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

Optimal. Leaf size=73 \[ -\frac{2 a^2 A}{\sqrt{x}}+2 a^2 B \sqrt{x}+\frac{4}{3} a A c x^{3/2}+\frac{4}{5} a B c x^{5/2}+\frac{2}{7} A c^2 x^{7/2}+\frac{2}{9} B c^2 x^{9/2} \]

[Out]

(-2*a^2*A)/Sqrt[x] + 2*a^2*B*Sqrt[x] + (4*a*A*c*x^(3/2))/3 + (4*a*B*c*x^(5/2))/5 + (2*A*c^2*x^(7/2))/7 + (2*B*
c^2*x^(9/2))/9

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Rubi [A]  time = 0.0246833, antiderivative size = 73, 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} \[ -\frac{2 a^2 A}{\sqrt{x}}+2 a^2 B \sqrt{x}+\frac{4}{3} a A c x^{3/2}+\frac{4}{5} a B c x^{5/2}+\frac{2}{7} A c^2 x^{7/2}+\frac{2}{9} B c^2 x^{9/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*A)/Sqrt[x] + 2*a^2*B*Sqrt[x] + (4*a*A*c*x^(3/2))/3 + (4*a*B*c*x^(5/2))/5 + (2*A*c^2*x^(7/2))/7 + (2*B*
c^2*x^(9/2))/9

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 )^2}{x^{3/2}} \, dx &=\int \left (\frac{a^2 A}{x^{3/2}}+\frac{a^2 B}{\sqrt{x}}+2 a A c \sqrt{x}+2 a B c x^{3/2}+A c^2 x^{5/2}+B c^2 x^{7/2}\right ) \, dx\\ &=-\frac{2 a^2 A}{\sqrt{x}}+2 a^2 B \sqrt{x}+\frac{4}{3} a A c x^{3/2}+\frac{4}{5} a B c x^{5/2}+\frac{2}{7} A c^2 x^{7/2}+\frac{2}{9} B c^2 x^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0198524, size = 52, normalized size = 0.71 \[ \frac{-630 a^2 (A-B x)+84 a c x^2 (5 A+3 B x)+10 c^2 x^4 (9 A+7 B x)}{315 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-630*a^2*(A - B*x) + 84*a*c*x^2*(5*A + 3*B*x) + 10*c^2*x^4*(9*A + 7*B*x))/(315*Sqrt[x])

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Maple [A]  time = 0.005, size = 54, normalized size = 0.7 \begin{align*} -{\frac{-70\,B{c}^{2}{x}^{5}-90\,A{c}^{2}{x}^{4}-252\,aBc{x}^{3}-420\,aAc{x}^{2}-630\,{a}^{2}Bx+630\,A{a}^{2}}{315}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(-35*B*c^2*x^5-45*A*c^2*x^4-126*B*a*c*x^3-210*A*a*c*x^2-315*B*a^2*x+315*A*a^2)/x^(1/2)

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Maxima [A]  time = 0.997461, size = 72, normalized size = 0.99 \begin{align*} \frac{2}{9} \, B c^{2} x^{\frac{9}{2}} + \frac{2}{7} \, A c^{2} x^{\frac{7}{2}} + \frac{4}{5} \, B a c x^{\frac{5}{2}} + \frac{4}{3} \, A a c x^{\frac{3}{2}} + 2 \, B a^{2} \sqrt{x} - \frac{2 \, A a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*B*c^2*x^(9/2) + 2/7*A*c^2*x^(7/2) + 4/5*B*a*c*x^(5/2) + 4/3*A*a*c*x^(3/2) + 2*B*a^2*sqrt(x) - 2*A*a^2/sqrt
(x)

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Fricas [A]  time = 1.25565, size = 139, normalized size = 1.9 \begin{align*} \frac{2 \,{\left (35 \, B c^{2} x^{5} + 45 \, A c^{2} x^{4} + 126 \, B a c x^{3} + 210 \, A a c x^{2} + 315 \, B a^{2} x - 315 \, A a^{2}\right )}}{315 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*B*c^2*x^5 + 45*A*c^2*x^4 + 126*B*a*c*x^3 + 210*A*a*c*x^2 + 315*B*a^2*x - 315*A*a^2)/sqrt(x)

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Sympy [A]  time = 2.69035, size = 76, normalized size = 1.04 \begin{align*} - \frac{2 A a^{2}}{\sqrt{x}} + \frac{4 A a c x^{\frac{3}{2}}}{3} + \frac{2 A c^{2} x^{\frac{7}{2}}}{7} + 2 B a^{2} \sqrt{x} + \frac{4 B a c x^{\frac{5}{2}}}{5} + \frac{2 B c^{2} x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**2/sqrt(x) + 4*A*a*c*x**(3/2)/3 + 2*A*c**2*x**(7/2)/7 + 2*B*a**2*sqrt(x) + 4*B*a*c*x**(5/2)/5 + 2*B*c**
2*x**(9/2)/9

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Giac [A]  time = 1.22124, size = 72, normalized size = 0.99 \begin{align*} \frac{2}{9} \, B c^{2} x^{\frac{9}{2}} + \frac{2}{7} \, A c^{2} x^{\frac{7}{2}} + \frac{4}{5} \, B a c x^{\frac{5}{2}} + \frac{4}{3} \, A a c x^{\frac{3}{2}} + 2 \, B a^{2} \sqrt{x} - \frac{2 \, A a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*B*c^2*x^(9/2) + 2/7*A*c^2*x^(7/2) + 4/5*B*a*c*x^(5/2) + 4/3*A*a*c*x^(3/2) + 2*B*a^2*sqrt(x) - 2*A*a^2/sqrt
(x)