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

Optimal. Leaf size=43 \[ -\frac{2 a A}{7 x^{7/2}}-\frac{2 a B}{5 x^{5/2}}-\frac{2 A c}{3 x^{3/2}}-\frac{2 B c}{\sqrt{x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0120947, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ -\frac{2 a A}{7 x^{7/2}}-\frac{2 a B}{5 x^{5/2}}-\frac{2 A c}{3 x^{3/2}}-\frac{2 B c}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0146186, size = 33, normalized size = 0.77 \[ \frac{-6 a (5 A+7 B x)-70 c x^2 (A+3 B x)}{105 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 30, normalized size = 0.7 \begin{align*} -{\frac{210\,Bc{x}^{3}+70\,Ac{x}^{2}+42\,aBx+30\,aA}{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)/x^(9/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00226, size = 39, normalized size = 0.91 \begin{align*} -\frac{2 \,{\left (105 \, B c x^{3} + 35 \, A c x^{2} + 21 \, B a x + 15 \, A a\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)/x^(9/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.28399, size = 85, normalized size = 1.98 \begin{align*} -\frac{2 \,{\left (105 \, B c x^{3} + 35 \, A c x^{2} + 21 \, B a x + 15 \, A a\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)/x^(9/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 4.12499, size = 46, normalized size = 1.07 \begin{align*} - \frac{2 A a}{7 x^{\frac{7}{2}}} - \frac{2 A c}{3 x^{\frac{3}{2}}} - \frac{2 B a}{5 x^{\frac{5}{2}}} - \frac{2 B c}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.32025, size = 39, normalized size = 0.91 \begin{align*} -\frac{2 \,{\left (105 \, B c x^{3} + 35 \, A c x^{2} + 21 \, B a x + 15 \, A a\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)/x^(9/2),x, algorithm="giac")

[Out]

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