3.5.63 \(\int \frac {x^5}{(a^2+2 a b x^2+b^2 x^4)^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac {a^2}{4 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a}{b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

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Rubi [A]  time = 0.10, antiderivative size = 113, 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 = {1111, 646, 43} \begin {gather*} -\frac {a^2}{4 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a}{b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 646

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

Rule 1111

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && Integ
erQ[(m - 1)/2] && (GtQ[m, 0] || LtQ[0, 4*p, -m - 1])

Rubi steps

\begin {align*} \int \frac {x^5}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {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 {a^2}{b^5 (a+b x)^3}-\frac {2 a}{b^5 (a+b x)^2}+\frac {1}{b^5 (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {a}{b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^2}{4 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 61, normalized size = 0.54 \begin {gather*} \frac {a \left (3 a+4 b x^2\right )+2 \left (a+b x^2\right )^2 \log \left (a+b x^2\right )}{4 b^3 \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 1.17, size = 1590, normalized size = 14.07 \begin {gather*} \frac {-2 \sqrt {b^2} \log \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^8-2 \sqrt {b^2} \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^8-\frac {4 a \sqrt {b^2} \log \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^6}{b}+2 \sqrt {b^2 x^4+2 a b x^2+a^2} \log \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^6-\frac {4 a \sqrt {b^2} \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^6}{b}+2 \sqrt {b^2 x^4+2 a b x^2+a^2} \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^6+\frac {6 a^2 \left (b^2\right )^{3/2} x^4}{b^4}-\frac {2 a^2 \left (b^2\right )^{3/2} \log \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^4}{b^4}+\frac {2 a \sqrt {b^2 x^4+2 a b x^2+a^2} \log \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^4}{b}-\frac {2 a^2 \left (b^2\right )^{3/2} \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^4}{b^4}+\frac {2 a \sqrt {b^2 x^4+2 a b x^2+a^2} \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^4}{b}+\frac {6 a^3 \sqrt {b^2} x^2}{b^3}-\frac {6 a^2 \sqrt {b^2 x^4+2 a b x^2+a^2} x^2}{b^2}+\frac {2 a^4 \sqrt {b^2}}{b^4}}{\left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2 \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2}+\frac {2 b \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^8-2 b \log \left (-\sqrt {b^2} x^2 b^3-a b^3+\sqrt {b^2 x^4+2 a b x^2+a^2} b^3\right ) x^8+4 a \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^6-\frac {2 b \sqrt {b^2 x^4+2 a b x^2+a^2} \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^6}{\sqrt {b^2}}-4 a \log \left (-\sqrt {b^2} x^2 b^3-a b^3+\sqrt {b^2 x^4+2 a b x^2+a^2} b^3\right ) x^6+\frac {2 b \sqrt {b^2 x^4+2 a b x^2+a^2} \log \left (-\sqrt {b^2} x^2 b^3-a b^3+\sqrt {b^2 x^4+2 a b x^2+a^2} b^3\right ) x^6}{\sqrt {b^2}}-\frac {8 a b x^6}{\sqrt {b^2}}-\frac {2 a \sqrt {b^2 x^4+2 a b x^2+a^2} \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^4}{\sqrt {b^2}}+\frac {2 a^2 \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) x^4}{b}+\frac {2 a \sqrt {b^2 x^4+2 a b x^2+a^2} \log \left (-\sqrt {b^2} x^2 b^3-a b^3+\sqrt {b^2 x^4+2 a b x^2+a^2} b^3\right ) x^4}{\sqrt {b^2}}-\frac {2 a^2 \log \left (-\sqrt {b^2} x^2 b^3-a b^3+\sqrt {b^2 x^4+2 a b x^2+a^2} b^3\right ) x^4}{b}+\frac {8 a \sqrt {b^2 x^4+2 a b x^2+a^2} x^4}{b}-\frac {12 a^2 x^4}{\sqrt {b^2}}+\frac {4 a^2 \sqrt {b^2 x^4+2 a b x^2+a^2} x^2}{b^2}-\frac {6 a^3 x^2}{b \sqrt {b^2}}+\frac {2 a^3 \sqrt {b^2 x^4+2 a b x^2+a^2}}{b^3}}{\left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2 \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a^4*Sqrt[b^2])/b^4 + (6*a^3*Sqrt[b^2]*x^2)/b^3 + (6*a^2*(b^2)^(3/2)*x^4)/b^4 - (6*a^2*x^2*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4])/b^2 - (2*a^2*(b^2)^(3/2)*x^4*Log[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/b^4 -
(4*a*Sqrt[b^2]*x^6*Log[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/b - 2*Sqrt[b^2]*x^8*Log[-a - Sq
rt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]] + (2*a*x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[-a - Sqrt[b^2]*x
^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/b + 2*x^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[-a - Sqrt[b^2]*x^2 + Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4]] - (2*a^2*(b^2)^(3/2)*x^4*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]]
)/b^4 - (4*a*Sqrt[b^2]*x^6*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/b - 2*Sqrt[b^2]*x^8*Log[a
- Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]] + (2*a*x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[a - Sqrt[b^
2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/b + 2*x^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[a - Sqrt[b^2]*x^2 + S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/((-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])^2*(a - Sqrt[b^2]*x^2
+ Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])^2) + ((-6*a^3*x^2)/(b*Sqrt[b^2]) - (12*a^2*x^4)/Sqrt[b^2] - (8*a*b*x^6)/Sqr
t[b^2] + (2*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/b^3 + (4*a^2*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/b^2 + (8*a*
x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/b + (2*a^2*x^4*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/
b + 4*a*x^6*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]] + 2*b*x^8*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^
2 + 2*a*b*x^2 + b^2*x^4]] - (2*a*x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*
x^2 + b^2*x^4]])/Sqrt[b^2] - (2*b*x^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4]])/Sqrt[b^2] - (2*a^2*x^4*Log[-(a*b^3) - b^3*Sqrt[b^2]*x^2 + b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4
]])/b - 4*a*x^6*Log[-(a*b^3) - b^3*Sqrt[b^2]*x^2 + b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]] - 2*b*x^8*Log[-(a*b^3)
- b^3*Sqrt[b^2]*x^2 + b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]] + (2*a*x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[-(a
*b^3) - b^3*Sqrt[b^2]*x^2 + b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/Sqrt[b^2] + (2*b*x^6*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4]*Log[-(a*b^3) - b^3*Sqrt[b^2]*x^2 + b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/Sqrt[b^2])/((-a - Sqrt[b^2]*
x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])^2*(a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])^2)

