3.827 \(\int \frac{x^{9/2} (A+B x)}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=357 \[ \frac{x^{11/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{9/2} (3 A b-11 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 x^{7/2} (3 A b-11 a B)}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 x^{5/2} (3 A b-11 a B)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 x^{3/2} (a+b x) (3 A b-11 a B)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 \sqrt{x} (a+b x) (3 A b-11 a B)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 \sqrt{a} (a+b x) (3 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(21*(3*A*b - 11*a*B)*x^(5/2))/(64*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*x^(11/2))/(4*a*b*(a + b*
x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((3*A*b - 11*a*B)*x^(9/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) + (3*(3*A*b - 11*a*B)*x^(7/2))/(32*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (105*(3*A*b - 11*a
*B)*Sqrt[x]*(a + b*x))/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(3*A*b - 11*a*B)*x^(3/2)*(a + b*x))/(64*a*
b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (105*Sqrt[a]*(3*A*b - 11*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]
)/(64*b^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.181959, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {770, 78, 47, 50, 63, 205} \[ \frac{x^{11/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{9/2} (3 A b-11 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 x^{7/2} (3 A b-11 a B)}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 x^{5/2} (3 A b-11 a B)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 x^{3/2} (a+b x) (3 A b-11 a B)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 \sqrt{x} (a+b x) (3 A b-11 a B)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 \sqrt{a} (a+b x) (3 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(21*(3*A*b - 11*a*B)*x^(5/2))/(64*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*x^(11/2))/(4*a*b*(a + b*
x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((3*A*b - 11*a*B)*x^(9/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) + (3*(3*A*b - 11*a*B)*x^(7/2))/(32*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (105*(3*A*b - 11*a
*B)*Sqrt[x]*(a + b*x))/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(3*A*b - 11*a*B)*x^(3/2)*(a + b*x))/(64*a*
b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (105*Sqrt[a]*(3*A*b - 11*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]
)/(64*b^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

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 )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{x^{9/2} (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{11/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (b^2 (3 A b-11 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{9/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{11/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(3 A b-11 a B) x^{9/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (3 (3 A b-11 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{11/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(3 A b-11 a B) x^{9/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (3 A b-11 a B) x^{7/2}}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (21 (3 A b-11 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{5/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{21 (3 A b-11 a B) x^{5/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{11/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(3 A b-11 a B) x^{9/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (3 A b-11 a B) x^{7/2}}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (105 (3 A b-11 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{3/2}}{a b+b^2 x} \, dx}{128 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{21 (3 A b-11 a B) x^{5/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{11/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(3 A b-11 a B) x^{9/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (3 A b-11 a B) x^{7/2}}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (3 A b-11 a B) x^{3/2} (a+b x)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (105 (3 A b-11 a B) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{128 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{21 (3 A b-11 a B) x^{5/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{11/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(3 A b-11 a B) x^{9/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (3 A b-11 a B) x^{7/2}}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 (3 A b-11 a B) \sqrt{x} (a+b x)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (3 A b-11 a B) x^{3/2} (a+b x)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (105 a (3 A b-11 a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{128 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{21 (3 A b-11 a B) x^{5/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{11/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(3 A b-11 a B) x^{9/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (3 A b-11 a B) x^{7/2}}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 (3 A b-11 a B) \sqrt{x} (a+b x)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (3 A b-11 a B) x^{3/2} (a+b x)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (105 a (3 A b-11 a B) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x^2} \, dx,x,\sqrt{x}\right )}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{21 (3 A b-11 a B) x^{5/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{11/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(3 A b-11 a B) x^{9/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (3 A b-11 a B) x^{7/2}}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 (3 A b-11 a B) \sqrt{x} (a+b x)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (3 A b-11 a B) x^{3/2} (a+b x)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 \sqrt{a} (3 A b-11 a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0355227, size = 80, normalized size = 0.22 \[ \frac{x^{11/2} \left (-11 a^4 (a B-A b)-(a+b x)^4 (3 A b-11 a B) \, _2F_1\left (4,\frac{11}{2};\frac{13}{2};-\frac{b x}{a}\right )\right )}{44 a^5 b (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x^(11/2)*(-11*a^4*(-(A*b) + a*B) - (3*A*b - 11*a*B)*(a + b*x)^4*Hypergeometric2F1[4, 11/2, 13/2, -((b*x)/a)])
)/(44*a^5*b*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.021, size = 407, normalized size = 1.1 \begin{align*}{\frac{bx+a}{192\,{b}^{6}} \left ( -1408\,B\sqrt{ab}{x}^{9/2}a{b}^{4}-945\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{5}b-3465\,B\sqrt{ab}\sqrt{x}{a}^{5}+3465\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{6}+20790\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{4}{b}^{2}-3780\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{4}{b}^{2}+13860\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{5}b+945\,A\sqrt{ab}\sqrt{x}{a}^{4}b+2511\,A\sqrt{ab}{x}^{7/2}a{b}^{4}-9207\,B\sqrt{ab}{x}^{7/2}{a}^{2}{b}^{3}+128\,B\sqrt{ab}{x}^{11/2}{b}^{5}+4599\,A\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{3}+384\,A\sqrt{ab}{x}^{9/2}{b}^{5}-945\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{4}a{b}^{5}-16863\,B\sqrt{ab}{x}^{5/2}{a}^{3}{b}^{2}+3465\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{4}{a}^{2}{b}^{4}-3780\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{4}+13860\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3}{a}^{3}{b}^{3}+3465\,A\sqrt{ab}{x}^{3/2}{a}^{3}{b}^{2}-5670\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{3}-12705\,B\sqrt{ab}{x}^{3/2}{a}^{4}b \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \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)^(5/2),x)

