3.3.74 \(\int \frac {(d+e x^2)^{3/2} (a+b \log (c x^n))}{x^{10}} \, dx\) [274]

Optimal. Leaf size=256 \[ -\frac {8 b e^4 n \sqrt {d+e x^2}}{315 d^3 x}-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}+\frac {8 b e^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{315 d^3}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5} \]

[Out]

-8/945*b*e^3*n*(e*x^2+d)^(3/2)/d^3/x^3-8/1575*b*e^2*n*(e*x^2+d)^(5/2)/d^3/x^5-1/81*b*n*(e*x^2+d)^(7/2)/d^2/x^9
+50/3969*b*e*n*(e*x^2+d)^(7/2)/d^3/x^7+8/315*b*e^(9/2)*n*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/d^3-1/9*(e*x^2+d)^
(5/2)*(a+b*ln(c*x^n))/d/x^9+4/63*e*(e*x^2+d)^(5/2)*(a+b*ln(c*x^n))/d^2/x^7-8/315*e^2*(e*x^2+d)^(5/2)*(a+b*ln(c
*x^n))/d^3/x^5-8/315*b*e^4*n*(e*x^2+d)^(1/2)/d^3/x

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Rubi [A]
time = 0.15, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {277, 270, 2392, 12, 1279, 462, 283, 223, 212} \begin {gather*} -\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {8 b e^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{315 d^3}-\frac {8 b e^4 n \sqrt {d+e x^2}}{315 d^3 x}-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/x^10,x]

[Out]

(-8*b*e^4*n*Sqrt[d + e*x^2])/(315*d^3*x) - (8*b*e^3*n*(d + e*x^2)^(3/2))/(945*d^3*x^3) - (8*b*e^2*n*(d + e*x^2
)^(5/2))/(1575*d^3*x^5) - (b*n*(d + e*x^2)^(7/2))/(81*d^2*x^9) + (50*b*e*n*(d + e*x^2)^(7/2))/(3969*d^3*x^7) +
(8*b*e^(9/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(315*d^3) - ((d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(9*d*
x^9) + (4*e*(d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(63*d^2*x^7) - (8*e^2*(d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/
(315*d^3*x^5)

Rule 12

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 462

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a,
b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (
(GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1]))

Rule 1279

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]

Rule 2392

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}-(b n) \int \frac {\left (d+e x^2\right )^{5/2} \left (-35 d^2+20 d e x^2-8 e^2 x^4\right )}{315 d^3 x^{10}} \, dx\\ &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}-\frac {(b n) \int \frac {\left (d+e x^2\right )^{5/2} \left (-35 d^2+20 d e x^2-8 e^2 x^4\right )}{x^{10}} \, dx}{315 d^3}\\ &=-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {(b n) \int \frac {\left (d+e x^2\right )^{5/2} \left (-250 d^2 e+72 d e^2 x^2\right )}{x^8} \, dx}{2835 d^4}\\ &=-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {\left (8 b e^2 n\right ) \int \frac {\left (d+e x^2\right )^{5/2}}{x^6} \, dx}{315 d^3}\\ &=-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {\left (8 b e^3 n\right ) \int \frac {\left (d+e x^2\right )^{3/2}}{x^4} \, dx}{315 d^3}\\ &=-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {\left (8 b e^4 n\right ) \int \frac {\sqrt {d+e x^2}}{x^2} \, dx}{315 d^3}\\ &=-\frac {8 b e^4 n \sqrt {d+e x^2}}{315 d^3 x}-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {\left (8 b e^5 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{315 d^3}\\ &=-\frac {8 b e^4 n \sqrt {d+e x^2}}{315 d^3 x}-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac {\left (8 b e^5 n\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{315 d^3}\\ &=-\frac {8 b e^4 n \sqrt {d+e x^2}}{315 d^3 x}-\frac {8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac {8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac {50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}+\frac {8 b e^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{315 d^3}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac {4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac {8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 178, normalized size = 0.70 \begin {gather*} -\frac {\sqrt {d+e x^2} \left (315 a \left (d+e x^2\right )^2 \left (35 d^2-20 d e x^2+8 e^2 x^4\right )+b n \left (1225 d^4+2425 d^3 e x^2+429 d^2 e^2 x^4-677 d e^3 x^6+2614 e^4 x^8\right )\right )+315 b \left (d+e x^2\right )^{5/2} \left (35 d^2-20 d e x^2+8 e^2 x^4\right ) \log \left (c x^n\right )-2520 b e^{9/2} n x^9 \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{99225 d^3 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/x^10,x]

