3.1136 \(\int \frac{(d+e x^2)^2 (a+b \tan ^{-1}(c x))}{x^8} \, dx\)

Optimal. Leaf size=186 \[ -\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{b c \left (15 c^4 d^2-42 c^2 d e+35 e^2\right )}{210 x^2}+\frac{1}{210} b c^3 \left (15 c^4 d^2-42 c^2 d e+35 e^2\right ) \log \left (c^2 x^2+1\right )-\frac{1}{105} b c^3 \log (x) \left (15 c^4 d^2-42 c^2 d e+35 e^2\right )+\frac{b c d \left (5 c^2 d-14 e\right )}{140 x^4}-\frac{b c d^2}{42 x^6} \]

[Out]

-(b*c*d^2)/(42*x^6) + (b*c*d*(5*c^2*d - 14*e))/(140*x^4) - (b*c*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2))/(210*x^2)
- (d^2*(a + b*ArcTan[c*x]))/(7*x^7) - (2*d*e*(a + b*ArcTan[c*x]))/(5*x^5) - (e^2*(a + b*ArcTan[c*x]))/(3*x^3)
- (b*c^3*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2)*Log[x])/105 + (b*c^3*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2)*Log[1 + c^
2*x^2])/210

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Rubi [A]  time = 0.232365, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {270, 4976, 12, 1251, 893} \[ -\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{b c \left (15 c^4 d^2-42 c^2 d e+35 e^2\right )}{210 x^2}+\frac{1}{210} b c^3 \left (15 c^4 d^2-42 c^2 d e+35 e^2\right ) \log \left (c^2 x^2+1\right )-\frac{1}{105} b c^3 \log (x) \left (15 c^4 d^2-42 c^2 d e+35 e^2\right )+\frac{b c d \left (5 c^2 d-14 e\right )}{140 x^4}-\frac{b c d^2}{42 x^6} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^8,x]

[Out]

-(b*c*d^2)/(42*x^6) + (b*c*d*(5*c^2*d - 14*e))/(140*x^4) - (b*c*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2))/(210*x^2)
- (d^2*(a + b*ArcTan[c*x]))/(7*x^7) - (2*d*e*(a + b*ArcTan[c*x]))/(5*x^5) - (e^2*(a + b*ArcTan[c*x]))/(3*x^3)
- (b*c^3*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2)*Log[x])/105 + (b*c^3*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2)*Log[1 + c^
2*x^2])/210

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 4976

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^
2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m +
2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] &&
!ILtQ[(m - 1)/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )}{x^8} \, dx &=-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac{-15 d^2-42 d e x^2-35 e^2 x^4}{105 x^7 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{105} (b c) \int \frac{-15 d^2-42 d e x^2-35 e^2 x^4}{x^7 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{210} (b c) \operatorname{Subst}\left (\int \frac{-15 d^2-42 d e x-35 e^2 x^2}{x^4 \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{210} (b c) \operatorname{Subst}\left (\int \left (-\frac{15 d^2}{x^4}+\frac{3 d \left (5 c^2 d-14 e\right )}{x^3}+\frac{-15 c^4 d^2+42 c^2 d e-35 e^2}{x^2}+\frac{15 c^6 d^2-42 c^4 d e+35 c^2 e^2}{x}+\frac{-15 c^8 d^2+42 c^6 d e-35 c^4 e^2}{1+c^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b c d^2}{42 x^6}+\frac{b c d \left (5 c^2 d-14 e\right )}{140 x^4}-\frac{b c \left (15 c^4 d^2-42 c^2 d e+35 e^2\right )}{210 x^2}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{105} b c^3 \left (15 c^4 d^2-42 c^2 d e+35 e^2\right ) \log (x)+\frac{1}{210} b c^3 \left (15 c^4 d^2-42 c^2 d e+35 e^2\right ) \log \left (1+c^2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.172982, size = 177, normalized size = 0.95 \[ \frac{1}{420} \left (-\frac{60 d^2 \left (a+b \tan ^{-1}(c x)\right )}{x^7}-\frac{168 d e \left (a+b \tan ^{-1}(c x)\right )}{x^5}-\frac{140 e^2 \left (a+b \tan ^{-1}(c x)\right )}{x^3}-5 b c d^2 \left (\frac{6 c^4 x^4-3 c^2 x^2+2}{x^6}-6 c^6 \log \left (c^2 x^2+1\right )+12 c^6 \log (x)\right )-42 b c d e \left (-\frac{2 c^2}{x^2}+2 c^4 \log \left (c^2 x^2+1\right )-4 c^4 \log (x)+\frac{1}{x^4}\right )-70 b c e^2 \left (-c^2 \log \left (c^2 x^2+1\right )+2 c^2 \log (x)+\frac{1}{x^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^8,x]

