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

Optimal. Leaf size=196 \[ -\frac {A b-a B}{2 a^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\log (x) (a+b x) (3 A b-a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (3 A b-a B) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 A b-a B}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.14, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 77} \begin {gather*} -\frac {A b-a B}{2 a^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 A b-a B}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\log (x) (a+b x) (3 A b-a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (3 A b-a B) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*A*b - a*B)/(a^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (A*b - a*B)/(2*a^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) - (A*(a + b*x))/(a^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((3*A*b - a*B)*(a + b*x)*Log[x])/(a^4*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + ((3*A*b - a*B)*(a + b*x)*Log[a + b*x])/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{x^2 \left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {A}{a^3 b^3 x^2}+\frac {-3 A b+a B}{a^4 b^3 x}+\frac {A b-a B}{a^2 b^2 (a+b x)^3}+\frac {2 A b-a B}{a^3 b^2 (a+b x)^2}+\frac {3 A b-a B}{a^4 b^2 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 A b-a B}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 a^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(3 A b-a B) (a+b x) \log (x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(3 A b-a B) (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 110, normalized size = 0.56 \begin {gather*} \frac {a \left (a^2 (3 B x-2 A)+a b x (2 B x-9 A)-6 A b^2 x^2\right )+2 x \log (x) (a+b x)^2 (a B-3 A b)+2 x (a+b x)^2 (3 A b-a B) \log (a+b x)}{2 a^4 x (a+b x) \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-6*A*b^2*x^2 + a*b*x*(-9*A + 2*B*x) + a^2*(-2*A + 3*B*x)) + 2*(-3*A*b + a*B)*x*(a + b*x)^2*Log[x] + 2*(3*A
*b - a*B)*x*(a + b*x)^2*Log[a + b*x])/(2*a^4*x*(a + b*x)*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [B]  time = 5.36, size = 1428, normalized size = 7.29 \begin {gather*} \frac {\left (-\frac {6 A b}{a^3}+\frac {2 \sqrt {b^2} B x \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right )}{a^3}-\frac {2 B \sqrt {a^2+2 b x a+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right )}{a^3}\right ) \left (\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x\right )^3}{a^4+4 b x a^3+12 b^2 x^2 a^2-4 \sqrt {b^2} x \sqrt {a^2+2 b x a+b^2 x^2} a^2+16 b^3 x^3 a-8 b \sqrt {b^2} x^2 \sqrt {a^2+2 b x a+b^2 x^2} a+8 b^4 x^4-8 \left (b^2\right )^{3/2} x^3 \sqrt {a^2+2 b x a+b^2 x^2}}+\frac {-16384 B x^{16} b^{17}-131072 a B x^{15} b^{16}+4096 a A x^{14} b^{16}-479232 a^2 B x^{14} b^{15}+32768 a^2 A x^{13} b^{15}-1064960 a^3 B x^{13} b^{14}+116736 a^3 A x^{12} b^{14}-1610752 a^4 B x^{12} b^{13}+247808 a^4 A x^{11} b^{13}-1757184 a^5 B x^{11} b^{12}+352000 a^5 A x^{10} b^{12}-1427712 a^6 B x^{10} b^{11}+354816 a^6 A x^9 b^{11}-878592 a^7 B x^9 b^{10}+261888 a^7 A x^8 b^{10}-411840 a^8 B x^8 b^9+143616 a^8 A x^7 b^9-146432 a^9 B x^7 b^8+58608 a^9 A x^6 b^8-38896 a^{10} B x^6 b^7+17600 a^{10} A x^5 b^7-7488 a^{11} B x^5 b^6+3784 a^{11} A x^4 b^6-988 a^{12} B x^4 b^5+552 a^{12} A x^3 b^5-80 a^{13} B x^3 b^4+49 a^{13} A x^2 b^4-3 a^{14} B x^2 b^3+2 a^{14} A x b^3-a^{15} A b^2+a^{16} B b+\sqrt {b^2} \sqrt {a^2+2 b x a+b^2 x^2} \left (B a^{15}-A b a^{14}-b B x a^{14}+4 b^2 B x^2 a^{13}-A b^2 x a^{13}+76 b^3 B x^3 a^{12}-48 A b^3 x^2 a^{12}+912 b^4 B x^4 