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

Optimal. Leaf size=219 \[ -\frac{63 (11 A b-a B)}{128 a^6 b \sqrt{x}}+\frac{21 (11 A b-a B)}{128 a^5 b \sqrt{x} (a+b x)}+\frac{21 (11 A b-a B)}{320 a^4 b \sqrt{x} (a+b x)^2}+\frac{3 (11 A b-a B)}{80 a^3 b \sqrt{x} (a+b x)^3}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}-\frac{63 (11 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{13/2} \sqrt{b}}+\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5} \]

[Out]

(-63*(11*A*b - a*B))/(128*a^6*b*Sqrt[x]) + (A*b - a*B)/(5*a*b*Sqrt[x]*(a + b*x)^5) + (11*A*b - a*B)/(40*a^2*b*
Sqrt[x]*(a + b*x)^4) + (3*(11*A*b - a*B))/(80*a^3*b*Sqrt[x]*(a + b*x)^3) + (21*(11*A*b - a*B))/(320*a^4*b*Sqrt
[x]*(a + b*x)^2) + (21*(11*A*b - a*B))/(128*a^5*b*Sqrt[x]*(a + b*x)) - (63*(11*A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt
[x])/Sqrt[a]])/(128*a^(13/2)*Sqrt[b])

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Rubi [A]  time = 0.0977657, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{63 (11 A b-a B)}{128 a^6 b \sqrt{x}}+\frac{21 (11 A b-a B)}{128 a^5 b \sqrt{x} (a+b x)}+\frac{21 (11 A b-a B)}{320 a^4 b \sqrt{x} (a+b x)^2}+\frac{3 (11 A b-a B)}{80 a^3 b \sqrt{x} (a+b x)^3}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}-\frac{63 (11 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{13/2} \sqrt{b}}+\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-63*(11*A*b - a*B))/(128*a^6*b*Sqrt[x]) + (A*b - a*B)/(5*a*b*Sqrt[x]*(a + b*x)^5) + (11*A*b - a*B)/(40*a^2*b*
Sqrt[x]*(a + b*x)^4) + (3*(11*A*b - a*B))/(80*a^3*b*Sqrt[x]*(a + b*x)^3) + (21*(11*A*b - a*B))/(320*a^4*b*Sqrt
[x]*(a + b*x)^2) + (21*(11*A*b - a*B))/(128*a^5*b*Sqrt[x]*(a + b*x)) - (63*(11*A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt
[x])/Sqrt[a]])/(128*a^(13/2)*Sqrt[b])

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^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{A+B x}{x^{3/2} (a+b x)^6} \, dx\\ &=\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5}+\frac{(11 A b-a B) \int \frac{1}{x^{3/2} (a+b x)^5} \, dx}{10 a b}\\ &=\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}+\frac{(9 (11 A b-a B)) \int \frac{1}{x^{3/2} (a+b x)^4} \, dx}{80 a^2 b}\\ &=\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}+\frac{3 (11 A b-a B)}{80 a^3 b \sqrt{x} (a+b x)^3}+\frac{(21 (11 A b-a B)) \int \frac{1}{x^{3/2} (a+b x)^3} \, dx}{160 a^3 b}\\ &=\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}+\frac{3 (11 A b-a B)}{80 a^3 b \sqrt{x} (a+b x)^3}+\frac{21 (11 A b-a B)}{320 a^4 b \sqrt{x} (a+b x)^2}+\frac{(21 (11 A b-a B)) \int \frac{1}{x^{3/2} (a+b x)^2} \, dx}{128 a^4 b}\\ &=\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}+\frac{3 (11 A b-a B)}{80 a^3 b \sqrt{x} (a+b x)^3}+\frac{21 (11 A b-a B)}{320 a^4 b \sqrt{x} (a+b x)^2}+\frac{21 (11 A b-a B)}{128 a^5 b \sqrt{x} (a+b x)}+\frac{(63 (11 A b-a B)) \int \frac{1}{x^{3/2} (a+b x)} \, dx}{256 a^5 b}\\ &=-\frac{63 (11 A b-a B)}{128 a^6 b \sqrt{x}}+\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}+\frac{3 (11 A b-a B)}{80 a^3 b \sqrt{x} (a+b x)^3}+\frac{21 (11 A b-a B)}{320 a^4 b \sqrt{x} (a+b x)^2}+\frac{21 (11 A b-a B)}{128 a^5 b \sqrt{x} (a+b x)}-\frac{(63 (11 A b-a B)) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{256 a^6}\\ &=-\frac{63 (11 A b-a B)}{128 a^6 b \sqrt{x}}+\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}+\frac{3 (11 A b-a B)}{80 a^3 b \sqrt{x} (a+b x)^3}+\frac{21 (11 A b-a B)}{320 a^4 b \sqrt{x} (a+b x)^2}+\frac{21 (11 A b-a B)}{128 a^5 b \sqrt{x} (a+b x)}-\frac{(63 (11 A b-a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{128 a^6}\\ &=-\frac{63 (11 A b-a B)}{128 a^6 b \sqrt{x}}+\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}+\frac{3 (11 A b-a B)}{80 a^3 b \sqrt{x} (a+b x)^3}+\frac{21 (11 A b-a B)}{320 a^4 b \sqrt{x} (a+b x)^2}+\frac{21 (11 A b-a B)}{128 a^5 b \sqrt{x} (a+b x)}-\frac{63 (11 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{13/2} \sqrt{b}}\\ \end{align*}