[Out]

((-60*d^2*(a + b*ArcTan[c*x]))/x^7 - (168*d*e*(a + b*ArcTan[c*x]))/x^5 - (140*e^2*(a + b*ArcTan[c*x]))/x^3 - 7
0*b*c*e^2*(x^(-2) + 2*c^2*Log[x] - c^2*Log[1 + c^2*x^2]) - 42*b*c*d*e*(x^(-4) - (2*c^2)/x^2 - 4*c^4*Log[x] + 2
*c^4*Log[1 + c^2*x^2]) - 5*b*c*d^2*((2 - 3*c^2*x^2 + 6*c^4*x^4)/x^6 + 12*c^6*Log[x] - 6*c^6*Log[1 + c^2*x^2]))
/420

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Maple [A]  time = 0.047, size = 224, normalized size = 1.2 \begin{align*} -{\frac{a{d}^{2}}{7\,{x}^{7}}}-{\frac{2\,aed}{5\,{x}^{5}}}-{\frac{a{e}^{2}}{3\,{x}^{3}}}-{\frac{b{d}^{2}\arctan \left ( cx \right ) }{7\,{x}^{7}}}-{\frac{2\,b\arctan \left ( cx \right ) ed}{5\,{x}^{5}}}-{\frac{b\arctan \left ( cx \right ){e}^{2}}{3\,{x}^{3}}}+{\frac{{c}^{7}b\ln \left ({c}^{2}{x}^{2}+1 \right ){d}^{2}}{14}}-{\frac{{c}^{5}b\ln \left ({c}^{2}{x}^{2}+1 \right ) ed}{5}}+{\frac{{c}^{3}b\ln \left ({c}^{2}{x}^{2}+1 \right ){e}^{2}}{6}}-{\frac{{c}^{5}b{d}^{2}}{14\,{x}^{2}}}+{\frac{{c}^{3}bed}{5\,{x}^{2}}}-{\frac{cb{e}^{2}}{6\,{x}^{2}}}-{\frac{{c}^{7}b{d}^{2}\ln \left ( cx \right ) }{7}}+{\frac{2\,{c}^{5}b\ln \left ( cx \right ) de}{5}}-{\frac{{c}^{3}b\ln \left ( cx \right ){e}^{2}}{3}}-{\frac{cb{d}^{2}}{42\,{x}^{6}}}+{\frac{{c}^{3}b{d}^{2}}{28\,{x}^{4}}}-{\frac{bced}{10\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^2*(a+b*arctan(c*x))/x^8,x)

[Out]

-1/7*a*d^2/x^7-2/5*a*e*d/x^5-1/3*a*e^2/x^3-1/7*b*arctan(c*x)*d^2/x^7-2/5*b*arctan(c*x)*e*d/x^5-1/3*b*arctan(c*
x)*e^2/x^3+1/14*c^7*b*ln(c^2*x^2+1)*d^2-1/5*c^5*b*ln(c^2*x^2+1)*e*d+1/6*c^3*b*ln(c^2*x^2+1)*e^2-1/14*c^5*b*d^2
/x^2+1/5*c^3*b*e*d/x^2-1/6*c*b*e^2/x^2-1/7*c^7*b*d^2*ln(c*x)+2/5*c^5*b*ln(c*x)*d*e-1/3*c^3*b*ln(c*x)*e^2-1/42*
b*c*d^2/x^6+1/28*c^3*b*d^2/x^4-1/10*c*b*e*d/x^4

