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

Optimal. Leaf size=243 \[ \frac {3 b \log (x) (a+b x) (2 A b-a B)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b (a+b x) (2 A b-a B) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (3 A b-2 a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (3 A b-a B)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (A b-a B)}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.16, antiderivative size = 243, 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 {b (3 A b-2 a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (3 A b-a B)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (A b-a B)}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b \log (x) (a+b x) (2 A b-a B)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b (a+b x) (2 A b-a B) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(3*A*b - 2*a*B))/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b*(A*b - a*B))/(2*a^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - (A*(a + b*x))/(2*a^3*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((3*A*b - a*B)*(a + b*x))/(a^4*x*Sqrt[
a^2 + 2*a*b*x + b^2*x^2]) + (3*b*(2*A*b - a*B)*(a + b*x)*Log[x])/(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b*(2
*A*b - a*B)*(a + b*x)*Log[a + b*x])/(a^5*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^3 \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^3 \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^3}+\frac {-3 A b+a B}{a^4 b^3 x^2}-\frac {3 (-2 A b+a B)}{a^5 b^2 x}+\frac {-A b+a B}{a^3 b (a+b x)^3}+\frac {-3 A b+2 a B}{a^4 b (a+b x)^2}+\frac {3 (-2 A b+a B)}{a^5 b (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {b (3 A b-2 a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (A b-a B)}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(3 A b-a B) (a+b x)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b (2 A b-a B) (a+b x) \log (x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b (2 A b-a B) (a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 133, normalized size = 0.55 \begin {gather*} \frac {-a \left (a^3 (A+2 B x)+a^2 b x (9 B x-4 A)+6 a b^2 x^2 (B x-3 A)-12 A b^3 x^3\right )+6 b x^2 \log (x) (a+b x)^2 (2 A b-a B)+6 b x^2 (a+b x)^2 (a B-2 A b) \log (a+b x)}{2 a^5 x^2 (a+b x) \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a*(-12*A*b^3*x^3 + 6*a*b^2*x^2*(-3*A + B*x) + a^3*(A + 2*B*x) + a^2*b*x*(-4*A + 9*B*x))) + 6*b*(2*A*b - a*B
)*x^2*(a + b*x)^2*Log[x] + 6*b*(-2*A*b + a*B)*x^2*(a + b*x)^2*Log[a + b*x])/(2*a^5*x^2*(a + b*x)*Sqrt[(a + b*x
)^2])

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IntegrateAlgebraic [B]  time = 5.97, size = 1948, normalized size = 8.02

