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

Optimal. Leaf size=195 \[ \frac {(3 a B+7 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{9/2} b^{5/2}}+\frac {\sqrt {x} (3 a B+7 A b)}{128 a^4 b^2 (a+b x)}+\frac {\sqrt {x} (3 a B+7 A b)}{192 a^3 b^2 (a+b x)^2}+\frac {\sqrt {x} (3 a B+7 A b)}{240 a^2 b^2 (a+b x)^3}-\frac {\sqrt {x} (3 a B+7 A b)}{40 a b^2 (a+b x)^4}+\frac {x^{3/2} (A b-a B)}{5 a b (a+b x)^5} \]

[Out]

1/5*(A*b-B*a)*x^(3/2)/a/b/(b*x+a)^5+1/128*(7*A*b+3*B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(9/2)/b^(5/2)-1/40*(
7*A*b+3*B*a)*x^(1/2)/a/b^2/(b*x+a)^4+1/240*(7*A*b+3*B*a)*x^(1/2)/a^2/b^2/(b*x+a)^3+1/192*(7*A*b+3*B*a)*x^(1/2)
/a^3/b^2/(b*x+a)^2+1/128*(7*A*b+3*B*a)*x^(1/2)/a^4/b^2/(b*x+a)

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Rubi [A]  time = 0.09, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {27, 78, 47, 51, 63, 205} \[ \frac {\sqrt {x} (3 a B+7 A b)}{128 a^4 b^2 (a+b x)}+\frac {\sqrt {x} (3 a B+7 A b)}{192 a^3 b^2 (a+b x)^2}+\frac {\sqrt {x} (3 a B+7 A b)}{240 a^2 b^2 (a+b x)^3}+\frac {(3 a B+7 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{9/2} b^{5/2}}-\frac {\sqrt {x} (3 a B+7 A b)}{40 a b^2 (a+b x)^4}+\frac {x^{3/2} (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*x^(3/2))/(5*a*b*(a + b*x)^5) - ((7*A*b + 3*a*B)*Sqrt[x])/(40*a*b^2*(a + b*x)^4) + ((7*A*b + 3*a*B
)*Sqrt[x])/(240*a^2*b^2*(a + b*x)^3) + ((7*A*b + 3*a*B)*Sqrt[x])/(192*a^3*b^2*(a + b*x)^2) + ((7*A*b + 3*a*B)*
Sqrt[x])/(128*a^4*b^2*(a + b*x)) + ((7*A*b + 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(9/2)*b^(5/2))

