3.774 \(\int \frac{x^{9/2} (A+B x)}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=213 \[ \frac{x^{9/2} (A b-11 a B)}{40 a b^2 (a+b x)^4}+\frac{3 x^{7/2} (A b-11 a B)}{80 a b^3 (a+b x)^3}+\frac{21 x^{5/2} (A b-11 a B)}{320 a b^4 (a+b x)^2}+\frac{21 x^{3/2} (A b-11 a B)}{128 a b^5 (a+b x)}-\frac{63 \sqrt{x} (A b-11 a B)}{128 a b^6}+\frac{63 (A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 \sqrt{a} b^{13/2}}+\frac{x^{11/2} (A b-a B)}{5 a b (a+b x)^5} \]

[Out]

(-63*(A*b - 11*a*B)*Sqrt[x])/(128*a*b^6) + ((A*b - a*B)*x^(11/2))/(5*a*b*(a + b*x)^5) + ((A*b - 11*a*B)*x^(9/2
))/(40*a*b^2*(a + b*x)^4) + (3*(A*b - 11*a*B)*x^(7/2))/(80*a*b^3*(a + b*x)^3) + (21*(A*b - 11*a*B)*x^(5/2))/(3
20*a*b^4*(a + b*x)^2) + (21*(A*b - 11*a*B)*x^(3/2))/(128*a*b^5*(a + b*x)) + (63*(A*b - 11*a*B)*ArcTan[(Sqrt[b]
*Sqrt[x])/Sqrt[a]])/(128*Sqrt[a]*b^(13/2))

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Rubi [A]  time = 0.095871, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {27, 78, 47, 50, 63, 205} \[ \frac{x^{9/2} (A b-11 a B)}{40 a b^2 (a+b x)^4}+\frac{3 x^{7/2} (A b-11 a B)}{80 a b^3 (a+b x)^3}+\frac{21 x^{5/2} (A b-11 a B)}{320 a b^4 (a+b x)^2}+\frac{21 x^{3/2} (A b-11 a B)}{128 a b^5 (a+b x)}-\frac{63 \sqrt{x} (A b-11 a B)}{128 a b^6}+\frac{63 (A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 \sqrt{a} b^{13/2}}+\frac{x^{11/2} (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(-63*(A*b - 11*a*B)*Sqrt[x])/(128*a*b^6) + ((A*b - a*B)*x^(11/2))/(5*a*b*(a + b*x)^5) + ((A*b - 11*a*B)*x^(9/2
))/(40*a*b^2*(a + b*x)^4) + (3*(A*b - 11*a*B)*x^(7/2))/(80*a*b^3*(a + b*x)^3) + (21*(A*b - 11*a*B)*x^(5/2))/(3
20*a*b^4*(a + b*x)^2) + (21*(A*b - 11*a*B)*x^(3/2))/(128*a*b^5*(a + b*x)) + (63*(A*b - 11*a*B)*ArcTan[(Sqrt[b]
*Sqrt[x])/Sqrt[a]])/(128*Sqrt[a]*b^(13/2))

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 47

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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{x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{x^{9/2} (A+B x)}{(a+b x)^6} \, dx\\ &=\frac{(A b-a B) x^{11/2}}{5 a b (a+b x)^5}-\frac{(A b-11 a B) \int \frac{x^{9/2}}{(a+b x)^5} \, dx}{10 a b}\\ &=\frac{(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac{(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}-\frac{(9 (A b-11 a B)) \int \frac{x^{7/2}}{(a+b x)^4} \, dx}{80 a b^2}\\ &=\frac{(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac{(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac{3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}-\frac{(21 (A b-11 a B)) \int \frac{x^{5/2}}{(a+b x)^3} \, dx}{160 a b^3}\\ &=\frac{(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac{(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac{3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac{21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}-\frac{(21 (A b-11 a B)) \int \frac{x^{3/2}}{(a+b x)^2} \, dx}{128 a b^4}\\ &=\frac{(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac{(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac{3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac{21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}+\frac{21 (A b-11 a B) x^{3/2}}{128 a b^5 (a+b x)}-\frac{(63 (A b-11 a B)) \int \frac{\sqrt{x}}{a+b x} \, dx}{256 a b^5}\\ &=-\frac{63 (A b-11 a B) \sqrt{x}}{128 a b^6}+\frac{(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac{(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac{3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac{21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}+\frac{21 (A b-11 a B) x^{3/2}}{128 a b^5 (a+b x)}+\frac{(63 (A b-11 a B)) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{256 b^6}\\ &=-\frac{63 (A b-11 a B) \sqrt{x}}{128 a b^6}+\frac{(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac{(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac{3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac{21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}+\frac{21 (A b-11 a B) x^{3/2}}{128 a b^5 (a+b x)}+\frac{(63 (A b-11 a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{128 b^6}\\ &=-\frac{63 (A b-11 a B) \sqrt{x}}{128 a b^6}+\frac{(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac{(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac{3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac{21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}+\frac{21 (A b-11 a B) x^{3/2}}{128 a b^5 (a+b x)}+\frac{63 (A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 \sqrt{a} b^{13/2}}\\ \end{align*}