Mathematica [C]  time = 0.0308618, size = 59, normalized size = 0.27 \[ \frac{\frac{a^5 (A b-a B)}{(a+b x)^5}+(a B-11 A b) \, _2F_1\left (-\frac{1}{2},5;\frac{1}{2};-\frac{b x}{a}\right )}{5 a^6 b \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^5*(A*b - a*B))/(a + b*x)^5 + (-11*A*b + a*B)*Hypergeometric2F1[-1/2, 5, 1/2, -((b*x)/a)])/(5*a^6*b*Sqrt[x]
)

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Maple [A]  time = 0.024, size = 239, normalized size = 1.1 \begin{align*} -2\,{\frac{A}{{a}^{6}\sqrt{x}}}-{\frac{437\,A{b}^{5}}{128\,{a}^{6} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}+{\frac{63\,{b}^{4}B}{128\,{a}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}-{\frac{977\,{b}^{4}A}{64\,{a}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}+{\frac{147\,{b}^{3}B}{64\,{a}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}-{\frac{131\,A{b}^{3}}{5\,{a}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}+{\frac{21\,{b}^{2}B}{5\,{a}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}-{\frac{1327\,A{b}^{2}}{64\,{a}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}+{\frac{237\,bB}{64\,{a}^{2} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}-{\frac{843\,Ab}{128\,{a}^{2} \left ( bx+a \right ) ^{5}}\sqrt{x}}+{\frac{193\,B}{128\,a \left ( bx+a \right ) ^{5}}\sqrt{x}}-{\frac{693\,Ab}{128\,{a}^{6}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{63\,B}{128\,{a}^{5}}\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^(3/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-2*A/a^6/x^(1/2)-437/128/a^6/(b*x+a)^5*x^(9/2)*A*b^5+63/128/a^5/(b*x+a)^5*x^(9/2)*B*b^4-977/64/a^5/(b*x+a)^5*A
*x^(7/2)*b^4+147/64/a^4/(b*x+a)^5*B*x^(7/2)*b^3-131/5/a^4/(b*x+a)^5*x^(5/2)*A*b^3+21/5/a^3/(b*x+a)^5*x^(5/2)*b
^2*B-1327/64/a^3/(b*x+a)^5*x^(3/2)*A*b^2+237/64/a^2/(b*x+a)^5*x^(3/2)*b*B-843/128/a^2/(b*x+a)^5*x^(1/2)*A*b+19
3/128/a/(b*x+a)^5*x^(1/2)*B-693/128/a^6/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A*b+63/128/a^5/(a*b)^(1/2)*a
rctan(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^(3/2)/(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 = 1.63652, size = 1503, normalized size = 6.86 \begin{align*} \left [\frac{315 \,{\left ({\left (B a b^{5} - 11 \, A b^{6}\right )} x^{6} + 5 \,{\left (B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + 10 \,{\left (B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4} + 10 \,{\left (B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{3} + 5 \,{\left (B a^{5} b - 11 \, A a^{4} b^{2}\right )} x^{2} +{\left (B a^{6} - 11 \, A a^{5} b\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x - a + 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right ) - 2 \,{\left (1280 \, A a^{6} b - 315 \,{\left (B a^{2} b^{5} - 11 \, A a b^{6}\right )} x^{5} - 1470 \,{\left (B a^{3} b^{4} - 11 \, A a^{2} b^{5}\right )} x^{4} - 2688 \,{\left (B a^{4} b^{3} - 11 \, A a^{3} b^{4}\right )} x^{3} - 2370 \,{\left (B a^{5} b^{2} - 11 \, A a^{4} b^{3}\right )} x^{2} - 965 \,{\left (B a^{6} b - 11 \, A a^{5} b^{2}\right )} x\right )} \sqrt{x}}{1280 \,{\left (a^{7} b^{6} x^{6} + 5 \, a^{8} b^{5} x^{5} + 10 \, a^{9} b^{4} x^{4} + 10 \, a^{10} b^{3} x^{3} + 5 \, a^{11} b^{2} x^{2} + a^{12} b x\right )}}, -\frac{315 \,{\left ({\left (B a b^{5} - 11 \, A b^{6}\right )} x^{6} + 5 \,{\left (B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + 10 \,{\left (B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4} + 10 \,{\left (B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{3} + 5 \,{\left (B a^{5} b - 11 \, A a^{4} b^{2}\right )} x^{2} +{\left (B a^{6} - 11 \, A a^{5} b\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (1280 \, A a^{6} b - 315 \,{\left (B a^{2} b^{5} - 11 \, A a b^{6}\right )} x^{5} - 1470 \,{\left (B a^{3} b^{4} - 11 \, A a^{2} b^{5}\right )} x^{4} - 2688 \,{\left (B a^{4} b^{3} - 11 \, A a^{3} b^{4}\right )} x^{3} - 2370 \,{\left (B a^{5} b^{2} - 11 \, A a^{4} b^{3}\right )} x^{2} - 965 \,{\left (B a^{6} b - 11 \, A a^{5} b^{2}\right )} x\right )} \sqrt{x}}{640 \,{\left (a^{7} b^{6} x^{6} + 5 \, a^{8} b^{5} x^{5} + 10 \, a^{9} b^{4} x^{4} + 10 \, a^{10} b^{3} x^{3} + 5 \, a^{11} b^{2} x^{2} + a^{12} b x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/1280*(315*((B*a*b^5 - 11*A*b^6)*x^6 + 5*(B*a^2*b^4 - 11*A*a*b^5)*x^5 + 10*(B*a^3*b^3 - 11*A*a^2*b^4)*x^4 +
10*(B*a^4*b^2 - 11*A*a^3*b^3)*x^3 + 5*(B*a^5*b - 11*A*a^4*b^2)*x^2 + (B*a^6 - 11*A*a^5*b)*x)*sqrt(-a*b)*log((b
*x - a + 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) - 2*(1280*A*a^6*b - 315*(B*a^2*b^5 - 11*A*a*b^6)*x^5 - 1470*(B*a^3*b
^4 - 11*A*a^2*b^5)*x^4 - 2688*(B*a^4*b^3 - 11*A*a^3*b^4)*x^3 - 2370*(B*a^5*b^2 - 11*A*a^4*b^3)*x^2 - 965*(B*a^
6*b - 11*A*a^5*b^2)*x)*sqrt(x))/(a^7*b^6*x^6 + 5*a^8*b^5*x^5 + 10*a^9*b^4*x^4 + 10*a^10*b^3*x^3 + 5*a^11*b^2*x
^2 + a^12*b*x), -1/640*(315*((B*a*b^5 - 11*A*b^6)*x^6 + 5*(B*a^2*b^4 - 11*A*a*b^5)*x^5 + 10*(B*a^3*b^3 - 11*A*
a^2*b^4)*x^4 + 10*(B*a^4*b^2 - 11*A*a^3*b^3)*x^3 + 5*(B*a^5*b - 11*A*a^4*b^2)*x^2 + (B*a^6 - 11*A*a^5*b)*x)*sq
rt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (1280*A*a^6*b - 315*(B*a^2*b^5 - 11*A*a*b^6)*x^5 - 1470*(B*a^3*b^4 - 1
1*A*a^2*b^5)*x^4 - 2688*(B*a^4*b^3 - 11*A*a^3*b^4)*x^3 - 2370*(B*a^5*b^2 - 11*A*a^4*b^3)*x^2 - 965*(B*a^6*b -
11*A*a^5*b^2)*x)*sqrt(x))/(a^7*b^6*x^6 + 5*a^8*b^5*x^5 + 10*a^9*b^4*x^4 + 10*a^10*b^3*x^3 + 5*a^11*b^2*x^2 + a
^12*b*x)]

