3.407 \(\int \frac{a+b \log (c (d+e \sqrt{x})^n)}{x^4} \, dx\)

Optimal. Leaf size=141 \[ -\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{3 x^3}-\frac{b e^3 n}{9 d^3 x^{3/2}}+\frac{b e^2 n}{12 d^2 x^2}-\frac{b e^5 n}{3 d^5 \sqrt{x}}+\frac{b e^4 n}{6 d^4 x}+\frac{b e^6 n \log \left (d+e \sqrt{x}\right )}{3 d^6}-\frac{b e^6 n \log (x)}{6 d^6}-\frac{b e n}{15 d x^{5/2}} \]

[Out]

-(b*e*n)/(15*d*x^(5/2)) + (b*e^2*n)/(12*d^2*x^2) - (b*e^3*n)/(9*d^3*x^(3/2)) + (b*e^4*n)/(6*d^4*x) - (b*e^5*n)
/(3*d^5*Sqrt[x]) + (b*e^6*n*Log[d + e*Sqrt[x]])/(3*d^6) - (a + b*Log[c*(d + e*Sqrt[x])^n])/(3*x^3) - (b*e^6*n*
Log[x])/(6*d^6)

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Rubi [A]  time = 0.0921506, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2454, 2395, 44} \[ -\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{3 x^3}-\frac{b e^3 n}{9 d^3 x^{3/2}}+\frac{b e^2 n}{12 d^2 x^2}-\frac{b e^5 n}{3 d^5 \sqrt{x}}+\frac{b e^4 n}{6 d^4 x}+\frac{b e^6 n \log \left (d+e \sqrt{x}\right )}{3 d^6}-\frac{b e^6 n \log (x)}{6 d^6}-\frac{b e n}{15 d x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*Sqrt[x])^n])/x^4,x]

[Out]

-(b*e*n)/(15*d*x^(5/2)) + (b*e^2*n)/(12*d^2*x^2) - (b*e^3*n)/(9*d^3*x^(3/2)) + (b*e^4*n)/(6*d^4*x) - (b*e^5*n)
/(3*d^5*Sqrt[x]) + (b*e^6*n*Log[d + e*Sqrt[x]])/(3*d^6) - (a + b*Log[c*(d + e*Sqrt[x])^n])/(3*x^3) - (b*e^6*n*
Log[x])/(6*d^6)

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{x^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^7} \, dx,x,\sqrt{x}\right )\\ &=-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{3 x^3}+\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \frac{1}{x^6 (d+e x)} \, dx,x,\sqrt{x}\right )\\ &=-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{3 x^3}+\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \left (\frac{1}{d x^6}-\frac{e}{d^2 x^5}+\frac{e^2}{d^3 x^4}-\frac{e^3}{d^4 x^3}+\frac{e^4}{d^5 x^2}-\frac{e^5}{d^6 x}+\frac{e^6}{d^6 (d+e x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{b e n}{15 d x^{5/2}}+\frac{b e^2 n}{12 d^2 x^2}-\frac{b e^3 n}{9 d^3 x^{3/2}}+\frac{b e^4 n}{6 d^4 x}-\frac{b e^5 n}{3 d^5 \sqrt{x}}+\frac{b e^6 n \log \left (d+e \sqrt{x}\right )}{3 d^6}-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{3 x^3}-\frac{b e^6 n \log (x)}{6 d^6}\\ \end{align*}

Mathematica [A]  time = 0.131117, size = 132, normalized size = 0.94 \[ -\frac{a}{3 x^3}-\frac{b \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{3 x^3}+\frac{1}{3} b e n \left (-\frac{e^2}{3 d^3 x^{3/2}}-\frac{e^4}{d^5 \sqrt{x}}+\frac{e^3}{2 d^4 x}+\frac{e^5 \log \left (d+e \sqrt{x}\right )}{d^6}-\frac{e^5 \log (x)}{2 d^6}+\frac{e}{4 d^2 x^2}-\frac{1}{5 d x^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*Sqrt[x])^n])/x^4,x]

[Out]