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Maxima [A]  time = 0.974972, size = 266, normalized size = 1.43 \begin{align*} \frac{1}{84} \,{\left ({\left (6 \, c^{6} \log \left (c^{2} x^{2} + 1\right ) - 6 \, c^{6} \log \left (x^{2}\right ) - \frac{6 \, c^{4} x^{4} - 3 \, c^{2} x^{2} + 2}{x^{6}}\right )} c - \frac{12 \, \arctan \left (c x\right )}{x^{7}}\right )} b d^{2} - \frac{1}{10} \,{\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac{2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac{4 \, \arctan \left (c x\right )}{x^{5}}\right )} b d e + \frac{1}{6} \,{\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac{1}{x^{2}}\right )} c - \frac{2 \, \arctan \left (c x\right )}{x^{3}}\right )} b e^{2} - \frac{a e^{2}}{3 \, x^{3}} - \frac{2 \, a d e}{5 \, x^{5}} - \frac{a d^{2}}{7 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arctan(c*x))/x^8,x, algorithm="maxima")

[Out]

1/84*((6*c^6*log(c^2*x^2 + 1) - 6*c^6*log(x^2) - (6*c^4*x^4 - 3*c^2*x^2 + 2)/x^6)*c - 12*arctan(c*x)/x^7)*b*d^
2 - 1/10*((2*c^4*log(c^2*x^2 + 1) - 2*c^4*log(x^2) - (2*c^2*x^2 - 1)/x^4)*c + 4*arctan(c*x)/x^5)*b*d*e + 1/6*(
(c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)*b*e^2 - 1/3*a*e^2/x^3 - 2/5*a*d*e/x^5 - 1
/7*a*d^2/x^7

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Fricas [A]  time = 1.56819, size = 462, normalized size = 2.48 \begin{align*} \frac{2 \,{\left (15 \, b c^{7} d^{2} - 42 \, b c^{5} d e + 35 \, b c^{3} e^{2}\right )} x^{7} \log \left (c^{2} x^{2} + 1\right ) - 4 \,{\left (15 \, b c^{7} d^{2} - 42 \, b c^{5} d e + 35 \, b c^{3} e^{2}\right )} x^{7} \log \left (x\right ) - 140 \, a e^{2} x^{4} - 2 \,{\left (15 \, b c^{5} d^{2} - 42 \, b c^{3} d e + 35 \, b c e^{2}\right )} x^{5} - 10 \, b c d^{2} x - 168 \, a d e x^{2} + 3 \,{\left (5 \, b c^{3} d^{2} - 14 \, b c d e\right )} x^{3} - 60 \, a d^{2} - 4 \,{\left (35 \, b e^{2} x^{4} + 42 \, b d e x^{2} + 15 \, b d^{2}\right )} \arctan \left (c x\right )}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arctan(c*x))/x^8,x, algorithm="fricas")

[Out]

1/420*(2*(15*b*c^7*d^2 - 42*b*c^5*d*e + 35*b*c^3*e^2)*x^7*log(c^2*x^2 + 1) - 4*(15*b*c^7*d^2 - 42*b*c^5*d*e +
35*b*c^3*e^2)*x^7*log(x) - 140*a*e^2*x^4 - 2*(15*b*c^5*d^2 - 42*b*c^3*d*e + 35*b*c*e^2)*x^5 - 10*b*c*d^2*x - 1
68*a*d*e*x^2 + 3*(5*b*c^3*d^2 - 14*b*c*d*e)*x^3 - 60*a*d^2 - 4*(35*b*e^2*x^4 + 42*b*d*e*x^2 + 15*b*d^2)*arctan
(c*x))/x^7