result too large to display

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-2*a^15*A*b^2 + a^16*b*B - 22*a^14*A*b^3*x - 3*a^15*b^2*B*x - 236*a^13*A*b^4
*x^2 - 62*a^14*b^3*B*x^2 - 1376*a^12*A*b^5*x^3 - 924*a^13*b^4*B*x^3 - 4864*a^11*A*b^6*x^4 - 8344*a^12*b^5*B*x^
4 - 8864*a^10*A*b^7*x^5 - 52080*a^11*b^6*B*x^5 + 4288*a^9*A*b^8*x^6 - 237664*a^10*b^7*B*x^6 + 78080*a^8*A*b^9*
x^7 - 819392*a^9*b^8*B*x^7 + 251392*a^7*A*b^10*x^8 - 2173312*a^8*b^9*B*x^8 + 480768*a^6*A*b^11*x^9 - 4471040*a
^7*b^10*B*x^9 + 617472*a^5*A*b^12*x^10 - 7133696*a^6*b^11*B*x^10 + 540672*a^4*A*b^13*x^11 - 8754176*a^5*b^12*B
*x^11 + 311296*a^3*A*b^14*x^12 - 8112128*a^4*b^13*B*x^12 + 106496*a^2*A*b^15*x^13 - 5492736*a^3*b^14*B*x^13 +
16384*a*A*b^16*x^14 - 2564096*a^2*b^15*B*x^14 - 737280*a*b^16*B*x^15 - 98304*b^17*B*x^16) + 4*Sqrt[b^2]*(a^17*
B + 24*a^15*A*b^2*x + 2*a^16*b*B*x + 258*a^14*A*b^3*x^2 + 65*a^15*b^2*B*x^2 + 1612*a^13*A*b^4*x^3 + 986*a^14*b
^3*B*x^3 + 6240*a^12*A*b^5*x^4 + 9268*a^13*b^4*B*x^4 + 13728*a^11*A*b^6*x^5 + 60424*a^12*b^5*B*x^5 + 4576*a^10
*A*b^7*x^6 + 289744*a^11*b^6*B*x^6 - 82368*a^9*A*b^8*x^7 + 1057056*a^10*b^7*B*x^7 - 329472*a^8*A*b^9*x^8 + 299
2704*a^9*b^8*B*x^8 - 732160*a^7*A*b^10*x^9 + 6644352*a^8*b^9*B*x^9 - 1098240*a^6*A*b^11*x^10 + 11604736*a^7*b^
10*B*x^10 - 1158144*a^5*A*b^12*x^11 + 15887872*a^6*b^11*B*x^11 - 851968*a^4*A*b^13*x^12 + 16866304*a^5*b^12*B*
x^12 - 417792*a^3*A*b^14*x^13 + 13604864*a^4*b^13*B*x^13 - 122880*a^2*A*b^15*x^14 + 8056832*a^3*b^14*B*x^14 -
16384*a*A*b^16*x^15 + 3301376*a^2*b^15*B*x^15 + 835584*a*b^16*B*x^16 + 98304*b^17*B*x^17))/(a^3*Sqrt[b^2]*Sqrt
[a^2 + 2*a*b*x + b^2*x^2]*(-8*a^15*b*x^2 - 232*a^14*b^2*x^3 - 3136*a^13*b^3*x^4 - 26208*a^12*b^4*x^5 - 151424*
a^11*b^5*x^6 - 640640*a^10*b^6*x^7 - 2050048*a^9*b^7*x^8 - 5051904*a^8*b^8*x^9 - 9664512*a^7*b^9*x^10 - 143503
36*a^6*b^10*x^11 - 16400384*a^5*b^11*x^12 - 14163968*a^4*b^12*x^13 - 8945664*a^3*b^13*x^14 - 3899392*a^2*b^14*
x^15 - 1048576*a*b^15*x^16 - 131072*b^16*x^17) + a^3*(8*a^16*b^2*x^2 + 240*a^15*b^3*x^3 + 3368*a^14*b^4*x^4 +
29344*a^13*b^5*x^5 + 177632*a^12*b^6*x^6 + 792064*a^11*b^7*x^7 + 2690688*a^10*b^8*x^8 + 7101952*a^9*b^9*x^9 +
14716416*a^8*b^10*x^10 + 24014848*a^7*b^11*x^11 + 30750720*a^6*b^12*x^12 + 30564352*a^5*b^13*x^13 + 23109632*a
^4*b^14*x^14 + 12845056*a^3*b^15*x^15 + 4947968*a^2*b^16*x^16 + 1179648*a*b^17*x^17 + 131072*b^18*x^18)) + ((-
36*A*(b^2)^(3/2)*x)/a^2 - (72*A*b^3*Sqrt[b^2]*x^2)/a^3 - (48*A*b^4*Sqrt[b^2]*x^3)/a^4 + (12*A*b^2*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/a^2 + (24*A*b^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a^3 + (48*A*b^4*x^2*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/a^4 + 6*b*B*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a] + (24*b^2*B*x*ArcTanh[(-(Sqrt
[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a])/a + (72*b^3*B*x^2*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/a])/a^2 + (96*b^4*B*x^3*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a])/a^3 + (48*b^5*
B*x^4*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a])/a^4 - (24*b*Sqrt[b^2]*B*x*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^2 - (48*(b^2)^(3/2)*B*x^2*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 - (48*b^3*Sqrt[b^2]*B*x
^3*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^4)/(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*Sq
rt[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]) + (12*A*b^2*ArcTa
nh[(Sqrt[b^2]*x)/a - Sqrt[a^2 + 2*a*b*x + b^2*x^2]/a])/a^5

