3.3.97 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^3}{x^{17}} \, dx\)

Optimal. Leaf size=40 \[ \frac {b \left (a+b x^2\right )^7}{112 a^2 x^{14}}-\frac {\left (a+b x^2\right )^7}{16 a x^{16}} \]

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Rubi [A]  time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 266, 45, 37} \begin {gather*} \frac {b \left (a+b x^2\right )^7}{112 a^2 x^{14}}-\frac {\left (a+b x^2\right )^7}{16 a x^{16}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^17,x]

[Out]

-(a + b*x^2)^7/(16*a*x^16) + (b*(a + b*x^2)^7)/(112*a^2*x^14)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 37

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] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

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*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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]]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{17}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^{17}} \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^9} \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac {\left (a+b x^2\right )^7}{16 a x^{16}}-\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^8} \, dx,x,x^2\right )}{16 a b^5}\\ &=-\frac {\left (a+b x^2\right )^7}{16 a x^{16}}+\frac {b \left (a+b x^2\right )^7}{112 a^2 x^{14}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 78, normalized size = 1.95 \begin {gather*} -\frac {a^6}{16 x^{16}}-\frac {3 a^5 b}{7 x^{14}}-\frac {5 a^4 b^2}{4 x^{12}}-\frac {2 a^3 b^3}{x^{10}}-\frac {15 a^2 b^4}{8 x^8}-\frac {a b^5}{x^6}-\frac {b^6}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^17,x]

[Out]

-1/16*a^6/x^16 - (3*a^5*b)/(7*x^14) - (5*a^4*b^2)/(4*x^12) - (2*a^3*b^3)/x^10 - (15*a^2*b^4)/(8*x^8) - (a*b^5)
/x^6 - b^6/(4*x^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{17}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^17,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^17, x]

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fricas [A]  time = 0.85, size = 70, normalized size = 1.75 \begin {gather*} -\frac {28 \, b^{6} x^{12} + 112 \, a b^{5} x^{10} + 210 \, a^{2} b^{4} x^{8} + 224 \, a^{3} b^{3} x^{6} + 140 \, a^{4} b^{2} x^{4} + 48 \, a^{5} b x^{2} + 7 \, a^{6}}{112 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^17,x, algorithm="fricas")

[Out]

-1/112*(28*b^6*x^12 + 112*a*b^5*x^10 + 210*a^2*b^4*x^8 + 224*a^3*b^3*x^6 + 140*a^4*b^2*x^4 + 48*a^5*b*x^2 + 7*
a^6)/x^16

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giac [A]  time = 0.17, size = 70, normalized size = 1.75 \begin {gather*} -\frac {28 \, b^{6} x^{12} + 112 \, a b^{5} x^{10} + 210 \, a^{2} b^{4} x^{8} + 224 \, a^{3} b^{3} x^{6} + 140 \, a^{4} b^{2} x^{4} + 48 \, a^{5} b x^{2} + 7 \, a^{6}}{112 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^17,x, algorithm="giac")

[Out]

-1/112*(28*b^6*x^12 + 112*a*b^5*x^10 + 210*a^2*b^4*x^8 + 224*a^3*b^3*x^6 + 140*a^4*b^2*x^4 + 48*a^5*b*x^2 + 7*
a^6)/x^16

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maple [A]  time = 0.01, size = 69, normalized size = 1.72 \begin {gather*} -\frac {b^{6}}{4 x^{4}}-\frac {a \,b^{5}}{x^{6}}-\frac {15 a^{2} b^{4}}{8 x^{8}}-\frac {2 a^{3} b^{3}}{x^{10}}-\frac {5 a^{4} b^{2}}{4 x^{12}}-\frac {3 a^{5} b}{7 x^{14}}-\frac {a^{6}}{16 x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^3/x^17,x)

[Out]

-2*a^3*b^3/x^10-1/16*a^6/x^16-5/4*a^4*b^2/x^12-a*b^5/x^6-15/8*a^2*b^4/x^8-3/7*a^5*b/x^14-1/4*b^6/x^4

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maxima [A]  time = 1.39, size = 70, normalized size = 1.75 \begin {gather*} -\frac {28 \, b^{6} x^{12} + 112 \, a b^{5} x^{10} + 210 \, a^{2} b^{4} x^{8} + 224 \, a^{3} b^{3} x^{6} + 140 \, a^{4} b^{2} x^{4} + 48 \, a^{5} b x^{2} + 7 \, a^{6}}{112 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^17,x, algorithm="maxima")

[Out]

-1/112*(28*b^6*x^12 + 112*a*b^5*x^10 + 210*a^2*b^4*x^8 + 224*a^3*b^3*x^6 + 140*a^4*b^2*x^4 + 48*a^5*b*x^2 + 7*
a^6)/x^16

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mupad [B]  time = 4.37, size = 69, normalized size = 1.72 \begin {gather*} -\frac {\frac {a^6}{16}+\frac {3\,a^5\,b\,x^2}{7}+\frac {5\,a^4\,b^2\,x^4}{4}+2\,a^3\,b^3\,x^6+\frac {15\,a^2\,b^4\,x^8}{8}+a\,b^5\,x^{10}+\frac {b^6\,x^{12}}{4}}{x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/x^17,x)

[Out]

-(a^6/16 + (b^6*x^12)/4 + (3*a^5*b*x^2)/7 + a*b^5*x^10 + (5*a^4*b^2*x^4)/4 + 2*a^3*b^3*x^6 + (15*a^2*b^4*x^8)/
8)/x^16

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sympy [B]  time = 0.68, size = 75, normalized size = 1.88 \begin {gather*} \frac {- 7 a^{6} - 48 a^{5} b x^{2} - 140 a^{4} b^{2} x^{4} - 224 a^{3} b^{3} x^{6} - 210 a^{2} b^{4} x^{8} - 112 a b^{5} x^{10} - 28 b^{6} x^{12}}{112 x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**17,x)

[Out]

(-7*a**6 - 48*a**5*b*x**2 - 140*a**4*b**2*x**4 - 224*a**3*b**3*x**6 - 210*a**2*b**4*x**8 - 112*a*b**5*x**10 -
28*b**6*x**12)/(112*x**16)

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