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fricas [A]  time = 1.03, size = 69, normalized size = 0.61 \begin {gather*} \frac {4 \, a b x^{2} + 3 \, a^{2} + 2 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/4*(4*a*b*x^2 + 3*a^2 + 2*(b^2*x^4 + 2*a*b*x^2 + a^2)*log(b*x^2 + a))/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)

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giac [A]  time = 0.23, size = 64, normalized size = 0.57 \begin {gather*} \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {4 \, a x^{2} + \frac {3 \, a^{2}}{b}}{4 \, {\left (b x^{2} + a\right )}^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.02, size = 81, normalized size = 0.72 \begin {gather*} \frac {\left (2 b^{2} x^{4} \ln \left (b \,x^{2}+a \right )+4 a b \,x^{2} \ln \left (b \,x^{2}+a \right )+4 a b \,x^{2}+2 a^{2} \ln \left (b \,x^{2}+a \right )+3 a^{2}\right ) \left (b \,x^{2}+a \right )}{4 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

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

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maxima [A]  time = 1.37, size = 55, normalized size = 0.49 \begin {gather*} \frac {4 \, a b x^{2} + 3 \, a^{2}}{4 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {\log \left (b x^{2} + a\right )}{2 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="maxima")

[Out]

1/4*(4*a*b*x^2 + 3*a^2)/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3) + 1/2*log(b*x^2 + a)/b^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^5}{{\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(x^5/(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2),x)

[Out]

int(x^5/(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 {x^{5}}{\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(x**5/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Integral(x**5/((a + b*x**2)**2)**(3/2), x)

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