3.400 \(\int \frac{(A+B x) (a+c x^2)^2}{x^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0255857, 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} \[ -\frac{2 a^2 A}{5 x^{5/2}}-\frac{2 a^2 B}{3 x^{3/2}}-\frac{4 a A c}{\sqrt{x}}+4 a B c \sqrt{x}+\frac{2}{3} A c^2 x^{3/2}+\frac{2}{5} B c^2 x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/x^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.0217361, size = 53, normalized size = 0.73 \[ \frac{-2 a^2 (3 A+5 B x)+60 a c x^2 (B x-A)+2 c^2 x^4 (5 A+3 B x)}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^2)/x^(7/2),x]

[Out]

(60*a*c*x^2*(-A + B*x) + 2*c^2*x^4*(5*A + 3*B*x) - 2*a^2*(3*A + 5*B*x))/(15*x^(5/2))

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Maple [A]  time = 0.006, size = 54, normalized size = 0.7 \begin{align*} -{\frac{-6\,B{c}^{2}{x}^{5}-10\,A{c}^{2}{x}^{4}-60\,aBc{x}^{3}+60\,aAc{x}^{2}+10\,{a}^{2}Bx+6\,A{a}^{2}}{15}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2/x^(7/2),x)

[Out]

-2/15*(-3*B*c^2*x^5-5*A*c^2*x^4-30*B*a*c*x^3+30*A*a*c*x^2+5*B*a^2*x+3*A*a^2)/x^(5/2)

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Maxima [A]  time = 1.02888, size = 73, normalized size = 1. \begin{align*} \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, A c^{2} x^{\frac{3}{2}} + 4 \, B a c \sqrt{x} - \frac{2 \,{\left (30 \, A a c x^{2} + 5 \, B a^{2} x + 3 \, A a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/x^(7/2),x, algorithm="maxima")

[Out]

2/5*B*c^2*x^(5/2) + 2/3*A*c^2*x^(3/2) + 4*B*a*c*sqrt(x) - 2/15*(30*A*a*c*x^2 + 5*B*a^2*x + 3*A*a^2)/x^(5/2)

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Fricas [A]  time = 1.28849, size = 127, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (3 \, B c^{2} x^{5} + 5 \, A c^{2} x^{4} + 30 \, B a c x^{3} - 30 \, A a c x^{2} - 5 \, B a^{2} x - 3 \, A a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/x^(7/2),x, algorithm="fricas")

[Out]

2/15*(3*B*c^2*x^5 + 5*A*c^2*x^4 + 30*B*a*c*x^3 - 30*A*a*c*x^2 - 5*B*a^2*x - 3*A*a^2)/x^(5/2)

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Sympy [A]  time = 4.97412, size = 76, normalized size = 1.04 \begin{align*} - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 A a c}{\sqrt{x}} + \frac{2 A c^{2} x^{\frac{3}{2}}}{3} - \frac{2 B a^{2}}{3 x^{\frac{3}{2}}} + 4 B a c \sqrt{x} + \frac{2 B c^{2} 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)**2/x**(7/2),x)

[Out]

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

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Giac [A]  time = 1.1547, size = 73, normalized size = 1. \begin{align*} \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, A c^{2} x^{\frac{3}{2}} + 4 \, B a c \sqrt{x} - \frac{2 \,{\left (30 \, A a c x^{2} + 5 \, B a^{2} x + 3 \, A a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/x^(7/2),x, algorithm="giac")

[Out]

2/5*B*c^2*x^(5/2) + 2/3*A*c^2*x^(3/2) + 4*B*a*c*sqrt(x) - 2/15*(30*A*a*c*x^2 + 5*B*a^2*x + 3*A*a^2)/x^(5/2)