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

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

[Out]

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

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Rubi [A]  time = 0.0243161, 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} \[ -\frac{2 a^2 A}{7 x^{7/2}}-\frac{2 a^2 B}{5 x^{5/2}}-\frac{4 a A c}{3 x^{3/2}}-\frac{4 a B c}{\sqrt{x}}+2 A c^2 \sqrt{x}+\frac{2}{3} B c^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0219592, size = 51, normalized size = 0.7 \[ \frac{-6 a^2 (5 A+7 B x)-140 a c x^2 (A+3 B x)+70 c^2 x^4 (3 A+B x)}{105 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 54, normalized size = 0.7 \begin{align*} -{\frac{-70\,B{c}^{2}{x}^{5}-210\,A{c}^{2}{x}^{4}+420\,aBc{x}^{3}+140\,aAc{x}^{2}+42\,{a}^{2}Bx+30\,A{a}^{2}}{105}{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)^2/x^(9/2),x)

[Out]

-2/105*(-35*B*c^2*x^5-105*A*c^2*x^4+210*B*a*c*x^3+70*A*a*c*x^2+21*B*a^2*x+15*A*a^2)/x^(7/2)

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Maxima [A]  time = 1.00801, size = 73, normalized size = 1. \begin{align*} \frac{2}{3} \, B c^{2} x^{\frac{3}{2}} + 2 \, A c^{2} \sqrt{x} - \frac{2 \,{\left (210 \, B a c x^{3} + 70 \, A a c x^{2} + 21 \, B a^{2} x + 15 \, A a^{2}\right )}}{105 \, 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)^2/x^(9/2),x, algorithm="maxima")

[Out]

2/3*B*c^2*x^(3/2) + 2*A*c^2*sqrt(x) - 2/105*(210*B*a*c*x^3 + 70*A*a*c*x^2 + 21*B*a^2*x + 15*A*a^2)/x^(7/2)

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Fricas [A]  time = 1.23627, size = 136, normalized size = 1.86 \begin{align*} \frac{2 \,{\left (35 \, B c^{2} x^{5} + 105 \, A c^{2} x^{4} - 210 \, B a c x^{3} - 70 \, A a c x^{2} - 21 \, B a^{2} x - 15 \, A a^{2}\right )}}{105 \, 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)^2/x^(9/2),x, algorithm="fricas")

[Out]

2/105*(35*B*c^2*x^5 + 105*A*c^2*x^4 - 210*B*a*c*x^3 - 70*A*a*c*x^2 - 21*B*a^2*x - 15*A*a^2)/x^(7/2)

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Sympy [A]  time = 6.57587, size = 76, normalized size = 1.04 \begin{align*} - \frac{2 A a^{2}}{7 x^{\frac{7}{2}}} - \frac{4 A a c}{3 x^{\frac{3}{2}}} + 2 A c^{2} \sqrt{x} - \frac{2 B a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 B a c}{\sqrt{x}} + \frac{2 B c^{2} x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1575, size = 73, normalized size = 1. \begin{align*} \frac{2}{3} \, B c^{2} x^{\frac{3}{2}} + 2 \, A c^{2} \sqrt{x} - \frac{2 \,{\left (210 \, B a c x^{3} + 70 \, A a c x^{2} + 21 \, B a^{2} x + 15 \, A a^{2}\right )}}{105 \, 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)^2/x^(9/2),x, algorithm="giac")

[Out]

2/3*B*c^2*x^(3/2) + 2*A*c^2*sqrt(x) - 2/105*(210*B*a*c*x^3 + 70*A*a*c*x^2 + 21*B*a^2*x + 15*A*a^2)/x^(7/2)