-a/(3*x^3) - (b*Log[c*(d + e*Sqrt[x])^n])/(3*x^3) + (b*e*n*(-1/(5*d*x^(5/2)) + e/(4*d^2*x^2) - e^2/(3*d^3*x^(3
/2)) + e^3/(2*d^4*x) - e^4/(d^5*Sqrt[x]) + (e^5*Log[d + e*Sqrt[x]])/d^6 - (e^5*Log[x])/(2*d^6)))/3

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Maple [F]  time = 0.1, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e*x^(1/2))^n))/x^4,x)

[Out]

int((a+b*ln(c*(d+e*x^(1/2))^n))/x^4,x)

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Maxima [A]  time = 1.06341, size = 143, normalized size = 1.01 \begin{align*} \frac{1}{180} \, b e n{\left (\frac{60 \, e^{5} \log \left (e \sqrt{x} + d\right )}{d^{6}} - \frac{30 \, e^{5} \log \left (x\right )}{d^{6}} - \frac{60 \, e^{4} x^{2} - 30 \, d e^{3} x^{\frac{3}{2}} + 20 \, d^{2} e^{2} x - 15 \, d^{3} e \sqrt{x} + 12 \, d^{4}}{d^{5} x^{\frac{5}{2}}}\right )} - \frac{b \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )}{3 \, x^{3}} - \frac{a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^4,x, algorithm="maxima")

[Out]

1/180*b*e*n*(60*e^5*log(e*sqrt(x) + d)/d^6 - 30*e^5*log(x)/d^6 - (60*e^4*x^2 - 30*d*e^3*x^(3/2) + 20*d^2*e^2*x
- 15*d^3*e*sqrt(x) + 12*d^4)/(d^5*x^(5/2))) - 1/3*b*log((e*sqrt(x) + d)^n*c)/x^3 - 1/3*a/x^3

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Fricas [A]  time = 2.08661, size = 308, normalized size = 2.18 \begin{align*} -\frac{60 \, b e^{6} n x^{3} \log \left (\sqrt{x}\right ) - 30 \, b d^{2} e^{4} n x^{2} - 15 \, b d^{4} e^{2} n x + 60 \, b d^{6} \log \left (c\right ) + 60 \, a d^{6} - 60 \,{\left (b e^{6} n x^{3} - b d^{6} n\right )} \log \left (e \sqrt{x} + d\right ) + 4 \,{\left (15 \, b d e^{5} n x^{2} + 5 \, b d^{3} e^{3} n x + 3 \, b d^{5} e n\right )} \sqrt{x}}{180 \, d^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^4,x, algorithm="fricas")

[Out]

-1/180*(60*b*e^6*n*x^3*log(sqrt(x)) - 30*b*d^2*e^4*n*x^2 - 15*b*d^4*e^2*n*x + 60*b*d^6*log(c) + 60*a*d^6 - 60*
(b*e^6*n*x^3 - b*d^6*n)*log(e*sqrt(x) + d) + 4*(15*b*d*e^5*n*x^2 + 5*b*d^3*e^3*n*x + 3*b*d^5*e*n)*sqrt(x))/(d^
6*x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d+e*x**(1/2))**n))/x**4,x)

[Out]

