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

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

[Out]

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

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Rubi [A]  time = 0.0235656, 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} \[ -\frac{2 a^2 A}{3 x^{3/2}}-\frac{2 a^2 B}{\sqrt{x}}+4 a A c \sqrt{x}+\frac{4}{3} a B c x^{3/2}+\frac{2}{5} A c^2 x^{5/2}+\frac{2}{7} B c^2 x^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0195033, size = 51, normalized size = 0.7 \[ \frac{-70 a^2 (A+3 B x)+140 a c x^2 (3 A+B x)+6 c^2 x^4 (7 A+5 B x)}{105 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(140*a*c*x^2*(3*A + B*x) - 70*a^2*(A + 3*B*x) + 6*c^2*x^4*(7*A + 5*B*x))/(105*x^(3/2))

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Maple [A]  time = 0.006, size = 54, normalized size = 0.7 \begin{align*} -{\frac{-30\,B{c}^{2}{x}^{5}-42\,A{c}^{2}{x}^{4}-140\,aBc{x}^{3}-420\,aAc{x}^{2}+210\,{a}^{2}Bx+70\,A{a}^{2}}{105}{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)^2/x^(5/2),x)

[Out]

-2/105*(-15*B*c^2*x^5-21*A*c^2*x^4-70*B*a*c*x^3-210*A*a*c*x^2+105*B*a^2*x+35*A*a^2)/x^(3/2)

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Maxima [A]  time = 1.00669, size = 72, normalized size = 0.99 \begin{align*} \frac{2}{7} \, B c^{2} x^{\frac{7}{2}} + \frac{2}{5} \, A c^{2} x^{\frac{5}{2}} + \frac{4}{3} \, B a c x^{\frac{3}{2}} + 4 \, A a c \sqrt{x} - \frac{2 \,{\left (3 \, B a^{2} x + A a^{2}\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)^2/x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.30192, size = 136, normalized size = 1.86 \begin{align*} \frac{2 \,{\left (15 \, B c^{2} x^{5} + 21 \, A c^{2} x^{4} + 70 \, B a c x^{3} + 210 \, A a c x^{2} - 105 \, B a^{2} x - 35 \, A a^{2}\right )}}{105 \, 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)^2/x^(5/2),x, algorithm="fricas")

[Out]

2/105*(15*B*c^2*x^5 + 21*A*c^2*x^4 + 70*B*a*c*x^3 + 210*A*a*c*x^2 - 105*B*a^2*x - 35*A*a^2)/x^(3/2)

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Sympy [A]  time = 4.16685, size = 76, normalized size = 1.04 \begin{align*} - \frac{2 A a^{2}}{3 x^{\frac{3}{2}}} + 4 A a c \sqrt{x} + \frac{2 A c^{2} x^{\frac{5}{2}}}{5} - \frac{2 B a^{2}}{\sqrt{x}} + \frac{4 B a c x^{\frac{3}{2}}}{3} + \frac{2 B c^{2} 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)**2/x**(5/2),x)

[Out]

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

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Giac [A]  time = 1.14818, size = 72, normalized size = 0.99 \begin{align*} \frac{2}{7} \, B c^{2} x^{\frac{7}{2}} + \frac{2}{5} \, A c^{2} x^{\frac{5}{2}} + \frac{4}{3} \, B a c x^{\frac{3}{2}} + 4 \, A a c \sqrt{x} - \frac{2 \,{\left (3 \, B a^{2} x + A a^{2}\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)^2/x^(5/2),x, algorithm="giac")

[Out]

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