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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {\sqrt {x} (A+B x)}{(a+b x)^6} \, dx\\ &=\frac {(A b-a B) x^{3/2}}{5 a b (a+b x)^5}+\frac {(7 A b+3 a B) \int \frac {\sqrt {x}}{(a+b x)^5} \, dx}{10 a b}\\ &=\frac {(A b-a B) x^{3/2}}{5 a b (a+b x)^5}-\frac {(7 A b+3 a B) \sqrt {x}}{40 a b^2 (a+b x)^4}+\frac {(7 A b+3 a B) \int \frac {1}{\sqrt {x} (a+b x)^4} \, dx}{80 a b^2}\\ &=\frac {(A b-a B) x^{3/2}}{5 a b (a+b x)^5}-\frac {(7 A b+3 a B) \sqrt {x}}{40 a b^2 (a+b x)^4}+\frac {(7 A b+3 a B) \sqrt {x}}{240 a^2 b^2 (a+b x)^3}+\frac {(7 A b+3 a B) \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx}{96 a^2 b^2}\\ &=\frac {(A b-a B) x^{3/2}}{5 a b (a+b x)^5}-\frac {(7 A b+3 a B) \sqrt {x}}{40 a b^2 (a+b x)^4}+\frac {(7 A b+3 a B) \sqrt {x}}{240 a^2 b^2 (a+b x)^3}+\frac {(7 A b+3 a B) \sqrt {x}}{192 a^3 b^2 (a+b x)^2}+\frac {(7 A b+3 a B) \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx}{128 a^3 b^2}\\ &=\frac {(A b-a B) x^{3/2}}{5 a b (a+b x)^5}-\frac {(7 A b+3 a B) \sqrt {x}}{40 a b^2 (a+b x)^4}+\frac {(7 A b+3 a B) \sqrt {x}}{240 a^2 b^2 (a+b x)^3}+\frac {(7 A b+3 a B) \sqrt {x}}{192 a^3 b^2 (a+b x)^2}+\frac {(7 A b+3 a B) \sqrt {x}}{128 a^4 b^2 (a+b x)}+\frac {(7 A b+3 a B) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 a^4 b^2}\\ &=\frac {(A b-a B) x^{3/2}}{5 a b (a+b x)^5}-\frac {(7 A b+3 a B) \sqrt {x}}{40 a b^2 (a+b x)^4}+\frac {(7 A b+3 a B) \sqrt {x}}{240 a^2 b^2 (a+b x)^3}+\frac {(7 A b+3 a B) \sqrt {x}}{192 a^3 b^2 (a+b x)^2}+\frac {(7 A b+3 a B) \sqrt {x}}{128 a^4 b^2 (a+b x)}+\frac {(7 A b+3 a B) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 a^4 b^2}\\ &=\frac {(A b-a B) x^{3/2}}{5 a b (a+b x)^5}-\frac {(7 A b+3 a B) \sqrt {x}}{40 a b^2 (a+b x)^4}+\frac {(7 A b+3 a B) \sqrt {x}}{240 a^2 b^2 (a+b x)^3}+\frac {(7 A b+3 a B) \sqrt {x}}{192 a^3 b^2 (a+b x)^2}+\frac {(7 A b+3 a B) \sqrt {x}}{128 a^4 b^2 (a+b x)}+\frac {(7 A b+3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{9/2} b^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 61, normalized size = 0.31 \[ \frac {x^{3/2} \left (\frac {3 a^5 (A b-a B)}{(a+b x)^5}+(3 a B+7 A b) \, _2F_1\left (\frac {3}{2},5;\frac {5}{2};-\frac {b x}{a}\right )\right )}{15 a^6 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(3/2)*((3*a^5*(A*b - a*B))/(a + b*x)^5 + (7*A*b + 3*a*B)*Hypergeometric2F1[3/2, 5, 5/2, -((b*x)/a)]))/(15*a
^6*b)

