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

Optimal. Leaf size=136 \[ -\frac{2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac{2 c^3 (b B-A c)}{b^5 \sqrt{x}}+\frac{2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{11/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{9 b x^{9/2}} \]

[Out]

(-2*A)/(9*b*x^(9/2)) - (2*(b*B - A*c))/(7*b^2*x^(7/2)) + (2*c*(b*B - A*c))/(5*b^3*x^(5/2)) - (2*c^2*(b*B - A*c
))/(3*b^4*x^(3/2)) + (2*c^3*(b*B - A*c))/(b^5*Sqrt[x]) + (2*c^(7/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[
b]])/b^(11/2)

________________________________________________________________________________________

Rubi [A]  time = 0.0800834, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {781, 78, 51, 63, 205} \[ -\frac{2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac{2 c^3 (b B-A c)}{b^5 \sqrt{x}}+\frac{2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{11/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{9 b x^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A)/(9*b*x^(9/2)) - (2*(b*B - A*c))/(7*b^2*x^(7/2)) + (2*c*(b*B - A*c))/(5*b^3*x^(5/2)) - (2*c^2*(b*B - A*c
))/(3*b^4*x^(3/2)) + (2*c^3*(b*B - A*c))/(b^5*Sqrt[x]) + (2*c^(7/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[
b]])/b^(11/2)

Rule 781

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/e^p, Int[(e
*x)^(m + p)*(f + g*x)*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m}, x] && IntegerQ[p]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{A+B x}{x^{9/2} \left (b x+c x^2\right )} \, dx &=\int \frac{A+B x}{x^{11/2} (b+c x)} \, dx\\ &=-\frac{2 A}{9 b x^{9/2}}+\frac{\left (2 \left (\frac{9 b B}{2}-\frac{9 A c}{2}\right )\right ) \int \frac{1}{x^{9/2} (b+c x)} \, dx}{9 b}\\ &=-\frac{2 A}{9 b x^{9/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{(c (b B-A c)) \int \frac{1}{x^{7/2} (b+c x)} \, dx}{b^2}\\ &=-\frac{2 A}{9 b x^{9/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}+\frac{\left (c^2 (b B-A c)\right ) \int \frac{1}{x^{5/2} (b+c x)} \, dx}{b^3}\\ &=-\frac{2 A}{9 b x^{9/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac{2 c^2 (b B-A c)}{3 b^4 x^{3/2}}-\frac{\left (c^3 (b B-A c)\right ) \int \frac{1}{x^{3/2} (b+c x)} \, dx}{b^4}\\ &=-\frac{2 A}{9 b x^{9/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac{2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac{2 c^3 (b B-A c)}{b^5 \sqrt{x}}+\frac{\left (c^4 (b B-A c)\right ) \int \frac{1}{\sqrt{x} (b+c x)} \, dx}{b^5}\\ &=-\frac{2 A}{9 b x^{9/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac{2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac{2 c^3 (b B-A c)}{b^5 \sqrt{x}}+\frac{\left (2 c^4 (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{1}{b+c x^2} \, dx,x,\sqrt{x}\right )}{b^5}\\ &=-\frac{2 A}{9 b x^{9/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac{2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac{2 c^3 (b B-A c)}{b^5 \sqrt{x}}+\frac{2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.0146791, size = 44, normalized size = 0.32 \[ \frac{2 \left (\, _2F_1\left (-\frac{7}{2},1;-\frac{5}{2};-\frac{c x}{b}\right ) (9 A c x-9 b B x)-7 A b\right )}{63 b^2 x^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-7*A*b + (-9*b*B*x + 9*A*c*x)*Hypergeometric2F1[-7/2, 1, -5/2, -((c*x)/b)]))/(63*b^2*x^(9/2))

________________________________________________________________________________________

Maple [A]  time = 0.014, size = 150, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{9\,b}{x}^{-{\frac{9}{2}}}}+{\frac{2\,Ac}{7\,{b}^{2}}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{7\,b}{x}^{-{\frac{7}{2}}}}-2\,{\frac{{c}^{4}A}{{b}^{5}\sqrt{x}}}+2\,{\frac{{c}^{3}B}{{b}^{4}\sqrt{x}}}-{\frac{2\,A{c}^{2}}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,Bc}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,{c}^{3}A}{3\,{b}^{4}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,{c}^{2}B}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{A{c}^{5}}{{b}^{5}\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \right ) }+2\,{\frac{{c}^{4}B}{{b}^{4}\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(9/2)/(c*x^2+b*x),x)