Mathematica [C]  time = 0.0333943, size = 61, normalized size = 0.29 \[ \frac{x^{11/2} \left (\frac{11 a^5 (A b-a B)}{(a+b x)^5}+(11 a B-A b) \, _2F_1\left (5,\frac{11}{2};\frac{13}{2};-\frac{b x}{a}\right )\right )}{55 a^6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(x^(11/2)*((11*a^5*(A*b - a*B))/(a + b*x)^5 + (-(A*b) + 11*a*B)*Hypergeometric2F1[5, 11/2, 13/2, -((b*x)/a)]))
/(55*a^6*b)

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Maple [A]  time = 0.02, size = 239, normalized size = 1.1 \begin{align*} 2\,{\frac{B\sqrt{x}}{{b}^{6}}}-{\frac{193\,A}{128\,b \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}+{\frac{843\,aB}{128\,{b}^{2} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}+{\frac{1327\,B{a}^{2}}{64\,{b}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}-{\frac{237\,aA}{64\,{b}^{2} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}+{\frac{131\,B{a}^{3}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}-{\frac{21\,A{a}^{2}}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}-{\frac{147\,A{a}^{3}}{64\,{b}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}+{\frac{977\,B{a}^{4}}{64\,{b}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}+{\frac{437\,B{a}^{5}}{128\,{b}^{6} \left ( bx+a \right ) ^{5}}\sqrt{x}}-{\frac{63\,A{a}^{4}}{128\,{b}^{5} \left ( bx+a \right ) ^{5}}\sqrt{x}}+{\frac{63\,A}{128\,{b}^{5}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{693\,aB}{128\,{b}^{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(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*B/b^6*x^(1/2)-193/128/b/(b*x+a)^5*x^(9/2)*A+843/128/b^2/(b*x+a)^5*x^(9/2)*B*a+1327/64/b^3/(b*x+a)^5*x^(7/2)*
B*a^2-237/64/b^2/(b*x+a)^5*x^(7/2)*A*a+131/5/b^4/(b*x+a)^5*x^(5/2)*B*a^3-21/5/b^3/(b*x+a)^5*x^(5/2)*A*a^2-147/
64/b^4/(b*x+a)^5*A*x^(3/2)*a^3+977/64/b^5/(b*x+a)^5*B*x^(3/2)*a^4+437/128/b^6/(b*x+a)^5*x^(1/2)*B*a^5-63/128/b
^5/(b*x+a)^5*x^(1/2)*A*a^4+63/128/b^5/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A-693/128/b^6/(a*b)^(1/2)*arct
an(x^(1/2)*b/(a*b)^(1/2))*a*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(x^(9/2)*(B*x+A)/(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 = 1.9625, size = 1474, normalized size = 6.92 \begin{align*} \left [\frac{315 \,{\left (11 \, B a^{6} - A a^{5} b +{\left (11 \, B a b^{5} - A b^{6}\right )} x^{5} + 5 \,{\left (11 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 10 \,{\left (11 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (11 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (11 \, B a^{5} b - A a^{4} b^{2}\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 \, B a b^{6} x^{5} + 3465 \, B a^{6} b - 315 \, A a^{5} b^{2} + 965 \,{\left (11 \, B a^{2} b^{5} - A a b^{6}\right )} x^{4} + 2370 \,{\left (11 \, B a^{3} b^{4} - A a^{2} b^{5}\right )} x^{3} + 2688 \,{\left (11 \, B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{2} + 1470 \,{\left (11 \, B a^{5} b^{2} - A a^{4} b^{3}\right )} x\right )} \sqrt{x}}{1280 \,{\left (a b^{12} x^{5} + 