3.5.67 \(\int \frac {1}{x^3 (a^2+2 a b x^2+b^2 x^4)^{3/2}} \, dx\)

Optimal. Leaf size=189 \[ -\frac {b}{4 a^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 b \log (x) \left (a+b x^2\right )}{a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{2 a^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

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Rubi [A]  time = 0.10, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1112, 266, 44} \begin {gather*} -\frac {b}{4 a^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{2 a^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 b \log (x) \left (a+b x^2\right )}{a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]

[Out]

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1112

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^3 \left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a b+b^2 x\right )^3} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^3 b^3 x^2}-\frac {3}{a^4 b^2 x}+\frac {1}{a^2 b (a+b x)^3}+\frac {2}{a^3 b (a+b x)^2}+\frac {3}{a^4 b (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {b}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{4 a^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{2 a^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 b \left (a+b x^2\right ) \log (x)}{a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 97, normalized size = 0.51 \begin {gather*} \frac {-a \left (2 a^2+9 a b x^2+6 b^2 x^4\right )-12 b x^2 \log (x) \left (a+b x^2\right )^2+6 b x^2 \left (a+b x^2\right )^2 \log \left (a+b x^2\right )}{4 a^4 x^2 \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]

[Out]

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

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IntegrateAlgebraic [B]  time = 3.30, size = 796, normalized size = 4.21 \begin {gather*} \frac {49152 b^{17} x^{32}+393216 a b^{16} x^{30}+1454080 a^2 b^{15} x^{28}+3301376 a^3 b^{14} x^{26}+5151744 a^4 b^{13} x^{24}+5857280 a^5 b^{12} x^{22}+5015296 a^6 b^{11} x^{20}+3294720 a^7 b^{10} x^{18}+1674816 a^8 b^9 x^{16}+658944 a^9 b^8 x^{14}+199056 a^{10} b^7 x^{12}+45344 a^{11} b^6 x^{10}+7540 a^{12} b^5 x^8+864 a^{13} b^4 x^6+61 a^{14} b^3 x^4+2 a^{15} b^2 x^2-a^{16} b+\sqrt {b^2} \sqrt {b^2 x^4+2 a b x^2+a^2} \left (-49152 b^{15} x^{30}-344064 a b^{14} x^{28}-1110016 a^2 b^{13} x^{26}-2191360 a^3 b^{12} x^{24}-2960384 a^4 b^{11} x^{22}-2896896 a^5 b^{10} x^{20}-2118400 a^6 b^9 x^{18}-1176320 a^7 b^8 x^{16}-498496 a^8 b^7 x^{14}-160448 a^9 b^6 x^{12}-38608 a^{10} b^5 x^{10}-6736 a^{11} b^4 x^8-804 a^{12} b^3 x^6-60 a^{13} b^2 x^4-a^{14} b x^2-a^{15}\right )}{2 a^3 \sqrt {b^2 x^4+2 a b x^2+a^2} \left (-16384 b^{16} x^{28}-122880 a b^{15} x^{26}-425984 a^2 b^{14} x^{24}-905216 a^3 b^{13} x^{22}-1317888 a^4 b^{12} x^{20}-1391104 a^5 b^{11} x^{18}-1098240 a^6 b^{10} x^{16}-658944 a^7 b^9 x^{14}-302016 a^8 b^8 x^{12}-105248 a^9 b^7 x^{10}-27456 a^{10} b^6 x^8-5200 a^{11} b^5 x^6-676 a^{12} b^4 x^4-54 a^{13} b^3 x^2-2 a^{14} b^2\right ) x^4+2 a^3 \sqrt {b^2} \left (16384 b^{16} x^{30}+139264 a b^{15} x^{28}+548864 a^2 b^{14} x^{26}+1331200 a^3 b^{13} x^{24}+2223104 a^4 b^{12} x^{22}+2708992 a^5 b^{11} x^{20}+2489344 a^6 b^{10} x^{18}+1757184 a^7 b^9 x^{16}+960960 a^8 b^8 x^{14}+407264 a^9 b^7 x^{12}+132704 a^{10} b^6 x^{10}+32656 a^{11} b^5 x^8+5876 a^{12} b^4 x^6+730 a^{13} b^3 x^4+56 a^{14} b^2 x^2+2 a^{15} b\right ) x^4}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {b^2} x^2}{a}-\frac {\sqrt {b^2 x^4+2 a b x^2+a^2}}{a}\right )}{a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]

[Out]