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fricas [A]  time = 0.83, size = 657, normalized size = 3.37 \[ \left [-\frac {15 \, {\left (3 \, B a^{6} + 7 \, A a^{5} b + {\left (3 \, B a b^{5} + 7 \, A b^{6}\right )} x^{5} + 5 \, {\left (3 \, B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{4} + 10 \, {\left (3 \, B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (3 \, B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (3 \, B a^{5} b + 7 \, A a^{4} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (45 \, B a^{6} b + 105 \, A a^{5} b^{2} - 15 \, {\left (3 \, B a^{2} b^{5} + 7 \, A a b^{6}\right )} x^{4} - 70 \, {\left (3 \, B a^{3} b^{4} + 7 \, A a^{2} b^{5}\right )} x^{3} - 128 \, {\left (3 \, B a^{4} b^{3} + 7 \, A a^{3} b^{4}\right )} x^{2} + 10 \, {\left (21 \, B a^{5} b^{2} - 79 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{5} b^{8} x^{5} + 5 \, a^{6} b^{7} x^{4} + 10 \, a^{7} b^{6} x^{3} + 10 \, a^{8} b^{5} x^{2} + 5 \, a^{9} b^{4} x + a^{10} b^{3}\right )}}, -\frac {15 \, {\left (3 \, B a^{6} + 7 \, A a^{5} b + {\left (3 \, B a b^{5} + 7 \, A b^{6}\right )} x^{5} + 5 \, {\left (3 \, B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{4} + 10 \, {\left (3 \, B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (3 \, B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (3 \, B a^{5} b + 7 \, A a^{4} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (45 \, B a^{6} b + 105 \, A a^{5} b^{2} - 15 \, {\left (3 \, B a^{2} b^{5} + 7 \, A a b^{6}\right )} x^{4} - 70 \, {\left (3 \, B a^{3} b^{4} + 7 \, A a^{2} b^{5}\right )} x^{3} - 128 \, {\left (3 \, B a^{4} b^{3} + 7 \, A a^{3} b^{4}\right )} x^{2} + 10 \, {\left (21 \, B a^{5} b^{2} - 79 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{5} b^{8} x^{5} + 5 \, a^{6} b^{7} x^{4} + 10 \, a^{7} b^{6} x^{3} + 10 \, a^{8} b^{5} x^{2} + 5 \, a^{9} b^{4} x + a^{10} b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(15*(3*B*a^6 + 7*A*a^5*b + (3*B*a*b^5 + 7*A*b^6)*x^5 + 5*(3*B*a^2*b^4 + 7*A*a*b^5)*x^4 + 10*(3*B*a^3*
b^3 + 7*A*a^2*b^4)*x^3 + 10*(3*B*a^4*b^2 + 7*A*a^3*b^3)*x^2 + 5*(3*B*a^5*b + 7*A*a^4*b^2)*x)*sqrt(-a*b)*log((b
*x - a - 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) + 2*(45*B*a^6*b + 105*A*a^5*b^2 - 15*(3*B*a^2*b^5 + 7*A*a*b^6)*x^4 -
70*(3*B*a^3*b^4 + 7*A*a^2*b^5)*x^3 - 128*(3*B*a^4*b^3 + 7*A*a^3*b^4)*x^2 + 10*(21*B*a^5*b^2 - 79*A*a^4*b^3)*x
)*sqrt(x))/(a^5*b^8*x^5 + 5*a^6*b^7*x^4 + 10*a^7*b^6*x^3 + 10*a^8*b^5*x^2 + 5*a^9*b^4*x + a^10*b^3), -1/1920*(
15*(3*B*a^6 + 7*A*a^5*b + (3*B*a*b^5 + 7*A*b^6)*x^5 + 5*(3*B*a^2*b^4 + 7*A*a*b^5)*x^4 + 10*(3*B*a^3*b^3 + 7*A*
a^2*b^4)*x^3 + 10*(3*B*a^4*b^2 + 7*A*a^3*b^3)*x^2 + 5*(3*B*a^5*b + 7*A*a^4*b^2)*x)*sqrt(a*b)*arctan(sqrt(a*b)/
(b*sqrt(x))) + (45*B*a^6*b + 105*A*a^5*b^2 - 15*(3*B*a^2*b^5 + 7*A*a*b^6)*x^4 - 70*(3*B*a^3*b^4 + 7*A*a^2*b^5)
*x^3 - 128*(3*B*a^4*b^3 + 7*A*a^3*b^4)*x^2 + 10*(21*B*a^5*b^2 - 79*A*a^4*b^3)*x)*sqrt(x))/(a^5*b^8*x^5 + 5*a^6
*b^7*x^4 + 10*a^7*b^6*x^3 + 10*a^8*b^5*x^2 + 5*a^9*b^4*x + a^10*b^3)]

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giac [A]  time = 0.18, size = 156, normalized size = 0.80 \[ \frac {{\left (3 \, B a + 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{4} b^{2}} + \frac {45 \, B a b^{4} x^{\frac {9}{2}} + 105 \, A b^{5} x^{\frac {9}{2}} + 210 \, B a^{2} b^{3} x^{\frac {7}{2}} + 490 \, A a b^{4} x^{\frac {7}{2}} + 384 \, B a^{3} b^{2} x^{\frac {5}{2}} + 896 \, A a^{2} b^{3} x^{\frac {5}{2}} - 210 \, B a^{4} b x^{\frac {3}{2}} + 790 \, A a^{3} b^{2} x^{\frac {3}{2}} - 45 \, B a^{5} \sqrt {x} - 105 \, A a^{4} b \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{4} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/128*(3*B*a + 7*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4*b^2) + 1/1920*(45*B*a*b^4*x^(9/2) + 105*A*b^5
*x^(9/2) + 210*B*a^2*b^3*x^(7/2) + 490*A*a*b^4*x^(7/2) + 384*B*a^3*b^2*x^(5/2) + 896*A*a^2*b^3*x^(5/2) - 210*B
*a^4*b*x^(3/2) + 790*A*a^3*b^2*x^(3/2) - 45*B*a^5*sqrt(x) - 105*A*a^4*b*sqrt(x))/((b*x + a)^5*a^4*b^2)