5 \, a^{2} b^{11} x^{4} + 10 \, a^{3} b^{10} x^{3} + 10 \, a^{4} b^{9} x^{2} + 5 \, a^{5} b^{8} x + a^{6} b^{7}\right )}}, \frac{315 \,{\left (11 \, B a^{6} - A a^{5} b +{\left (11 \, B a b^{5} - A b^{6}\right )} x^{5} + 5 \,{\left (11 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 10 \,{\left (11 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (11 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (11 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (1280 \, B a b^{6} x^{5} + 3465 \, B a^{6} b - 315 \, A a^{5} b^{2} + 965 \,{\left (11 \, B a^{2} b^{5} - A a b^{6}\right )} x^{4} + 2370 \,{\left (11 \, B a^{3} b^{4} - A a^{2} b^{5}\right )} x^{3} + 2688 \,{\left (11 \, B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{2} + 1470 \,{\left (11 \, B a^{5} b^{2} - A a^{4} b^{3}\right )} x\right )} \sqrt{x}}{640 \,{\left (a b^{12} x^{5} + 5 \, a^{2} b^{11} x^{4} + 10 \, a^{3} b^{10} x^{3} + 10 \, a^{4} b^{9} x^{2} + 5 \, a^{5} b^{8} x + a^{6} b^{7}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(315*(11*B*a^6 - A*a^5*b + (11*B*a*b^5 - A*b^6)*x^5 + 5*(11*B*a^2*b^4 - A*a*b^5)*x^4 + 10*(11*B*a^3*b^
3 - A*a^2*b^4)*x^3 + 10*(11*B*a^4*b^2 - A*a^3*b^3)*x^2 + 5*(11*B*a^5*b - A*a^4*b^2)*x)*sqrt(-a*b)*log((b*x - a
- 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) + 2*(1280*B*a*b^6*x^5 + 3465*B*a^6*b - 315*A*a^5*b^2 + 965*(11*B*a^2*b^5 -
A*a*b^6)*x^4 + 2370*(11*B*a^3*b^4 - A*a^2*b^5)*x^3 + 2688*(11*B*a^4*b^3 - A*a^3*b^4)*x^2 + 1470*(11*B*a^5*b^2
- A*a^4*b^3)*x)*sqrt(x))/(a*b^12*x^5 + 5*a^2*b^11*x^4 + 10*a^3*b^10*x^3 + 10*a^4*b^9*x^2 + 5*a^5*b^8*x + a^6*
b^7), 1/640*(315*(11*B*a^6 - A*a^5*b + (11*B*a*b^5 - A*b^6)*x^5 + 5*(11*B*a^2*b^4 - A*a*b^5)*x^4 + 10*(11*B*a^
3*b^3 - A*a^2*b^4)*x^3 + 10*(11*B*a^4*b^2 - A*a^3*b^3)*x^2 + 5*(11*B*a^5*b - A*a^4*b^2)*x)*sqrt(a*b)*arctan(sq
rt(a*b)/(b*sqrt(x))) + (1280*B*a*b^6*x^5 + 3465*B*a^6*b - 315*A*a^5*b^2 + 965*(11*B*a^2*b^5 - A*a*b^6)*x^4 + 2
370*(11*B*a^3*b^4 - A*a^2*b^5)*x^3 + 2688*(11*B*a^4*b^3 - A*a^3*b^4)*x^2 + 1470*(11*B*a^5*b^2 - A*a^4*b^3)*x)*
sqrt(x))/(a*b^12*x^5 + 5*a^2*b^11*x^4 + 10*a^3*b^10*x^3 + 10*a^4*b^9*x^2 + 5*a^5*b^8*x + a^6*b^7)]

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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(x**(9/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.23552, size = 215, normalized size = 1.01 \begin{align*} \frac{2 \, B \sqrt{x}}{b^{6}} - \frac{63 \,{\left (11 \, B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} b^{6}} + \frac{4215 \, B a b^{4} x^{\frac{9}{2}} - 965 \, A b^{5} x^{\frac{9}{2}} + 13270 \, B a^{2} b^{3} x^{\frac{7}{2}} - 2370 \, A a b^{4} x^{\frac{7}{2}} + 16768 \, B a^{3} b^{2} x^{\frac{5}{2}} - 2688 \, A a^{2} b^{3} x^{\frac{5}{2}} + 9770 \, B a^{4} b x^{\frac{3}{2}} - 1470 \, A a^{3} b^{2} x^{\frac{3}{2}} + 2185 \, B a^{5} \sqrt{x} - 315 \, A a^{4} b \sqrt{x}}{640 \,{\left (b x + a\right )}^{5} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

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