(-(a^16*b) + 2*a^15*b^2*x^2 + 61*a^14*b^3*x^4 + 864*a^13*b^4*x^6 + 7540*a^12*b^5*x^8 + 45344*a^11*b^6*x^10 + 1
99056*a^10*b^7*x^12 + 658944*a^9*b^8*x^14 + 1674816*a^8*b^9*x^16 + 3294720*a^7*b^10*x^18 + 5015296*a^6*b^11*x^
20 + 5857280*a^5*b^12*x^22 + 5151744*a^4*b^13*x^24 + 3301376*a^3*b^14*x^26 + 1454080*a^2*b^15*x^28 + 393216*a*
b^16*x^30 + 49152*b^17*x^32 + Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-a^15 - a^14*b*x^2 - 60*a^13*b^2*x^4
- 804*a^12*b^3*x^6 - 6736*a^11*b^4*x^8 - 38608*a^10*b^5*x^10 - 160448*a^9*b^6*x^12 - 498496*a^8*b^7*x^14 - 117
6320*a^7*b^8*x^16 - 2118400*a^6*b^9*x^18 - 2896896*a^5*b^10*x^20 - 2960384*a^4*b^11*x^22 - 2191360*a^3*b^12*x^
24 - 1110016*a^2*b^13*x^26 - 344064*a*b^14*x^28 - 49152*b^15*x^30))/(2*a^3*x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]
*(-2*a^14*b^2 - 54*a^13*b^3*x^2 - 676*a^12*b^4*x^4 - 5200*a^11*b^5*x^6 - 27456*a^10*b^6*x^8 - 105248*a^9*b^7*x
^10 - 302016*a^8*b^8*x^12 - 658944*a^7*b^9*x^14 - 1098240*a^6*b^10*x^16 - 1391104*a^5*b^11*x^18 - 1317888*a^4*
b^12*x^20 - 905216*a^3*b^13*x^22 - 425984*a^2*b^14*x^24 - 122880*a*b^15*x^26 - 16384*b^16*x^28) + 2*a^3*Sqrt[b
^2]*x^4*(2*a^15*b + 56*a^14*b^2*x^2 + 730*a^13*b^3*x^4 + 5876*a^12*b^4*x^6 + 32656*a^11*b^5*x^8 + 132704*a^10*
b^6*x^10 + 407264*a^9*b^7*x^12 + 960960*a^8*b^8*x^14 + 1757184*a^7*b^9*x^16 + 2489344*a^6*b^10*x^18 + 2708992*
a^5*b^11*x^20 + 2223104*a^4*b^12*x^22 + 1331200*a^3*b^13*x^24 + 548864*a^2*b^14*x^26 + 139264*a*b^15*x^28 + 16
384*b^16*x^30)) - (3*b*ArcTanh[(Sqrt[b^2]*x^2)/a - Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/a])/a^4

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fricas [A]  time = 1.06, size = 119, normalized size = 0.63 \begin {gather*} -\frac {6 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3} - 6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \relax (x)}{4 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="fricas")

[Out]

-1/4*(6*a*b^2*x^4 + 9*a^2*b*x^2 + 2*a^3 - 6*(b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*log(b*x^2 + a) + 12*(b^3*x^6 +
2*a*b^2*x^4 + a^2*b*x^2)*log(x))/(a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^2)

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giac [A]  time = 0.24, size = 96, normalized size = 0.51 \begin {gather*} \frac {3 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {3 \, b \log \left ({\left | x \right |}\right )}{a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {6 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3}}{4 \, {\left (b x^{2} + a\right )}^{2} a^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")

[Out]

3/2*b*log(abs(b*x^2 + a))/(a^4*sgn(b*x^2 + a)) - 3*b*log(abs(x))/(a^4*sgn(b*x^2 + a)) - 1/4*(6*a*b^2*x^4 + 9*a
^2*b*x^2 + 2*a^3)/((b*x^2 + a)^2*a^4*x^2*sgn(b*x^2 + a))

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maple [A]  time = 0.02, size = 133, normalized size = 0.70 \begin {gather*} -\frac {\left (12 b^{3} x^{6} \ln \relax (x )-6 b^{3} x^{6} \ln \left (b \,x^{2}+a \right )+24 a \,b^{2} x^{4} \ln \relax (x )-12 a \,b^{2} x^{4} \ln \left (b \,x^{2}+a \right )+6 a \,b^{2} x^{4}+12 a^{2} b \,x^{2} \ln \relax (x )-6 a^{2} b \,x^{2} \ln \left (b \,x^{2}+a \right )+9 a^{2} b \,x^{2}+2 a^{3}\right ) \left (b \,x^{2}+a \right )}{4 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} a^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

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

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maxima [A]  time = 1.37, size = 75, normalized size = 0.40 \begin {gather*} -\frac {6 \, b^{2} x^{4} + 9 \, a b x^{2} + 2 \, a^{2}}{4 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} + \frac {3 \, b \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac {3 \, b \log \relax (x)}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="maxima")

[Out]

-1/4*(6*b^2*x^4 + 9*a*b*x^2 + 2*a^2)/(a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2) + 3/2*b*log(b*x^2 + a)/a^4 - 3*b*lo
g(x)/a^4

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^3\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2)),x)

[Out]

int(1/(x^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Integral(1/(x**3*((a + b*x**2)**2)**(3/2)), x)

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