3.5.7 \(\int \frac {a+b \log (c (d+e \sqrt {x})^n)}{x^4} \, dx\) [407]

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

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 141, 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 = {2504, 2442, 46} \begin {gather*} -\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{3 x^3}+\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^5 n}{3 d^5 \sqrt {x}}+\frac {b e^4 n}{6 d^4 x}-\frac {b e^3 n}{9 d^3 x^{3/2}}+\frac {b e^2 n}{12 d^2 x^2}-\frac {b e n}{15 d x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*(b*e*n)/(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 46

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] && Lt
Q[m + n + 2, 0])

Rule 2442

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 2504

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+e \sqrt {x}\right )^n\right )}{x^4} \, dx &=2 \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.09, size = 132, normalized size = 0.94 \begin {gather*} -\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 {1}{5 d x^{5/2}}+\frac {e}{4 d^2 x^2}-\frac {e^2}{3 d^3 x^{3/2}}+\frac {e^3}{2 d^4 x}-\frac {e^4}{d^5 \sqrt {x}}+\frac {e^5 \log \left (d+e \sqrt {x}\right )}{d^6}-\frac {e^5 \log (x)}{2 d^6}\right ) \end {gather*}

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*Log[c*(d + e*Sqrt[x])^n])/(3*x^3) + (b*e*n*(-1/5*1/(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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )}{x^{4}}\, dx\]

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 = 0.27, size = 104, normalized size = 0.74 \begin {gather*} \frac {1}{180} \, b n {\left (\frac {60 \, e^{5} \log \left (\sqrt {x} e + d\right )}{d^{6}} - \frac {30 \, e^{5} \log \left (x\right )}{d^{6}} + \frac {15 \, d^{3} \sqrt {x} e - 12 \, d^{4} - 20 \, d^{2} x e^{2} + 30 \, d x^{\frac {3}{2}} e^{3} - 60 \, x^{2} e^{4}}{d^{5} x^{\frac {5}{2}}}\right )} e - \frac {b \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \end {gather*}

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

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Fricas [A]
time = 0.37, size = 120, normalized size = 0.85 \begin {gather*} \frac {15 \, b d^{4} n x e^{2} - 60 \, b d^{6} \log \left (c\right ) - 60 \, a d^{6} + 30 \, b d^{2} n x^{2} e^{4} - 60 \, b n x^{3} e^{6} \log \left (\sqrt {x}\right ) - 60 \, {\left (b d^{6} n - b n x^{3} e^{6}\right )} \log \left (\sqrt {x} e + d\right ) - 4 \, {\left (3 \, b d^{5} n e + 5 \, b d^{3} n x e^{3} + 15 \, b d n x^{2} e^{5}\right )} \sqrt {x}}{180 \, d^{6} x^{3}} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (112) = 224\).
time = 4.91, size = 542, normalized size = 3.84 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.76, size = 110, normalized size = 0.78 \begin {gather*} \frac {2\,b\,e^6\,n\,\mathrm {atanh}\left (\frac {2\,e\,\sqrt {x}}{d}+1\right )}{3\,d^6}-\frac {\frac {b\,e\,n}{5\,d}+\frac {b\,e^3\,n\,x}{3\,d^3}-\frac {b\,e^2\,n\,\sqrt {x}}{4\,d^2}+\frac {b\,e^5\,n\,x^2}{d^5}-\frac {b\,e^4\,n\,x^{3/2}}{2\,d^4}}{3\,x^{5/2}}-\frac {b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3\,x^3}-\frac {a}{3\,x^3} \end {gather*}

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]

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

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