sqrt{x}} + \frac{2 A c^{3} x^{\frac{3}{2}}}{3} - \frac{2 B a^{3}}{7 x^{\frac{7}{2}}} - \frac{2 B a^{2} c}{x^{\frac{3}{2}}} + 6 B a c^{2} \sqrt{x} + \frac{2 B c^{3} x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/x**(11/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.16012, size = 105, normalized size = 1.02 \begin{align*} \frac{2}{5} \, B c^{3} x^{\frac{5}{2}} + \frac{2}{3} \, A c^{3} x^{\frac{3}{2}} + 6 \, B a c^{2} \sqrt{x} - \frac{2 \,{\left (945 \, A a c^{2} x^{4} + 315 \, B a^{2} c x^{3} + 189 \, A a^{2} c x^{2} + 45 \, B a^{3} x + 35 \, A a^{3}\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(11/2),x, algorithm="giac")

[Out]

2/5*B*c^3*x^(5/2) + 2/3*A*c^3*x^(3/2) + 6*B*a*c^2*sqrt(x) - 2/315*(945*A*a*c^2*x^4 + 315*B*a^2*c*x^3 + 189*A*a
^2*c*x^2 + 45*B*a^3*x + 35*A*a^3)/x^(9/2)