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Sympy [A]  time = 5.4085, size = 289, normalized size = 1.55 \begin{align*} \begin{cases} - \frac{a d^{2}}{7 x^{7}} - \frac{2 a d e}{5 x^{5}} - \frac{a e^{2}}{3 x^{3}} - \frac{b c^{7} d^{2} \log{\left (x \right )}}{7} + \frac{b c^{7} d^{2} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{14} - \frac{b c^{5} d^{2}}{14 x^{2}} + \frac{2 b c^{5} d e \log{\left (x \right )}}{5} - \frac{b c^{5} d e \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{5} + \frac{b c^{3} d^{2}}{28 x^{4}} + \frac{b c^{3} d e}{5 x^{2}} - \frac{b c^{3} e^{2} \log{\left (x \right )}}{3} + \frac{b c^{3} e^{2} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{6} - \frac{b c d^{2}}{42 x^{6}} - \frac{b c d e}{10 x^{4}} - \frac{b c e^{2}}{6 x^{2}} - \frac{b d^{2} \operatorname{atan}{\left (c x \right )}}{7 x^{7}} - \frac{2 b d e \operatorname{atan}{\left (c x \right )}}{5 x^{5}} - \frac{b e^{2} \operatorname{atan}{\left (c x \right )}}{3 x^{3}} & \text{for}\: c \neq 0 \\a \left (- \frac{d^{2}}{7 x^{7}} - \frac{2 d e}{5 x^{5}} - \frac{e^{2}}{3 x^{3}}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**2*(a+b*atan(c*x))/x**8,x)

[Out]

Piecewise((-a*d**2/(7*x**7) - 2*a*d*e/(5*x**5) - a*e**2/(3*x**3) - b*c**7*d**2*log(x)/7 + b*c**7*d**2*log(x**2
+ c**(-2))/14 - b*c**5*d**2/(14*x**2) + 2*b*c**5*d*e*log(x)/5 - b*c**5*d*e*log(x**2 + c**(-2))/5 + b*c**3*d**
2/(28*x**4) + b*c**3*d*e/(5*x**2) - b*c**3*e**2*log(x)/3 + b*c**3*e**2*log(x**2 + c**(-2))/6 - b*c*d**2/(42*x*
*6) - b*c*d*e/(10*x**4) - b*c*e**2/(6*x**2) - b*d**2*atan(c*x)/(7*x**7) - 2*b*d*e*atan(c*x)/(5*x**5) - b*e**2*
atan(c*x)/(3*x**3), Ne(c, 0)), (a*(-d**2/(7*x**7) - 2*d*e/(5*x**5) - e**2/(3*x**3)), True))

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Giac [A]  time = 1.09629, size = 315, normalized size = 1.69 \begin{align*} \frac{30 \, b c^{7} d^{2} x^{7} \log \left (c^{2} x^{2} + 1\right ) - 60 \, b c^{7} d^{2} x^{7} \log \left (x\right ) - 84 \, b c^{5} d x^{7} e \log \left (c^{2} x^{2} + 1\right ) + 168 \, b c^{5} d x^{7} e \log \left (x\right ) - 30 \, b c^{5} d^{2} x^{5} + 70 \, b c^{3} x^{7} e^{2} \log \left (c^{2} x^{2} + 1\right ) - 140 \, b c^{3} x^{7} e^{2} \log \left (x\right ) + 84 \, b c^{3} d x^{5} e + 15 \, b c^{3} d^{2} x^{3} - 70 \, b c x^{5} e^{2} - 140 \, b x^{4} \arctan \left (c x\right ) e^{2} - 42 \, b c d x^{3} e - 140 \, a x^{4} e^{2} - 168 \, b d x^{2} \arctan \left (c x\right ) e - 10 \, b c d^{2} x - 168 \, a d x^{2} e - 60 \, b d^{2} \arctan \left (c x\right ) - 60 \, a d^{2}}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arctan(c*x))/x^8,x, algorithm="giac")

[Out]

1/420*(30*b*c^7*d^2*x^7*log(c^2*x^2 + 1) - 60*b*c^7*d^2*x^7*log(x) - 84*b*c^5*d*x^7*e*log(c^2*x^2 + 1) + 168*b
*c^5*d*x^7*e*log(x) - 30*b*c^5*d^2*x^5 + 70*b*c^3*x^7*e^2*log(c^2*x^2 + 1) - 140*b*c^3*x^7*e^2*log(x) + 84*b*c
^3*d*x^5*e + 15*b*c^3*d^2*x^3 - 70*b*c*x^5*e^2 - 140*b*x^4*arctan(c*x)*e^2 - 42*b*c*d*x^3*e - 140*a*x^4*e^2 -
168*b*d*x^2*arctan(c*x)*e - 10*b*c*d^2*x - 168*a*d*x^2*e - 60*b*d^2*arctan(c*x) - 60*a*d^2)/x^7