3.781 \(\int \frac{A+B x}{x^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=240 \[ \frac{231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}-\frac{77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac{33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{231 (13 A b-3 a B)}{128 a^7 \sqrt{x}}+\frac{231 \sqrt{b} (13 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{15/2}}+\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5} \]

[Out]

(-77*(13*A*b - 3*a*B))/(128*a^6*b*x^(3/2)) + (231*(13*A*b - 3*a*B))/(128*a^7*Sqrt[x]) + (A*b - a*B)/(5*a*b*x^(
3/2)*(a + b*x)^5) + (13*A*b - 3*a*B)/(40*a^2*b*x^(3/2)*(a + b*x)^4) + (11*(13*A*b - 3*a*B))/(240*a^3*b*x^(3/2)
*(a + b*x)^3) + (33*(13*A*b - 3*a*B))/(320*a^4*b*x^(3/2)*(a + b*x)^2) + (231*(13*A*b - 3*a*B))/(640*a^5*b*x^(3
/2)*(a + b*x)) + (231*Sqrt[b]*(13*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(15/2))

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Rubi [A]  time = 0.116044, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {27, 78, 51, 63, 205} \[ \frac{231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}-\frac{77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac{33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{231 (13 A b-3 a B)}{128 a^7 \sqrt{x}}+\frac{231 \sqrt{b} (13 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{15/2}}+\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-77*(13*A*b - 3*a*B))/(128*a^6*b*x^(3/2)) + (231*(13*A*b - 3*a*B))/(128*a^7*Sqrt[x]) + (A*b - a*B)/(5*a*b*x^(
3/2)*(a + b*x)^5) + (13*A*b - 3*a*B)/(40*a^2*b*x^(3/2)*(a + b*x)^4) + (11*(13*A*b - 3*a*B))/(240*a^3*b*x^(3/2)
*(a + b*x)^3) + (33*(13*A*b - 3*a*B))/(320*a^4*b*x^(3/2)*(a + b*x)^2) + (231*(13*A*b - 3*a*B))/(640*a^5*b*x^(3
/2)*(a + b*x)) + (231*Sqrt[b]*(13*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(15/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && 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^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{A+B x}{x^{5/2} (a+b x)^6} \, dx\\ &=\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5}-\frac{\left (-\frac{13 A b}{2}+\frac{3 a B}{2}\right ) \int \frac{1}{x^{5/2} (a+b x)^5} \, dx}{5 a b}\\ &=\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{(11 (13 A b-3 a B)) \int \frac{1}{x^{5/2} (a+b x)^4} \, dx}{80 a^2 b}\\ &=\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{(33 (13 A b-3 a B)) \int \frac{1}{x^{5/2} (a+b x)^3} \, dx}{160 a^3 b}\\ &=\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac{(231 (13 A b-3 a B)) \int \frac{1}{x^{5/2} (a+b x)^2} \, dx}{640 a^4 b}\\ &=\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac{231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac{(231 (13 A b-3 a B)) \int \frac{1}{x^{5/2} (a+b x)} \, dx}{256 a^5 b}\\ &=-\frac{77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac{231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}-\frac{(231 (13 A b-3 a B)) \int \frac{1}{x^{3/2} (a+b x)} \, dx}{256 a^6}\\ &=-\frac{77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac{231 (13 A b-3 a B)}{128 a^7 \sqrt{x}}+\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac{231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac{(231 b (13 A b-3 a B)) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{256 a^7}\\ &=-\frac{77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac{231 (13 A b-3 a B)}{128 a^7 \sqrt{x}}+\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac{231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac{(231 b (13 A b-3 a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{128 a^7}\\ &=-\frac{77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac{231 (13 A b-3 a B)}{128 a^7 \sqrt{x}}+\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac{231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac{231 \sqrt{b} (13 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{15/2}}\\ \end{align*}

Mathematica [C]  time = 0.0337671, size = 61, normalized size = 0.25 \[ \frac{\frac{3 a^5 (A b-a B)}{(a+b x)^5}+(3 a B-13 A b) \, _2F_1\left (-\frac{3}{2},5;-\frac{1}{2};-\frac{b x}{a}\right )}{15 a^6 b x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((3*a^5*(A*b - a*B))/(a + b*x)^5 + (-13*A*b + 3*a*B)*Hypergeometric2F1[-3/2, 5, -1/2, -((b*x)/a)])/(15*a^6*b*x
^(3/2))

