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

Optimal. Leaf size=136 \[ -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 e^6}-\frac {b d^5 n}{3 e^5 \sqrt {x}}+\frac {b d^4 n}{6 e^4 x}-\frac {b d^3 n}{9 e^3 x^{3/2}}+\frac {b d^2 n}{12 e^2 x^2}-\frac {b d n}{15 e x^{5/2}}+\frac {b n}{18 x^3} \]

[Out]

1/18*b*n/x^3-1/15*b*d*n/e/x^(5/2)+1/12*b*d^2*n/e^2/x^2-1/9*b*d^3*n/e^3/x^(3/2)+1/6*b*d^4*n/e^4/x+1/3*b*d^6*n*l
n(d+e/x^(1/2))/e^6+1/3*(-a-b*ln(c*(d+e/x^(1/2))^n))/x^3-1/3*b*d^5*n/e^5/x^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 136, normalized size of antiderivative = 1.00, 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, 43} \[ -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3}-\frac {b d^3 n}{9 e^3 x^{3/2}}+\frac {b d^2 n}{12 e^2 x^2}-\frac {b d^5 n}{3 e^5 \sqrt {x}}+\frac {b d^4 n}{6 e^4 x}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 e^6}-\frac {b d n}{15 e x^{5/2}}+\frac {b n}{18 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*n)/(18*x^3) - (b*d*n)/(15*e*x^(5/2)) + (b*d^2*n)/(12*e^2*x^2) - (b*d^3*n)/(9*e^3*x^(3/2)) + (b*d^4*n)/(6*e^
4*x) - (b*d^5*n)/(3*e^5*Sqrt[x]) + (b*d^6*n*Log[d + e/Sqrt[x]])/(3*e^6) - (a + b*Log[c*(d + e/Sqrt[x])^n])/(3*
x^3)

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])

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 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])

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx &=-\left (2 \operatorname {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3}+\frac {1}{3} (b e n) \operatorname {Subst}\left (\int \frac {x^6}{d+e x} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3}+\frac {1}{3} (b e n) \operatorname {Subst}\left (\int \left (-\frac {d^5}{e^6}+\frac {d^4 x}{e^5}-\frac {d^3 x^2}{e^4}+\frac {d^2 x^3}{e^3}-\frac {d x^4}{e^2}+\frac {x^5}{e}+\frac {d^6}{e^6 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=\frac {b n}{18 x^3}-\frac {b d n}{15 e x^{5/2}}+\frac {b d^2 n}{12 e^2 x^2}-\frac {b d^3 n}{9 e^3 x^{3/2}}+\frac {b d^4 n}{6 e^4 x}-\frac {b d^5 n}{3 e^5 \sqrt {x}}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 e^6}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 133, normalized size = 0.98 \[ -\frac {a}{3 x^3}-\frac {b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3}+\frac {1}{3} b e n \left (\frac {d^6 \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^7}-\frac {d^5}{e^6 \sqrt {x}}+\frac {d^4}{2 e^5 x}-\frac {d^3}{3 e^4 x^{3/2}}+\frac {d^2}{4 e^3 x^2}-\frac {d}{5 e^2 x^{5/2}}+\frac {1}{6 e x^3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*a/x^3 + (b*e*n*(1/(6*e*x^3) - d/(5*e^2*x^(5/2)) + d^2/(4*e^3*x^2) - d^3/(3*e^4*x^(3/2)) + d^4/(2*e^5*x) -
d^5/(e^6*Sqrt[x]) + (d^6*Log[d + e/Sqrt[x]])/e^7))/3 - (b*Log[c*(d + e/Sqrt[x])^n])/(3*x^3)

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fricas [A]  time = 0.42, size = 123, normalized size = 0.90 \[ \frac {30 \, b d^{4} e^{2} n x^{2} + 15 \, b d^{2} e^{4} n x + 10 \, b e^{6} n - 60 \, b e^{6} \log \relax (c) - 60 \, a e^{6} + 60 \, {\left (b d^{6} n x^{3} - b e^{6} n\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) - 4 \, {\left (15 \, b d^{5} e n x^{2} + 5 \, b d^{3} e^{3} n x + 3 \, b d e^{5} n\right )} \sqrt {x}}{180 \, e^{6} x^{3}} \]

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*(30*b*d^4*e^2*n*x^2 + 15*b*d^2*e^4*n*x + 10*b*e^6*n - 60*b*e^6*log(c) - 60*a*e^6 + 60*(b*d^6*n*x^3 - b*e
^6*n)*log((d*x + e*sqrt(x))/x) - 4*(15*b*d^5*e*n*x^2 + 5*b*d^3*e^3*n*x + 3*b*d*e^5*n)*sqrt(x))/(e^6*x^3)