a^{11}-504 A b^4 x^3 a^{11}+6576 b^5 B x^5 a^{10}-3280 A b^5 x^4 a^{10}+32320 b^6 B x^6 a^9-14320 A b^6 x^5 a^9+114112 b^7 B x^7 a^8-44288 A b^7 x^6 a^8+297728 b^8 B x^8 a^7-99328 A b^8 x^7 a^7+580864 b^9 B x^9 a^6-162560 A b^9 x^8 a^6+846848 b^{10} B x^{10} a^5-192256 A b^{10} x^9 a^5+910336 b^{11} B x^{11} a^4-159744 A b^{11} x^{10} a^4+700416 b^{12} B x^{12} a^3-88064 A b^{12} x^{11} a^3+364544 b^{13} B x^{13} a^2-28672 A b^{13} x^{12} a^2+114688 b^{14} B x^{14} a-4096 A b^{14} x^{13} a+16384 b^{15} B x^{15}\right )}{a^2 b \sqrt {a^2+2 b x a+b^2 x^2} \left (-16384 x^{14} b^{16}-122880 a x^{13} b^{15}-425984 a^2 x^{12} b^{14}-905216 a^3 x^{11} b^{13}-1317888 a^4 x^{10} b^{12}-1391104 a^5 x^9 b^{11}-1098240 a^6 x^8 b^{10}-658944 a^7 x^7 b^9-302016 a^8 x^6 b^8-105248 a^9 x^5 b^7-27456 a^{10} x^4 b^6-5200 a^{11} x^3 b^5-676 a^{12} x^2 b^4-54 a^{13} x b^3-2 a^{14} b^2\right ) x^2+a^2 b \sqrt {b^2} \left (16384 x^{15} b^{16}+139264 a x^{14} b^{15}+548864 a^2 x^{13} b^{14}+1331200 a^3 x^{12} b^{13}+2223104 a^4 x^{11} b^{12}+2708992 a^5 x^{10} b^{11}+2489344 a^6 x^9 b^{10}+1757184 a^7 x^8 b^9+960960 a^8 x^7 b^8+407264 a^9 x^6 b^7+132704 a^{10} x^5 b^6+32656 a^{11} x^4 b^5+5876 a^{12} x^3 b^4+730 a^{13} x^2 b^3+56 a^{14} x b^2+2 a^{15} b\right ) x^2}-\frac {6 A b \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 b x a+b^2 x^2}}{a}\right )}{a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^15*A*b^2) + a^16*b*B + 2*a^14*A*b^3*x + 49*a^13*A*b^4*x^2 - 3*a^14*b^3*B*x^2 + 552*a^12*A*b^5*x^3 - 80*a^
13*b^4*B*x^3 + 3784*a^11*A*b^6*x^4 - 988*a^12*b^5*B*x^4 + 17600*a^10*A*b^7*x^5 - 7488*a^11*b^6*B*x^5 + 58608*a
^9*A*b^8*x^6 - 38896*a^10*b^7*B*x^6 + 143616*a^8*A*b^9*x^7 - 146432*a^9*b^8*B*x^7 + 261888*a^7*A*b^10*x^8 - 41
1840*a^8*b^9*B*x^8 + 354816*a^6*A*b^11*x^9 - 878592*a^7*b^10*B*x^9 + 352000*a^5*A*b^12*x^10 - 1427712*a^6*b^11
*B*x^10 + 247808*a^4*A*b^13*x^11 - 1757184*a^5*b^12*B*x^11 + 116736*a^3*A*b^14*x^12 - 1610752*a^4*b^13*B*x^12
+ 32768*a^2*A*b^15*x^13 - 1064960*a^3*b^14*B*x^13 + 4096*a*A*b^16*x^14 - 479232*a^2*b^15*B*x^14 - 131072*a*b^1
6*B*x^15 - 16384*b^17*B*x^16 + Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-(a^14*A*b) + a^15*B - a^13*A*b^2*x -
a^14*b*B*x - 48*a^12*A*b^3*x^2 + 4*a^13*b^2*B*x^2 - 504*a^11*A*b^4*x^3 + 76*a^12*b^3*B*x^3 - 3280*a^10*A*b^5*x
^4 + 912*a^11*b^4*B*x^4 - 14320*a^9*A*b^6*x^5 + 6576*a^10*b^5*B*x^5 - 44288*a^8*A*b^7*x^6 + 32320*a^9*b^6*B*x^
6 - 99328*a^7*A*b^8*x^7 + 114112*a^8*b^7*B*x^7 - 162560*a^6*A*b^9*x^8 + 297728*a^7*b^8*B*x^8 - 192256*a^5*A*b^
10*x^9 + 580864*a^6*b^9*B*x^9 - 159744*a^4*A*b^11*x^10 + 846848*a^5*b^10*B*x^10 - 88064*a^3*A*b^12*x^11 + 9103
36*a^4*b^11*B*x^11 - 28672*a^2*A*b^13*x^12 + 700416*a^3*b^12*B*x^12 - 4096*a*A*b^14*x^13 + 364544*a^2*b^13*B*x
^13 + 114688*a*b^14*B*x^14 + 16384*b^15*B*x^15))/(a^2*b*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-2*a^14*b^2 - 54*a^
13*b^3*x - 676*a^12*b^4*x^2 - 5200*a^11*b^5*x^3 - 27456*a^10*b^6*x^4 - 105248*a^9*b^7*x^5 - 302016*a^8*b^8*x^6
- 658944*a^7*b^9*x^7 - 1098240*a^6*b^10*x^8 - 1391104*a^5*b^11*x^9 - 1317888*a^4*b^12*x^10 - 905216*a^3*b^13*
x^11 - 425984*a^2*b^14*x^12 - 122880*a*b^15*x^13 - 16384*b^16*x^14) + a^2*b*Sqrt[b^2]*x^2*(2*a^15*b + 56*a^14*
b^2*x + 730*a^13*b^3*x^2 + 5876*a^12*b^4*x^3 + 32656*a^11*b^5*x^4 + 132704*a^10*b^6*x^5 + 407264*a^9*b^7*x^6 +
960960*a^8*b^8*x^7 + 1757184*a^7*b^9*x^8 + 2489344*a^6*b^10*x^9 + 2708992*a^5*b^11*x^10 + 2223104*a^4*b^12*x^
11 + 1331200*a^3*b^13*x^12 + 548864*a^2*b^14*x^13 + 139264*a*b^15*x^14 + 16384*b^16*x^15)) + ((-(Sqrt[b^2]*x)