[Out]

1/192*(-1408*B*(a*b)^(1/2)*x^(9/2)*a*b^4-945*A*arctan(x^(1/2)*b/(a*b)^(1/2))*a^5*b-3465*B*(a*b)^(1/2)*x^(1/2)*
a^5+3465*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^6+20790*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^4*b^2-3780*A*arctan(x
^(1/2)*b/(a*b)^(1/2))*x*a^4*b^2+13860*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^5*b+945*A*(a*b)^(1/2)*x^(1/2)*a^4*b+
2511*A*(a*b)^(1/2)*x^(7/2)*a*b^4-9207*B*(a*b)^(1/2)*x^(7/2)*a^2*b^3+128*B*(a*b)^(1/2)*x^(11/2)*b^5+4599*A*(a*b
)^(1/2)*x^(5/2)*a^2*b^3+384*A*(a*b)^(1/2)*x^(9/2)*b^5-945*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^4*a*b^5-16863*B*(a
*b)^(1/2)*x^(5/2)*a^3*b^2+3465*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^4*a^2*b^4-3780*A*arctan(x^(1/2)*b/(a*b)^(1/2)
)*x^3*a^2*b^4+13860*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^3*a^3*b^3+3465*A*(a*b)^(1/2)*x^(3/2)*a^3*b^2-5670*A*arct
an(x^(1/2)*b/(a*b)^(1/2))*x^2*a^3*b^3-12705*B*(a*b)^(1/2)*x^(3/2)*a^4*b)*(b*x+a)/(a*b)^(1/2)/b^6/((b*x+a)^2)^(
5/2)

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48185, size = 1318, normalized size = 3.69 \begin{align*} \left [-\frac{315 \,{\left (11 \, B a^{5} - 3 \, A a^{4} b +{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 6 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (128 \, B b^{5} x^{5} - 3465 \, B a^{5} + 945 \, A a^{4} b - 128 \,{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} - 837 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} - 1533 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} - 1155 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{x}}{384 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}, \frac{315 \,{\left (11 \, B a^{5} - 3 \, A a^{4} b +{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 6 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) +{\left (128 \, B b^{5} x^{5} - 3465 \, B a^{5} + 945 \, A a^{4} b - 128 \,{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} - 837 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} - 1533 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} - 1155 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{x}}{192 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\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)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(315*(11*B*a^5 - 3*A*a^4*b + (11*B*a*b^4 - 3*A*b^5)*x^4 + 4*(11*B*a^2*b^3 - 3*A*a*b^4)*x^3 + 6*(11*B*a
^3*b^2 - 3*A*a^2*b^3)*x^2 + 4*(11*B*a^4*b - 3*A*a^3*b^2)*x)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)/
(b*x + a)) - 2*(128*B*b^5*x^5 - 3465*B*a^5 + 945*A*a^4*b - 128*(11*B*a*b^4 - 3*A*b^5)*x^4 - 837*(11*B*a^2*b^3
- 3*A*a*b^4)*x^3 - 1533*(11*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 - 1155*(11*B*a^4*b - 3*A*a^3*b^2)*x)*sqrt(x))/(b^10*x
^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6), 1/192*(315*(11*B*a^5 - 3*A*a^4*b + (11*B*a*b^4 - 3*
A*b^5)*x^4 + 4*(11*B*a^2*b^3 - 3*A*a*b^4)*x^3 + 6*(11*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 4*(11*B*a^4*b - 3*A*a^3*b
^2)*x)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) + (128*B*b^5*x^5 - 3465*B*a^5 + 945*A*a^4*b - 128*(11*B*a*b^4 -
3*A*b^5)*x^4 - 837*(11*B*a^2*b^3 - 3*A*a*b^4)*x^3 - 1533*(11*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 - 1155*(11*B*a^4*b
- 3*A*a^3*b^2)*x)*sqrt(x))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6)]

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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)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.26618, size = 258, normalized size = 0.72 \begin{align*} \frac{105 \,{\left (11 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} b^{6} \mathrm{sgn}\left (b x + a\right )} - \frac{2295 \, B a^{2} b^{3} x^{\frac{7}{2}} - 975 \, A a b^{4} x^{\frac{7}{2}} + 5855 \, B a^{3} b^{2} x^{\frac{5}{2}} - 2295 \, A a^{2} b^{3} x^{\frac{5}{2}} + 5153 \, B a^{4} b x^{\frac{3}{2}} - 1929 \, A a^{3} b^{2} x^{\frac{3}{2}} + 1545 \, B a^{5} \sqrt{x} - 561 \, A a^{4} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} b^{6} \mathrm{sgn}\left (b x + a\right )} + \frac{2 \,{\left (B b^{10} x^{\frac{3}{2}} - 15 \, B a b^{9} \sqrt{x} + 3 \, A b^{10} \sqrt{x}\right )}}{3 \, b^{15} \mathrm{sgn}\left (b x + a\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)^(5/2),x, algorithm="giac")

[Out]

105/64*(11*B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^6*sgn(b*x + a)) - 1/192*(2295*B*a^2*b^3*x
^(7/2) - 975*A*a*b^4*x^(7/2) + 5855*B*a^3*b^2*x^(5/2) - 2295*A*a^2*b^3*x^(5/2) + 5153*B*a^4*b*x^(3/2) - 1929*A
*a^3*b^2*x^(3/2) + 1545*B*a^5*sqrt(x) - 561*A*a^4*b*sqrt(x))/((b*x + a)^4*b^6*sgn(b*x + a)) + 2/3*(B*b^10*x^(3
/2) - 15*B*a*b^9*sqrt(x) + 3*A*b^10*sqrt(x))/(b^15*sgn(b*x + a))