[Out]

-2/9*A/b/x^(9/2)+2/7/b^2/x^(7/2)*A*c-2/7/b/x^(7/2)*B-2/b^5*c^4/x^(1/2)*A+2/b^4*c^3/x^(1/2)*B-2/5/b^3*c^2/x^(5/
2)*A+2/5/b^2*c/x^(5/2)*B+2/3/b^4*c^3/x^(3/2)*A-2/3/b^3*c^2/x^(3/2)*B-2*c^5/b^5/(b*c)^(1/2)*arctan(x^(1/2)*c/(b
*c)^(1/2))*A+2*c^4/b^4/(b*c)^(1/2)*arctan(x^(1/2)*c/(b*c)^(1/2))*B

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(c*x^2+b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.67705, size = 641, normalized size = 4.71 \begin{align*} \left [-\frac{315 \,{\left (B b c^{3} - A c^{4}\right )} x^{5} \sqrt{-\frac{c}{b}} \log \left (\frac{c x - 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 2 \,{\left (35 \, A b^{4} - 315 \,{\left (B b c^{3} - A c^{4}\right )} x^{4} + 105 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 63 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} x^{2} + 45 \,{\left (B b^{4} - A b^{3} c\right )} x\right )} \sqrt{x}}{315 \, b^{5} x^{5}}, -\frac{2 \,{\left (315 \,{\left (B b c^{3} - A c^{4}\right )} x^{5} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) +{\left (35 \, A b^{4} - 315 \,{\left (B b c^{3} - A c^{4}\right )} x^{4} + 105 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 63 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} x^{2} + 45 \,{\left (B b^{4} - A b^{3} c\right )} x\right )} \sqrt{x}\right )}}{315 \, b^{5} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(c*x^2+b*x),x, algorithm="fricas")

[Out]

[-1/315*(315*(B*b*c^3 - A*c^4)*x^5*sqrt(-c/b)*log((c*x - 2*b*sqrt(x)*sqrt(-c/b) - b)/(c*x + b)) + 2*(35*A*b^4
- 315*(B*b*c^3 - A*c^4)*x^4 + 105*(B*b^2*c^2 - A*b*c^3)*x^3 - 63*(B*b^3*c - A*b^2*c^2)*x^2 + 45*(B*b^4 - A*b^3
*c)*x)*sqrt(x))/(b^5*x^5), -2/315*(315*(B*b*c^3 - A*c^4)*x^5*sqrt(c/b)*arctan(b*sqrt(c/b)/(c*sqrt(x))) + (35*A
*b^4 - 315*(B*b*c^3 - A*c^4)*x^4 + 105*(B*b^2*c^2 - A*b*c^3)*x^3 - 63*(B*b^3*c - A*b^2*c^2)*x^2 + 45*(B*b^4 -
A*b^3*c)*x)*sqrt(x))/(b^5*x^5)]

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.13942, size = 173, normalized size = 1.27 \begin{align*} \frac{2 \,{\left (B b c^{4} - A c^{5}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{5}} + \frac{2 \,{\left (315 \, B b c^{3} x^{4} - 315 \, A c^{4} x^{4} - 105 \, B b^{2} c^{2} x^{3} + 105 \, A b c^{3} x^{3} + 63 \, B b^{3} c x^{2} - 63 \, A b^{2} c^{2} x^{2} - 45 \, B b^{4} x + 45 \, A b^{3} c x - 35 \, A b^{4}\right )}}{315 \, b^{5} x^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(c*x^2+b*x),x, algorithm="giac")

[Out]

2*(B*b*c^4 - A*c^5)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b^5) + 2/315*(315*B*b*c^3*x^4 - 315*A*c^4*x^4 - 105
*B*b^2*c^2*x^3 + 105*A*b*c^3*x^3 + 63*B*b^3*c*x^2 - 63*A*b^2*c^2*x^2 - 45*B*b^4*x + 45*A*b^3*c*x - 35*A*b^4)/(
b^5*x^(9/2))