3.168 \(\int \frac{x^{15}}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=114 \[ \frac{a^2 x^6}{b^5}-\frac{5 a^3 x^4}{2 b^6}+\frac{15 a^4 x^2}{2 b^7}-\frac{7 a^6}{2 b^8 \left (a+b x^2\right )}+\frac{a^7}{4 b^8 \left (a+b x^2\right )^2}-\frac{21 a^5 \log \left (a+b x^2\right )}{2 b^8}-\frac{3 a x^8}{8 b^4}+\frac{x^{10}}{10 b^3} \]

[Out]

(15*a^4*x^2)/(2*b^7) - (5*a^3*x^4)/(2*b^6) + (a^2*x^6)/b^5 - (3*a*x^8)/(8*b^4) + x^10/(10*b^3) + a^7/(4*b^8*(a
+ b*x^2)^2) - (7*a^6)/(2*b^8*(a + b*x^2)) - (21*a^5*Log[a + b*x^2])/(2*b^8)

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Rubi [A]  time = 0.103378, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^2 x^6}{b^5}-\frac{5 a^3 x^4}{2 b^6}+\frac{15 a^4 x^2}{2 b^7}-\frac{7 a^6}{2 b^8 \left (a+b x^2\right )}+\frac{a^7}{4 b^8 \left (a+b x^2\right )^2}-\frac{21 a^5 \log \left (a+b x^2\right )}{2 b^8}-\frac{3 a x^8}{8 b^4}+\frac{x^{10}}{10 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^15/(a + b*x^2)^3,x]

[Out]

(15*a^4*x^2)/(2*b^7) - (5*a^3*x^4)/(2*b^6) + (a^2*x^6)/b^5 - (3*a*x^8)/(8*b^4) + x^10/(10*b^3) + a^7/(4*b^8*(a
+ b*x^2)^2) - (7*a^6)/(2*b^8*(a + b*x^2)) - (21*a^5*Log[a + b*x^2])/(2*b^8)

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 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])

Rubi steps

\begin{align*} \int \frac{x^{15}}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^7}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{15 a^4}{b^7}-\frac{10 a^3 x}{b^6}+\frac{6 a^2 x^2}{b^5}-\frac{3 a x^3}{b^4}+\frac{x^4}{b^3}-\frac{a^7}{b^7 (a+b x)^3}+\frac{7 a^6}{b^7 (a+b x)^2}-\frac{21 a^5}{b^7 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{15 a^4 x^2}{2 b^7}-\frac{5 a^3 x^4}{2 b^6}+\frac{a^2 x^6}{b^5}-\frac{3 a x^8}{8 b^4}+\frac{x^{10}}{10 b^3}+\frac{a^7}{4 b^8 \left (a+b x^2\right )^2}-\frac{7 a^6}{2 b^8 \left (a+b x^2\right )}-\frac{21 a^5 \log \left (a+b x^2\right )}{2 b^8}\\ \end{align*}

Mathematica [A]  time = 0.0294884, size = 97, normalized size = 0.85 \[ \frac{40 a^2 b^3 x^6-100 a^3 b^2 x^4+300 a^4 b x^2-\frac{140 a^6}{a+b x^2}+\frac{10 a^7}{\left (a+b x^2\right )^2}-420 a^5 \log \left (a+b x^2\right )-15 a b^4 x^8+4 b^5 x^{10}}{40 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^15/(a + b*x^2)^3,x]

[Out]

(300*a^4*b*x^2 - 100*a^3*b^2*x^4 + 40*a^2*b^3*x^6 - 15*a*b^4*x^8 + 4*b^5*x^10 + (10*a^7)/(a + b*x^2)^2 - (140*
a^6)/(a + b*x^2) - 420*a^5*Log[a + b*x^2])/(40*b^8)

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Maple [A]  time = 0.01, size = 101, normalized size = 0.9 \begin{align*}{\frac{15\,{a}^{4}{x}^{2}}{2\,{b}^{7}}}-{\frac{5\,{a}^{3}{x}^{4}}{2\,{b}^{6}}}+{\frac{{a}^{2}{x}^{6}}{{b}^{5}}}-{\frac{3\,a{x}^{8}}{8\,{b}^{4}}}+{\frac{{x}^{10}}{10\,{b}^{3}}}+{\frac{{a}^{7}}{4\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{7\,{a}^{6}}{2\,{b}^{8} \left ( b{x}^{2}+a \right ) }}-{\frac{21\,{a}^{5}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15/(b*x^2+a)^3,x)

[Out]

