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

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

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Rubi [A]  time = 0.18, antiderivative size = 357, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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 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 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]

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]

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*}

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Mathematica [C]  time = 0.03, size = 80, normalized size = 0.22 \begin {gather*} \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}} \end {gather*}

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)]))/(4
4*a^5*b*(a + b*x)^3*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [A]  time = 42.60, size = 187, normalized size = 0.52 \begin {gather*} \frac {(a+b x) \left (\frac {105 \left (11 a^{3/2} B-3 \sqrt {a} A b\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 b^{13/2}}+\frac {\sqrt {x} \left (-3465 a^5 B+945 a^4 A b-12705 a^4 b B x+3465 a^3 A b^2 x-16863 a^3 b^2 B x^2+4599 a^2 A b^3 x^2-9207 a^2 b^3 B x^3+2511 a A b^4 x^3-1408 a b^4 B x^4+384 A b^5 x^4+128 b^5 B x^5\right )}{192 b^6 (a+b x)^4}\right )}{\sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*((Sqrt[x]*(945*a^4*A*b - 3465*a^5*B + 3465*a^3*A*b^2*x - 12705*a^4*b*B*x + 4599*a^2*A*b^3*x^2 - 168
63*a^3*b^2*B*x^2 + 2511*a*A*b^4*x^3 - 9207*a^2*b^3*B*x^3 + 384*A*b^5*x^4 - 1408*a*b^4*B*x^4 + 128*b^5*B*x^5))/
(192*b^6*(a + b*x)^4) + (105*(-3*Sqrt[a]*A*b + 11*a^(3/2)*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*b^(13/2)))
)/Sqrt[(a + b*x)^2]

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fricas [A]  time = 0.56, size = 585, normalized size = 1.64 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.23, size = 191, normalized size = 0.54 \begin {gather*} \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 {gather*}

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))

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maple [A]  time = 0.08, size = 407, normalized size = 1.14 \begin {gather*} \frac {\left (-945 A a \,b^{5} x^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3465 B \,a^{2} b^{4} x^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+128 \sqrt {a b}\, B \,b^{5} x^{\frac {11}{2}}-3780 A \,a^{2} b^{4} x^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+13860 B \,a^{3} b^{3} x^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+384 \sqrt {a b}\, A \,b^{5} x^{\frac {9}{2}}-1408 \sqrt {a b}\, B a \,b^{4} x^{\frac {9}{2}}-5670 A \,a^{3} b^{3} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+20790 B \,a^{4} b^{2} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+2511 \sqrt {a b}\, A a \,b^{4} x^{\frac {7}{2}}-9207 \sqrt {a b}\, B \,a^{2} b^{3} x^{\frac {7}{2}}-3780 A \,a^{4} b^{2} x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+13860 B \,a^{5} b x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+4599 \sqrt {a b}\, A \,a^{2} b^{3} x^{\frac {5}{2}}-16863 \sqrt {a b}\, B \,a^{3} b^{2} x^{\frac {5}{2}}-945 A \,a^{5} b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3465 B \,a^{6} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3465 \sqrt {a b}\, A \,a^{3} b^{2} x^{\frac {3}{2}}-12705 \sqrt {a b}\, B \,a^{4} b \,x^{\frac {3}{2}}+945 \sqrt {a b}\, A \,a^{4} b \sqrt {x}-3465 \sqrt {a b}\, B \,a^{5} \sqrt {x}\right ) \left (b x +a \right )}{192 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{6}} \end {gather*}

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+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+1
28*B*(a*b)^(1/2)*x^(11/2)*b^5+4599*A*(a*b)^(1/2)*x^(5/2)*a^2*b^3-945*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*a^5*b-3
465*B*(a*b)^(1/2)*x^(1/2)*a^5+3465*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*a^6-945*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))
*x^4*a*b^5-16863*B*(a*b)^(1/2)*x^(5/2)*a^3*b^2+3465*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^4*a^2*b^4-3780*A*arcta
n(1/(a*b)^(1/2)*b*x^(1/2))*x^3*a^2*b^4+13860*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^3*a^3*b^3+3465*A*(a*b)^(1/2)*
x^(3/2)*a^3*b^2-5670*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^2*a^3*b^3-12705*B*(a*b)^(1/2)*x^(3/2)*a^4*b+20790*B*a
rctan(1/(a*b)^(1/2)*b*x^(1/2))*x^2*a^4*b^2-3780*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x*a^4*b^2+13860*B*arctan(1/(
a*b)^(1/2)*b*x^(1/2))*x*a^5*b+945*A*(a*b)^(1/2)*x^(1/2)*a^4*b+384*A*(a*b)^(1/2)*x^(9/2)*b^5)*(b*x+a)/(a*b)^(1/
2)/b^6/((b*x+a)^2)^(5/2)

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maxima [A]  time = 1.76, size = 381, normalized size = 1.07 \begin {gather*} -\frac {5 \, {\left ({\left (2747 \, B a b^{5} - 693 \, A b^{6}\right )} x^{2} + 3 \, {\left (437 \, B a^{2} b^{4} - 63 \, A a b^{5}\right )} x\right )} x^{\frac {9}{2}} + 10 \, {\left (359 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x^{2} + 183 \, {\left (11 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x\right )} x^{\frac {7}{2}} + 20 \, {\left (242 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{2} + 117 \, {\left (11 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x\right )} x^{\frac {5}{2}} + 198 \, {\left (15 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} + 7 \, {\left (11 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} x^{\frac {3}{2}} + 63 \, {\left (11 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (11 \, B a^{6} - A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (a b^{10} x^{5} + 5 \, a^{2} b^{9} x^{4} + 10 \, a^{3} b^{8} x^{3} + 10 \, a^{4} b^{7} x^{2} + 5 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} + \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}} + \frac {7 \, {\left (11 \, {\left (13 \, B a b - 3 \, A b^{2}\right )} x^{\frac {3}{2}} - 30 \, {\left (11 \, B a^{2} - 3 \, A a b\right )} \sqrt {x}\right )}}{128 \, a b^{6}} \end {gather*}

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]

-1/1920*(5*((2747*B*a*b^5 - 693*A*b^6)*x^2 + 3*(437*B*a^2*b^4 - 63*A*a*b^5)*x)*x^(9/2) + 10*(359*(13*B*a^2*b^4
- 3*A*a*b^5)*x^2 + 183*(11*B*a^3*b^3 - A*a^2*b^4)*x)*x^(7/2) + 20*(242*(13*B*a^3*b^3 - 3*A*a^2*b^4)*x^2 + 117
*(11*B*a^4*b^2 - A*a^3*b^3)*x)*x^(5/2) + 198*(15*(13*B*a^4*b^2 - 3*A*a^3*b^3)*x^2 + 7*(11*B*a^5*b - A*a^4*b^2)
*x)*x^(3/2) + 63*(11*(13*B*a^5*b - 3*A*a^4*b^2)*x^2 + 5*(11*B*a^6 - A*a^5*b)*x)*sqrt(x))/(a*b^10*x^5 + 5*a^2*b
^9*x^4 + 10*a^3*b^8*x^3 + 10*a^4*b^7*x^2 + 5*a^5*b^6*x + a^6*b^5) + 105/64*(11*B*a^2 - 3*A*a*b)*arctan(b*sqrt(
x)/sqrt(a*b))/(sqrt(a*b)*b^6) + 7/128*(11*(13*B*a*b - 3*A*b^2)*x^(3/2) - 30*(11*B*a^2 - 3*A*a*b)*sqrt(x))/(a*b
^6)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^{9/2}\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(9/2)*(A + B*x))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((x^(9/2)*(A + B*x))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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