[Out]

-1/99225*(Sqrt[d + e*x^2]*(315*a*(d + e*x^2)^2*(35*d^2 - 20*d*e*x^2 + 8*e^2*x^4) + b*n*(1225*d^4 + 2425*d^3*e*
x^2 + 429*d^2*e^2*x^4 - 677*d*e^3*x^6 + 2614*e^4*x^8)) + 315*b*(d + e*x^2)^(5/2)*(35*d^2 - 20*d*e*x^2 + 8*e^2*
x^4)*Log[c*x^n] - 2520*b*e^(9/2)*n*x^9*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(d^3*x^9)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{10}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/x^10,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/x^10,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*log(c*x^n))/x^10,x, algorithm="maxima")

[Out]

-1/315*a*(8*(x^2*e + d)^(5/2)*e^2/(d^3*x^5) - 20*(x^2*e + d)^(5/2)*e/(d^2*x^7) + 35*(x^2*e + d)^(5/2)/(d*x^9))
+ b*integrate((x^2*e*log(c) + d*log(c) + (x^2*e + d)*log(x^n))*sqrt(x^2*e + d)/x^10, x)

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Fricas [A]
time = 0.51, size = 250, normalized size = 0.98 \begin {gather*} \frac {1260 \, b n x^{9} e^{\frac {9}{2}} \log \left (-2 \, x^{2} e - 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) - {\left (2 \, {\left (1307 \, b n + 1260 \, a\right )} x^{8} e^{4} - {\left (677 \, b d n + 1260 \, a d\right )} x^{6} e^{3} + 1225 \, b d^{4} n + 3 \, {\left (143 \, b d^{2} n + 315 \, a d^{2}\right )} x^{4} e^{2} + 11025 \, a d^{4} + 25 \, {\left (97 \, b d^{3} n + 630 \, a d^{3}\right )} x^{2} e + 315 \, {\left (8 \, b x^{8} e^{4} - 4 \, b d x^{6} e^{3} + 3 \, b d^{2} x^{4} e^{2} + 50 \, b d^{3} x^{2} e + 35 \, b d^{4}\right )} \log \left (c\right ) + 315 \, {\left (8 \, b n x^{8} e^{4} - 4 \, b d n x^{6} e^{3} + 3 \, b d^{2} n x^{4} e^{2} + 50 \, b d^{3} n x^{2} e + 35 \, b d^{4} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{99225 \, d^{3} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*log(c*x^n))/x^10,x, algorithm="fricas")

[Out]

1/99225*(1260*b*n*x^9*e^(9/2)*log(-2*x^2*e - 2*sqrt(x^2*e + d)*x*e^(1/2) - d) - (2*(1307*b*n + 1260*a)*x^8*e^4
- (677*b*d*n + 1260*a*d)*x^6*e^3 + 1225*b*d^4*n + 3*(143*b*d^2*n + 315*a*d^2)*x^4*e^2 + 11025*a*d^4 + 25*(97*
b*d^3*n + 630*a*d^3)*x^2*e + 315*(8*b*x^8*e^4 - 4*b*d*x^6*e^3 + 3*b*d^2*x^4*e^2 + 50*b*d^3*x^2*e + 35*b*d^4)*l
og(c) + 315*(8*b*n*x^8*e^4 - 4*b*d*n*x^6*e^3 + 3*b*d^2*n*x^4*e^2 + 50*b*d^3*n*x^2*e + 35*b*d^4*n)*log(x))*sqrt
(x^2*e + d))/(d^3*x^9)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(3/2)*(a+b*ln(c*x**n))/x**10,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3061 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*log(c*x^n))/x^10,x, algorithm="giac")

[Out]

integrate((x^2*e + d)^(3/2)*(b*log(c*x^n) + a)/x^10, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^{10}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^(3/2)*(a + b*log(c*x^n)))/x^10,x)

[Out]

int(((d + e*x^2)^(3/2)*(a + b*log(c*x^n)))/x^10, x)

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