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

[Out]

Timed out

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Giac [A]  time = 1.16222, size = 213, normalized size = 0.97 \begin{align*} \frac{63 \,{\left (B a - 11 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} a^{6}} - \frac{2 \, A}{a^{6} \sqrt{x}} + \frac{315 \, B a b^{4} x^{\frac{9}{2}} - 2185 \, A b^{5} x^{\frac{9}{2}} + 1470 \, B a^{2} b^{3} x^{\frac{7}{2}} - 9770 \, A a b^{4} x^{\frac{7}{2}} + 2688 \, B a^{3} b^{2} x^{\frac{5}{2}} - 16768 \, A a^{2} b^{3} x^{\frac{5}{2}} + 2370 \, B a^{4} b x^{\frac{3}{2}} - 13270 \, A a^{3} b^{2} x^{\frac{3}{2}} + 965 \, B a^{5} \sqrt{x} - 4215 \, A a^{4} b \sqrt{x}}{640 \,{\left (b x + a\right )}^{5} a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

63/128*(B*a - 11*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^6) - 2*A/(a^6*sqrt(x)) + 1/640*(315*B*a*b^4*x^(
9/2) - 2185*A*b^5*x^(9/2) + 1470*B*a^2*b^3*x^(7/2) - 9770*A*a*b^4*x^(7/2) + 2688*B*a^3*b^2*x^(5/2) - 16768*A*a
^2*b^3*x^(5/2) + 2370*B*a^4*b*x^(3/2) - 13270*A*a^3*b^2*x^(3/2) + 965*B*a^5*sqrt(x) - 4215*A*a^4*b*sqrt(x))/((
b*x + a)^5*a^6)