+ Sqrt[a^2 + 2*a*b*x + b^2*x^2])^3*((-6*A*b)/a^3 + (2*Sqrt[b^2]*B*x*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/a])/a^3 - (2*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/a])/a^3))/(a^4 + 4*a^3*b*x + 12*a^2*b^2*x^2 + 16*a*b^3*x^3 + 8*b^4*x^4 - 4*a^2*Sqrt[b^2]*x*Sqrt[a^2 + 2
*a*b*x + b^2*x^2] - 8*a*b*Sqrt[b^2]*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2] - 8*(b^2)^(3/2)*x^3*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) - (6*A*b*ArcTanh[(Sqrt[b^2]*x)/a - Sqrt[a^2 + 2*a*b*x + b^2*x^2]/a])/a^4

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fricas [A]  time = 0.44, size = 187, normalized size = 0.95 \begin {gather*} -\frac {2 \, A a^{3} - 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x + 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \relax (x)}{2 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*A*a^3 - 2*(B*a^2*b - 3*A*a*b^2)*x^2 - 3*(B*a^3 - 3*A*a^2*b)*x + 2*((B*a*b^2 - 3*A*b^3)*x^3 + 2*(B*a^2*
b - 3*A*a*b^2)*x^2 + (B*a^3 - 3*A*a^2*b)*x)*log(b*x + a) - 2*((B*a*b^2 - 3*A*b^3)*x^3 + 2*(B*a^2*b - 3*A*a*b^2
)*x^2 + (B*a^3 - 3*A*a^2*b)*x)*log(x))/(a^4*b^2*x^3 + 2*a^5*b*x^2 + a^6*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.07, size = 221, normalized size = 1.13 \begin {gather*} \frac {\left (-6 A \,b^{3} x^{3} \ln \relax (x )+6 A \,b^{3} x^{3} \ln \left (b x +a \right )+2 B a \,b^{2} x^{3} \ln \relax (x )-2 B a \,b^{2} x^{3} \ln \left (b x +a \right )-12 A a \,b^{2} x^{2} \ln \relax (x )+12 A a \,b^{2} x^{2} \ln \left (b x +a \right )+4 B \,a^{2} b \,x^{2} \ln \relax (x )-4 B \,a^{2} b \,x^{2} \ln \left (b x +a \right )-6 A \,a^{2} b x \ln \relax (x )+6 A \,a^{2} b x \ln \left (b x +a \right )-6 A a \,b^{2} x^{2}+2 B \,a^{3} x \ln \relax (x )-2 B \,a^{3} x \ln \left (b x +a \right )+2 B \,a^{2} b \,x^{2}-9 A \,a^{2} b x +3 B \,a^{3} x -2 A \,a^{3}\right ) \left (b x +a \right )}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} a^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(6*A*b^3*x^3*ln(b*x+a)-6*A*b^3*x^3*ln(x)-2*B*a*b^2*x^3*ln(b*x+a)+2*B*a*b^2*x^3*ln(x)+12*A*ln(b*x+a)*x^2*a*
b^2-12*A*a*b^2*x^2*ln(x)-4*B*ln(b*x+a)*x^2*a^2*b+4*B*a^2*b*x^2*ln(x)+6*A*ln(b*x+a)*x*a^2*b-6*A*ln(x)*x*a^2*b-6
*A*a*b^2*x^2-2*B*ln(b*x+a)*x*a^3+2*B*a^3*x*ln(x)+2*B*a^2*b*x^2-9*A*a^2*b*x+3*B*a^3*x-2*A*a^3)*(b*x+a)/x/a^4/((
b*x+a)^2)^(3/2)

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maxima [A]  time = 0.52, size = 191, normalized size = 0.97 \begin {gather*} -\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} B \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{3}} + \frac {3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} + \frac {B}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}} - \frac {3 \, A b}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}} - \frac {A}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x} + \frac {B}{2 \, a b^{2} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {A}{2 \, a^{2} b {\left (x + \frac {a}{b}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1)^(2*a*b*x + 2*a^2)*B*log(2*a*b*x/abs(x) + 2*a^2/abs(x))/a^3 + 3*(-1)^(2*a*b*x + 2*a^2)*A*b*log(2*a*b*x/ab
s(x) + 2*a^2/abs(x))/a^4 + B/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2) - 3*A*b/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3) -
A/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*x) + 1/2*B/(a*b^2*(x + a/b)^2) - 1/2*A/(a^2*b*(x + a/b)^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {A+B\,x}{x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{x^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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