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fricas [A]  time = 0.42, size = 225, normalized size = 0.93 \begin {gather*} -\frac {A a^{4} + 6 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 9 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 2 \, {\left (B a^{4} - 2 \, A a^{3} b\right )} x - 6 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 6 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(A*a^4 + 6*(B*a^2*b^2 - 2*A*a*b^3)*x^3 + 9*(B*a^3*b - 2*A*a^2*b^2)*x^2 + 2*(B*a^4 - 2*A*a^3*b)*x - 6*((B*
a*b^3 - 2*A*b^4)*x^4 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x^3 + (B*a^3*b - 2*A*a^2*b^2)*x^2)*log(b*x + a) + 6*((B*a*b^3
- 2*A*b^4)*x^4 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x^3 + (B*a^3*b - 2*A*a^2*b^2)*x^2)*log(x))/(a^5*b^2*x^4 + 2*a^6*b*
x^3 + a^7*x^2)

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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^3/(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 = 0.06, size = 262, normalized size = 1.08 \begin {gather*} -\frac {\left (-12 A \,b^{4} x^{4} \ln \relax (x )+12 A \,b^{4} x^{4} \ln \left (b x +a \right )+6 B a \,b^{3} x^{4} \ln \relax (x )-6 B a \,b^{3} x^{4} \ln \left (b x +a \right )-24 A a \,b^{3} x^{3} \ln \relax (x )+24 A a \,b^{3} x^{3} \ln \left (b x +a \right )+12 B \,a^{2} b^{2} x^{3} \ln \relax (x )-12 B \,a^{2} b^{2} x^{3} \ln \left (b x +a \right )-12 A \,a^{2} b^{2} x^{2} \ln \relax (x )+12 A \,a^{2} b^{2} x^{2} \ln \left (b x +a \right )-12 A a \,b^{3} x^{3}+6 B \,a^{3} b \,x^{2} \ln \relax (x )-6 B \,a^{3} b \,x^{2} \ln \left (b x +a \right )+6 B \,a^{2} b^{2} x^{3}-18 A \,a^{2} b^{2} x^{2}+9 B \,a^{3} b \,x^{2}-4 A \,a^{3} b x +2 B \,a^{4} x +A \,a^{4}\right ) \left (b x +a \right )}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} a^{5} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(12*A*b^4*x^4*ln(b*x+a)-12*A*b^4*x^4*ln(x)-6*B*a*b^3*x^4*ln(b*x+a)+6*B*a*b^3*x^4*ln(x)+24*A*ln(b*x+a)*x^3
*a*b^3-24*A*ln(x)*x^3*a*b^3-12*B*ln(b*x+a)*x^3*a^2*b^2+12*B*ln(x)*x^3*a^2*b^2+12*A*ln(b*x+a)*x^2*a^2*b^2-12*A*
ln(x)*x^2*a^2*b^2-12*A*a*b^3*x^3-6*B*ln(b*x+a)*x^2*a^3*b+6*B*ln(x)*x^2*a^3*b+6*B*a^2*b^2*x^3-18*A*a^2*b^2*x^2+
9*B*a^3*b*x^2-4*A*a^3*b*x+2*B*a^4*x+A*a^4)*(b*x+a)/x^2/a^5/((b*x+a)^2)^(3/2)

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maxima [A]  time = 0.49, size = 250, normalized size = 1.03 \begin {gather*} \frac {3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} - \frac {6 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{5}} - \frac {3 \, B b}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}} + \frac {6 \, A b^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4}} - \frac {B}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x} + \frac {5 \, A b}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} x} - \frac {A}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x^{2}} + \frac {A}{2 \, a^{3} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {B}{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^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

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

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

[Out]

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

[Out]

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

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