15/2*a^4*x^2/b^7-5/2*a^3*x^4/b^6+a^2*x^6/b^5-3/8*a*x^8/b^4+1/10*x^10/b^3+1/4*a^7/b^8/(b*x^2+a)^2-7/2*a^6/b^8/(
b*x^2+a)-21/2*a^5*ln(b*x^2+a)/b^8

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Maxima [A]  time = 1.66549, size = 150, normalized size = 1.32 \begin{align*} -\frac{14 \, a^{6} b x^{2} + 13 \, a^{7}}{4 \,{\left (b^{10} x^{4} + 2 \, a b^{9} x^{2} + a^{2} b^{8}\right )}} - \frac{21 \, a^{5} \log \left (b x^{2} + a\right )}{2 \, b^{8}} + \frac{4 \, b^{4} x^{10} - 15 \, a b^{3} x^{8} + 40 \, a^{2} b^{2} x^{6} - 100 \, a^{3} b x^{4} + 300 \, a^{4} x^{2}}{40 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/4*(14*a^6*b*x^2 + 13*a^7)/(b^10*x^4 + 2*a*b^9*x^2 + a^2*b^8) - 21/2*a^5*log(b*x^2 + a)/b^8 + 1/40*(4*b^4*x^
10 - 15*a*b^3*x^8 + 40*a^2*b^2*x^6 - 100*a^3*b*x^4 + 300*a^4*x^2)/b^7

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Fricas [A]  time = 1.23095, size = 298, normalized size = 2.61 \begin{align*} \frac{4 \, b^{7} x^{14} - 7 \, a b^{6} x^{12} + 14 \, a^{2} b^{5} x^{10} - 35 \, a^{3} b^{4} x^{8} + 140 \, a^{4} b^{3} x^{6} + 500 \, a^{5} b^{2} x^{4} + 160 \, a^{6} b x^{2} - 130 \, a^{7} - 420 \,{\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{2} + a^{7}\right )} \log \left (b x^{2} + a\right )}{40 \,{\left (b^{10} x^{4} + 2 \, a b^{9} x^{2} + a^{2} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/40*(4*b^7*x^14 - 7*a*b^6*x^12 + 14*a^2*b^5*x^10 - 35*a^3*b^4*x^8 + 140*a^4*b^3*x^6 + 500*a^5*b^2*x^4 + 160*a
^6*b*x^2 - 130*a^7 - 420*(a^5*b^2*x^4 + 2*a^6*b*x^2 + a^7)*log(b*x^2 + a))/(b^10*x^4 + 2*a*b^9*x^2 + a^2*b^8)

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Sympy [A]  time = 0.668941, size = 117, normalized size = 1.03 \begin{align*} - \frac{21 a^{5} \log{\left (a + b x^{2} \right )}}{2 b^{8}} + \frac{15 a^{4} x^{2}}{2 b^{7}} - \frac{5 a^{3} x^{4}}{2 b^{6}} + \frac{a^{2} x^{6}}{b^{5}} - \frac{3 a x^{8}}{8 b^{4}} - \frac{13 a^{7} + 14 a^{6} b x^{2}}{4 a^{2} b^{8} + 8 a b^{9} x^{2} + 4 b^{10} x^{4}} + \frac{x^{10}}{10 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15/(b*x**2+a)**3,x)

[Out]

-21*a**5*log(a + b*x**2)/(2*b**8) + 15*a**4*x**2/(2*b**7) - 5*a**3*x**4/(2*b**6) + a**2*x**6/b**5 - 3*a*x**8/(
8*b**4) - (13*a**7 + 14*a**6*b*x**2)/(4*a**2*b**8 + 8*a*b**9*x**2 + 4*b**10*x**4) + x**10/(10*b**3)

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Giac [A]  time = 2.40065, size = 154, normalized size = 1.35 \begin{align*} -\frac{21 \, a^{5} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{8}} + \frac{63 \, a^{5} b^{2} x^{4} + 112 \, a^{6} b x^{2} + 50 \, a^{7}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{8}} + \frac{4 \, b^{12} x^{10} - 15 \, a b^{11} x^{8} + 40 \, a^{2} b^{10} x^{6} - 100 \, a^{3} b^{9} x^{4} + 300 \, a^{4} b^{8} x^{2}}{40 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^2+a)^3,x, algorithm="giac")

[Out]

-21/2*a^5*log(abs(b*x^2 + a))/b^8 + 1/4*(63*a^5*b^2*x^4 + 112*a^6*b*x^2 + 50*a^7)/((b*x^2 + a)^2*b^8) + 1/40*(
4*b^12*x^10 - 15*a*b^11*x^8 + 40*a^2*b^10*x^6 - 100*a^3*b^9*x^4 + 300*a^4*b^8*x^2)/b^15