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Maple [A]  time = 0.025, size = 266, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{3\,{a}^{6}}{x}^{-{\frac{3}{2}}}}+12\,{\frac{Ab}{{a}^{7}\sqrt{x}}}-2\,{\frac{B}{{a}^{6}\sqrt{x}}}+{\frac{1467\,{b}^{6}A}{128\,{a}^{7} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}-{\frac{437\,{b}^{5}B}{128\,{a}^{6} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}+{\frac{9629\,A{b}^{5}}{192\,{a}^{6} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}-{\frac{977\,{b}^{4}B}{64\,{a}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}+{\frac{1253\,{b}^{4}A}{15\,{a}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}-{\frac{131\,{b}^{3}B}{5\,{a}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}+{\frac{12131\,A{b}^{3}}{192\,{a}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}-{\frac{1327\,{b}^{2}B}{64\,{a}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}+{\frac{2373\,A{b}^{2}}{128\,{a}^{3} \left ( bx+a \right ) ^{5}}\sqrt{x}}-{\frac{843\,bB}{128\,{a}^{2} \left ( bx+a \right ) ^{5}}\sqrt{x}}+{\frac{3003\,A{b}^{2}}{128\,{a}^{7}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{693\,bB}{128\,{a}^{6}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-2/3*A/a^6/x^(3/2)+12/a^7/x^(1/2)*A*b-2/a^6/x^(1/2)*B+1467/128/a^7*b^6/(b*x+a)^5*x^(9/2)*A-437/128/a^6*b^5/(b*
x+a)^5*x^(9/2)*B+9629/192/a^6*b^5/(b*x+a)^5*A*x^(7/2)-977/64/a^5*b^4/(b*x+a)^5*B*x^(7/2)+1253/15/a^5*b^4/(b*x+
a)^5*x^(5/2)*A-131/5/a^4*b^3/(b*x+a)^5*x^(5/2)*B+12131/192/a^4*b^3/(b*x+a)^5*x^(3/2)*A-1327/64/a^3*b^2/(b*x+a)
^5*x^(3/2)*B+2373/128/a^3*b^2/(b*x+a)^5*x^(1/2)*A-843/128/a^2*b/(b*x+a)^5*x^(1/2)*B+3003/128/a^7*b^2/(a*b)^(1/
2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A-693/128/a^6*b/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69816, size = 1658, normalized size = 6.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(3465*((3*B*a*b^5 - 13*A*b^6)*x^7 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*x^6 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4
)*x^5 + 10*(3*B*a^4*b^2 - 13*A*a^3*b^3)*x^4 + 5*(3*B*a^5*b - 13*A*a^4*b^2)*x^3 + (3*B*a^6 - 13*A*a^5*b)*x^2)*s
qrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(1280*A*a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6
+ 16170*(3*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 29568*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 26070*(3*B*a^4*b^2 - 13*A*a^
3*b^3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280*(3*B*a^6 - 13*A*a^5*b)*x)*sqrt(x))/(a^7*b^5*x^7 + 5*a
^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10*a^10*b^2*x^4 + 5*a^11*b*x^3 + a^12*x^2), 1/1920*(3465*((3*B*a*b^5 - 13*A*b^6)
*x^7 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*x^6 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^5 + 10*(3*B*a^4*b^2 - 13*A*a^3*b^3
)*x^4 + 5*(3*B*a^5*b - 13*A*a^4*b^2)*x^3 + (3*B*a^6 - 13*A*a^5*b)*x^2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x)
)) - (1280*A*a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6 + 16170*(3*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 29568*(3*B*a^3*b^3
- 13*A*a^2*b^4)*x^4 + 26070*(3*B*a^4*b^2 - 13*A*a^3*b^3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280*(3
*B*a^6 - 13*A*a^5*b)*x)*sqrt(x))/(a^7*b^5*x^7 + 5*a^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10*a^10*b^2*x^4 + 5*a^11*b*x^
3 + a^12*x^2)]

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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**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.14449, size = 243, normalized size = 1.01 \begin{align*} -\frac{231 \,{\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} a^{7}} - \frac{2 \,{\left (3 \, B a x - 18 \, A b x + A a\right )}}{3 \, a^{7} x^{\frac{3}{2}}} - \frac{6555 \, B a b^{5} x^{\frac{9}{2}} - 22005 \, A b^{6} x^{\frac{9}{2}} + 29310 \, B a^{2} b^{4} x^{\frac{7}{2}} - 96290 \, A a b^{5} x^{\frac{7}{2}} + 50304 \, B a^{3} b^{3} x^{\frac{5}{2}} - 160384 \, A a^{2} b^{4} x^{\frac{5}{2}} + 39810 \, B a^{4} b^{2} x^{\frac{3}{2}} - 121310 \, A a^{3} b^{3} x^{\frac{3}{2}} + 12645 \, B a^{5} b \sqrt{x} - 35595 \, A a^{4} b^{2} \sqrt{x}}{1920 \,{\left (b x + a\right )}^{5} a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-231/128*(3*B*a*b - 13*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7) - 2/3*(3*B*a*x - 18*A*b*x + A*a)/(a^
7*x^(3/2)) - 1/1920*(6555*B*a*b^5*x^(9/2) - 22005*A*b^6*x^(9/2) + 29310*B*a^2*b^4*x^(7/2) - 96290*A*a*b^5*x^(7
/2) + 50304*B*a^3*b^3*x^(5/2) - 160384*A*a^2*b^4*x^(5/2) + 39810*B*a^4*b^2*x^(3/2) - 121310*A*a^3*b^3*x^(3/2)
+ 12645*B*a^5*b*sqrt(x) - 35595*A*a^4*b^2*sqrt(x))/((b*x + a)^5*a^7)