Timed out

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Giac [B]  time = 1.3381, size = 732, normalized size = 5.19 \begin{align*} \frac{{\left (60 \,{\left (\sqrt{x} e + d\right )}^{6} b n e^{7} \log \left (\sqrt{x} e + d\right ) - 360 \,{\left (\sqrt{x} e + d\right )}^{5} b d n e^{7} \log \left (\sqrt{x} e + d\right ) + 900 \,{\left (\sqrt{x} e + d\right )}^{4} b d^{2} n e^{7} \log \left (\sqrt{x} e + d\right ) - 1200 \,{\left (\sqrt{x} e + d\right )}^{3} b d^{3} n e^{7} \log \left (\sqrt{x} e + d\right ) + 900 \,{\left (\sqrt{x} e + d\right )}^{2} b d^{4} n e^{7} \log \left (\sqrt{x} e + d\right ) - 360 \,{\left (\sqrt{x} e + d\right )} b d^{5} n e^{7} \log \left (\sqrt{x} e + d\right ) - 60 \,{\left (\sqrt{x} e + d\right )}^{6} b n e^{7} \log \left (\sqrt{x} e\right ) + 360 \,{\left (\sqrt{x} e + d\right )}^{5} b d n e^{7} \log \left (\sqrt{x} e\right ) - 900 \,{\left (\sqrt{x} e + d\right )}^{4} b d^{2} n e^{7} \log \left (\sqrt{x} e\right ) + 1200 \,{\left (\sqrt{x} e + d\right )}^{3} b d^{3} n e^{7} \log \left (\sqrt{x} e\right ) - 900 \,{\left (\sqrt{x} e + d\right )}^{2} b d^{4} n e^{7} \log \left (\sqrt{x} e\right ) + 360 \,{\left (\sqrt{x} e + d\right )} b d^{5} n e^{7} \log \left (\sqrt{x} e\right ) - 60 \, b d^{6} n e^{7} \log \left (\sqrt{x} e\right ) - 60 \,{\left (\sqrt{x} e + d\right )}^{5} b d n e^{7} + 330 \,{\left (\sqrt{x} e + d\right )}^{4} b d^{2} n e^{7} - 740 \,{\left (\sqrt{x} e + d\right )}^{3} b d^{3} n e^{7} + 855 \,{\left (\sqrt{x} e + d\right )}^{2} b d^{4} n e^{7} - 522 \,{\left (\sqrt{x} e + d\right )} b d^{5} n e^{7} + 137 \, b d^{6} n e^{7} - 60 \, b d^{6} e^{7} \log \left (c\right ) - 60 \, a d^{6} e^{7}\right )} e^{\left (-1\right )}}{180 \,{\left ({\left (\sqrt{x} e + d\right )}^{6} d^{6} - 6 \,{\left (\sqrt{x} e + d\right )}^{5} d^{7} + 15 \,{\left (\sqrt{x} e + d\right )}^{4} d^{8} - 20 \,{\left (\sqrt{x} e + d\right )}^{3} d^{9} + 15 \,{\left (\sqrt{x} e + d\right )}^{2} d^{10} - 6 \,{\left (\sqrt{x} e + d\right )} d^{11} + d^{12}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^4,x, algorithm="giac")

[Out]

1/180*(60*(sqrt(x)*e + d)^6*b*n*e^7*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)^5*b*d*n*e^7*log(sqrt(x)*e + d) +
900*(sqrt(x)*e + d)^4*b*d^2*n*e^7*log(sqrt(x)*e + d) - 1200*(sqrt(x)*e + d)^3*b*d^3*n*e^7*log(sqrt(x)*e + d) +
900*(sqrt(x)*e + d)^2*b*d^4*n*e^7*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)*b*d^5*n*e^7*log(sqrt(x)*e + d) - 6
0*(sqrt(x)*e + d)^6*b*n*e^7*log(sqrt(x)*e) + 360*(sqrt(x)*e + d)^5*b*d*n*e^7*log(sqrt(x)*e) - 900*(sqrt(x)*e +
d)^4*b*d^2*n*e^7*log(sqrt(x)*e) + 1200*(sqrt(x)*e + d)^3*b*d^3*n*e^7*log(sqrt(x)*e) - 900*(sqrt(x)*e + d)^2*b
*d^4*n*e^7*log(sqrt(x)*e) + 360*(sqrt(x)*e + d)*b*d^5*n*e^7*log(sqrt(x)*e) - 60*b*d^6*n*e^7*log(sqrt(x)*e) - 6
0*(sqrt(x)*e + d)^5*b*d*n*e^7 + 330*(sqrt(x)*e + d)^4*b*d^2*n*e^7 - 740*(sqrt(x)*e + d)^3*b*d^3*n*e^7 + 855*(s
qrt(x)*e + d)^2*b*d^4*n*e^7 - 522*(sqrt(x)*e + d)*b*d^5*n*e^7 + 137*b*d^6*n*e^7 - 60*b*d^6*e^7*log(c) - 60*a*d
^6*e^7)*e^(-1)/((sqrt(x)*e + d)^6*d^6 - 6*(sqrt(x)*e + d)^5*d^7 + 15*(sqrt(x)*e + d)^4*d^8 - 20*(sqrt(x)*e + d
)^3*d^9 + 15*(sqrt(x)*e + d)^2*d^10 - 6*(sqrt(x)*e + d)*d^11 + d^12)