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giac [B]  time = 0.27, size = 535, normalized size = 3.93 \[ \frac {1}{180} \, {\left (\frac {360 \, {\left (d \sqrt {x} + e\right )} b d^{5} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{4} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {360 \, {\left (d \sqrt {x} + e\right )} b d^{5} n}{\sqrt {x}} + \frac {360 \, {\left (d \sqrt {x} + e\right )} b d^{5} \log \relax (c)}{\sqrt {x}} + \frac {1200 \, {\left (d \sqrt {x} + e\right )}^{3} b d^{3} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{\frac {3}{2}}} + \frac {450 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{4} n}{x} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{4} \log \relax (c)}{x} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{4} b d^{2} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{2}} - \frac {400 \, {\left (d \sqrt {x} + e\right )}^{3} b d^{3} n}{x^{\frac {3}{2}}} + \frac {360 \, {\left (d \sqrt {x} + e\right )} a d^{5}}{\sqrt {x}} + \frac {1200 \, {\left (d \sqrt {x} + e\right )}^{3} b d^{3} \log \relax (c)}{x^{\frac {3}{2}}} + \frac {360 \, {\left (d \sqrt {x} + e\right )}^{5} b d n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{\frac {5}{2}}} + \frac {225 \, {\left (d \sqrt {x} + e\right )}^{4} b d^{2} n}{x^{2}} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{2} a d^{4}}{x} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{4} b d^{2} \log \relax (c)}{x^{2}} - \frac {60 \, {\left (d \sqrt {x} + e\right )}^{6} b n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{3}} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{5} b d n}{x^{\frac {5}{2}}} + \frac {1200 \, {\left (d \sqrt {x} + e\right )}^{3} a d^{3}}{x^{\frac {3}{2}}} + \frac {360 \, {\left (d \sqrt {x} + e\right )}^{5} b d \log \relax (c)}{x^{\frac {5}{2}}} + \frac {10 \, {\left (d \sqrt {x} + e\right )}^{6} b n}{x^{3}} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{4} a d^{2}}{x^{2}} - \frac {60 \, {\left (d \sqrt {x} + e\right )}^{6} b \log \relax (c)}{x^{3}} + \frac {360 \, {\left (d \sqrt {x} + e\right )}^{5} a d}{x^{\frac {5}{2}}} - \frac {60 \, {\left (d \sqrt {x} + e\right )}^{6} a}{x^{3}}\right )} e^{\left (-6\right )} \]

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*(360*(d*sqrt(x) + e)*b*d^5*n*log((d*sqrt(x) + e)/sqrt(x))/sqrt(x) - 900*(d*sqrt(x) + e)^2*b*d^4*n*log((d
*sqrt(x) + e)/sqrt(x))/x - 360*(d*sqrt(x) + e)*b*d^5*n/sqrt(x) + 360*(d*sqrt(x) + e)*b*d^5*log(c)/sqrt(x) + 12
00*(d*sqrt(x) + e)^3*b*d^3*n*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) + 450*(d*sqrt(x) + e)^2*b*d^4*n/x - 900*(d*s
qrt(x) + e)^2*b*d^4*log(c)/x - 900*(d*sqrt(x) + e)^4*b*d^2*n*log((d*sqrt(x) + e)/sqrt(x))/x^2 - 400*(d*sqrt(x)
+ e)^3*b*d^3*n/x^(3/2) + 360*(d*sqrt(x) + e)*a*d^5/sqrt(x) + 1200*(d*sqrt(x) + e)^3*b*d^3*log(c)/x^(3/2) + 36
0*(d*sqrt(x) + e)^5*b*d*n*log((d*sqrt(x) + e)/sqrt(x))/x^(5/2) + 225*(d*sqrt(x) + e)^4*b*d^2*n/x^2 - 900*(d*sq
rt(x) + e)^2*a*d^4/x - 900*(d*sqrt(x) + e)^4*b*d^2*log(c)/x^2 - 60*(d*sqrt(x) + e)^6*b*n*log((d*sqrt(x) + e)/s
qrt(x))/x^3 - 72*(d*sqrt(x) + e)^5*b*d*n/x^(5/2) + 1200*(d*sqrt(x) + e)^3*a*d^3/x^(3/2) + 360*(d*sqrt(x) + e)^
5*b*d*log(c)/x^(5/2) + 10*(d*sqrt(x) + e)^6*b*n/x^3 - 900*(d*sqrt(x) + e)^4*a*d^2/x^2 - 60*(d*sqrt(x) + e)^6*b
*log(c)/x^3 + 360*(d*sqrt(x) + e)^5*a*d/x^(5/2) - 60*(d*sqrt(x) + e)^6*a/x^3)*e^(-6)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )+a}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.81, size = 117, normalized size = 0.86 \[ \frac {1}{180} \, b e n {\left (\frac {60 \, d^{6} \log \left (d \sqrt {x} + e\right )}{e^{7}} - \frac {30 \, d^{6} \log \relax (x)}{e^{7}} - \frac {60 \, d^{5} x^{\frac {5}{2}} - 30 \, d^{4} e x^{2} + 20 \, d^{3} e^{2} x^{\frac {3}{2}} - 15 \, d^{2} e^{3} x + 12 \, d e^{4} \sqrt {x} - 10 \, e^{5}}{e^{6} x^{3}}\right )} - \frac {b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \]

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*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x
^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x) - 10*e^5)/(e^6*x^3)) - 1/3*b*log(c*(d + e/sqrt(x))^n)/x^3 - 1/3*a/x^3

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mupad [B]  time = 0.44, size = 113, normalized size = 0.83 \[ \frac {b\,n}{18\,x^3}-\frac {a}{3\,x^3}-\frac {b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{3\,x^3}-\frac {b\,d\,n}{15\,e\,x^{5/2}}+\frac {b\,d^6\,n\,\ln \left (d+\frac {e}{\sqrt {x}}\right )}{3\,e^6}+\frac {b\,d^2\,n}{12\,e^2\,x^2}+\frac {b\,d^4\,n}{6\,e^4\,x}-\frac {b\,d^3\,n}{9\,e^3\,x^{3/2}}-\frac {b\,d^5\,n}{3\,e^5\,\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*n)/(18*x^3) - a/(3*x^3) - (b*log(c*(d + e/x^(1/2))^n))/(3*x^3) - (b*d*n)/(15*e*x^(5/2)) + (b*d^6*n*log(d +
e/x^(1/2)))/(3*e^6) + (b*d^2*n)/(12*e^2*x^2) + (b*d^4*n)/(6*e^4*x) - (b*d^3*n)/(9*e^3*x^(3/2)) - (b*d^5*n)/(3*
e^5*x^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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