3.410 \(\int \frac{(A+B x) (a+c x^2)^3}{x^{9/2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{2 a^2 A c}{x^{3/2}}-\frac{2 a^3 A}{7 x^{7/2}}-\frac{6 a^2 B c}{\sqrt{x}}-\frac{2 a^3 B}{5 x^{5/2}}+6 a A c^2 \sqrt{x}+2 a B c^2 x^{3/2}+\frac{2}{5} A c^3 x^{5/2}+\frac{2}{7} B c^3 x^{7/2} \]

[Out]

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

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Rubi [A]  time = 0.036794, antiderivative size = 101, 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} \[ -\frac{2 a^2 A c}{x^{3/2}}-\frac{2 a^3 A}{7 x^{7/2}}-\frac{6 a^2 B c}{\sqrt{x}}-\frac{2 a^3 B}{5 x^{5/2}}+6 a A c^2 \sqrt{x}+2 a B c^2 x^{3/2}+\frac{2}{5} A c^3 x^{5/2}+\frac{2}{7} B c^3 x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/x^(9/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0289716, size = 70, normalized size = 0.69 \[ \frac{-70 a^2 c x^2 (A+3 B x)-2 a^3 (5 A+7 B x)+70 a c^2 x^4 (3 A+B x)+2 c^3 x^6 (7 A+5 B x)}{35 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/x^(9/2),x]

[Out]

(70*a*c^2*x^4*(3*A + B*x) - 70*a^2*c*x^2*(A + 3*B*x) + 2*c^3*x^6*(7*A + 5*B*x) - 2*a^3*(5*A + 7*B*x))/(35*x^(7
/2))

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Maple [A]  time = 0.005, size = 78, normalized size = 0.8 \begin{align*} -{\frac{-10\,B{c}^{3}{x}^{7}-14\,A{c}^{3}{x}^{6}-70\,aB{c}^{2}{x}^{5}-210\,aA{c}^{2}{x}^{4}+210\,{a}^{2}Bc{x}^{3}+70\,{a}^{2}Ac{x}^{2}+14\,{a}^{3}Bx+10\,A{a}^{3}}{35}{x}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/x^(9/2),x)

[Out]

-2/35*(-5*B*c^3*x^7-7*A*c^3*x^6-35*B*a*c^2*x^5-105*A*a*c^2*x^4+105*B*a^2*c*x^3+35*A*a^2*c*x^2+7*B*a^3*x+5*A*a^
3)/x^(7/2)

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Maxima [A]  time = 0.987694, size = 105, normalized size = 1.04 \begin{align*} \frac{2}{7} \, B c^{3} x^{\frac{7}{2}} + \frac{2}{5} \, A c^{3} x^{\frac{5}{2}} + 2 \, B a c^{2} x^{\frac{3}{2}} + 6 \, A a c^{2} \sqrt{x} - \frac{2 \,{\left (105 \, B a^{2} c x^{3} + 35 \, A a^{2} c x^{2} + 7 \, B a^{3} x + 5 \, A a^{3}\right )}}{35 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(9/2),x, algorithm="maxima")

[Out]

2/7*B*c^3*x^(7/2) + 2/5*A*c^3*x^(5/2) + 2*B*a*c^2*x^(3/2) + 6*A*a*c^2*sqrt(x) - 2/35*(105*B*a^2*c*x^3 + 35*A*a
^2*c*x^2 + 7*B*a^3*x + 5*A*a^3)/x^(7/2)

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Fricas [A]  time = 1.35794, size = 181, normalized size = 1.79 \begin{align*} \frac{2 \,{\left (5 \, B c^{3} x^{7} + 7 \, A c^{3} x^{6} + 35 \, B a c^{2} x^{5} + 105 \, A a c^{2} x^{4} - 105 \, B a^{2} c x^{3} - 35 \, A a^{2} c x^{2} - 7 \, B a^{3} x - 5 \, A a^{3}\right )}}{35 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(9/2),x, algorithm="fricas")

[Out]

2/35*(5*B*c^3*x^7 + 7*A*c^3*x^6 + 35*B*a*c^2*x^5 + 105*A*a*c^2*x^4 - 105*B*a^2*c*x^3 - 35*A*a^2*c*x^2 - 7*B*a^
3*x - 5*A*a^3)/x^(7/2)

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Sympy [A]  time = 12.5567, size = 107, normalized size = 1.06 \begin{align*} - \frac{2 A a^{3}}{7 x^{\frac{7}{2}}} - \frac{2 A a^{2} c}{x^{\frac{3}{2}}} + 6 A a c^{2} \sqrt{x} + \frac{2 A c^{3} x^{\frac{5}{2}}}{5} - \frac{2 B a^{3}}{5 x^{\frac{5}{2}}} - \frac{6 B a^{2} c}{\sqrt{x}} + 2 B a c^{2} x^{\frac{3}{2}} + \frac{2 B c^{3} x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/x**(9/2),x)

[Out]

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

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Giac [A]  time = 1.10923, size = 105, normalized size = 1.04 \begin{align*} \frac{2}{7} \, B c^{3} x^{\frac{7}{2}} + \frac{2}{5} \, A c^{3} x^{\frac{5}{2}} + 2 \, B a c^{2} x^{\frac{3}{2}} + 6 \, A a c^{2} \sqrt{x} - \frac{2 \,{\left (105 \, B a^{2} c x^{3} + 35 \, A a^{2} c x^{2} + 7 \, B a^{3} x + 5 \, A a^{3}\right )}}{35 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(9/2),x, algorithm="giac")

[Out]

2/7*B*c^3*x^(7/2) + 2/5*A*c^3*x^(5/2) + 2*B*a*c^2*x^(3/2) + 6*A*a*c^2*sqrt(x) - 2/35*(105*B*a^2*c*x^3 + 35*A*a
^2*c*x^2 + 7*B*a^3*x + 5*A*a^3)/x^(7/2)