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maple [A]  time = 0.08, size = 154, normalized size = 0.79 \[ \frac {7 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, a^{4} b}+\frac {3 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, a^{3} b^{2}}+\frac {\frac {\left (7 A b +3 B a \right ) b^{2} x^{\frac {9}{2}}}{128 a^{4}}+\frac {7 \left (7 A b +3 B a \right ) b \,x^{\frac {7}{2}}}{192 a^{3}}+\frac {\left (7 A b +3 B a \right ) x^{\frac {5}{2}}}{15 a^{2}}+\frac {\left (79 A b -21 B a \right ) x^{\frac {3}{2}}}{192 a b}-\frac {\left (7 A b +3 B a \right ) \sqrt {x}}{128 b^{2}}}{\left (b x +a \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*x^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*(1/256*(7*A*b+3*B*a)/a^4*b^2*x^(9/2)+7/384/a^3*b*(7*A*b+3*B*a)*x^(7/2)+1/30/a^2*(7*A*b+3*B*a)*x^(5/2)+1/384*
(79*A*b-21*B*a)/a/b*x^(3/2)-1/256*(7*A*b+3*B*a)/b^2*x^(1/2))/(b*x+a)^5+7/128/a^4/b/(a*b)^(1/2)*arctan(1/(a*b)^
(1/2)*b*x^(1/2))*A+3/128/a^3/b^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))*B

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maxima [A]  time = 1.38, size = 205, normalized size = 1.05 \[ \frac {15 \, {\left (3 \, B a b^{4} + 7 \, A b^{5}\right )} x^{\frac {9}{2}} + 70 \, {\left (3 \, B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{\frac {7}{2}} + 128 \, {\left (3 \, B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{\frac {5}{2}} - 10 \, {\left (21 \, B a^{4} b - 79 \, A a^{3} b^{2}\right )} x^{\frac {3}{2}} - 15 \, {\left (3 \, B a^{5} + 7 \, A a^{4} b\right )} \sqrt {x}}{1920 \, {\left (a^{4} b^{7} x^{5} + 5 \, a^{5} b^{6} x^{4} + 10 \, a^{6} b^{5} x^{3} + 10 \, a^{7} b^{4} x^{2} + 5 \, a^{8} b^{3} x + a^{9} b^{2}\right )}} + \frac {{\left (3 \, B a + 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{4} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/1920*(15*(3*B*a*b^4 + 7*A*b^5)*x^(9/2) + 70*(3*B*a^2*b^3 + 7*A*a*b^4)*x^(7/2) + 128*(3*B*a^3*b^2 + 7*A*a^2*b
^3)*x^(5/2) - 10*(21*B*a^4*b - 79*A*a^3*b^2)*x^(3/2) - 15*(3*B*a^5 + 7*A*a^4*b)*sqrt(x))/(a^4*b^7*x^5 + 5*a^5*
b^6*x^4 + 10*a^6*b^5*x^3 + 10*a^7*b^4*x^2 + 5*a^8*b^3*x + a^9*b^2) + 1/128*(3*B*a + 7*A*b)*arctan(b*sqrt(x)/sq
rt(a*b))/(sqrt(a*b)*a^4*b^2)

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mupad [B]  time = 1.26, size = 174, normalized size = 0.89 \[ \frac {\frac {x^{5/2}\,\left (7\,A\,b+3\,B\,a\right )}{15\,a^2}-\frac {\sqrt {x}\,\left (7\,A\,b+3\,B\,a\right )}{128\,b^2}+\frac {b^2\,x^{9/2}\,\left (7\,A\,b+3\,B\,a\right )}{128\,a^4}+\frac {x^{3/2}\,\left (79\,A\,b-21\,B\,a\right )}{192\,a\,b}+\frac {7\,b\,x^{7/2}\,\left (7\,A\,b+3\,B\,a\right )}{192\,a^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (7\,A\,b+3\,B\,a\right )}{128\,a^{9/2}\,b^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/2)*(A + B*x))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((x^(5/2)*(7*A*b + 3*B*a))/(15*a^2) - (x^(1/2)*(7*A*b + 3*B*a))/(128*b^2) + (b^2*x^(9/2)*(7*A*b + 3*B*a))/(128
*a^4) + (x^(3/2)*(79*A*b - 21*B*a))/(192*a*b) + (7*b*x^(7/2)*(7*A*b + 3*B*a))/(192*a^3))/(a^5 + b^5*x^5 + 5*a*
b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x) + (atan((b^(1/2)*x^(1/2))/a^(1/2))*(7*A*b + 3*B*a))/(12
8*a^(9/2)*b^(5/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x**(1/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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