3.406 \(\int \frac {a+b \log (c (d+e \sqrt {x})^n)}{x^3} \, dx\)

Optimal. Leaf size=109 \[ -\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{2 x^2}+\frac {b e^4 n \log \left (d+e \sqrt {x}\right )}{2 d^4}-\frac {b e^4 n \log (x)}{4 d^4}-\frac {b e^3 n}{2 d^3 \sqrt {x}}+\frac {b e^2 n}{4 d^2 x}-\frac {b e n}{6 d x^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 109, 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, 44} \[ -\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{2 x^2}-\frac {b e^3 n}{2 d^3 \sqrt {x}}+\frac {b e^2 n}{4 d^2 x}+\frac {b e^4 n \log \left (d+e \sqrt {x}\right )}{2 d^4}-\frac {b e^4 n \log (x)}{4 d^4}-\frac {b e n}{6 d x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 104, normalized size = 0.95 \[ -\frac {a}{2 x^2}-\frac {b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{2 x^2}+\frac {1}{2} b e n \left (\frac {e^3 \log \left (d+e \sqrt {x}\right )}{d^4}-\frac {e^3 \log (x)}{2 d^4}-\frac {e^2}{d^3 \sqrt {x}}+\frac {e}{2 d^2 x}-\frac {1}{3 d x^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 97, normalized size = 0.89 \[ -\frac {6 \, b e^{4} n x^{2} \log \left (\sqrt {x}\right ) - 3 \, b d^{2} e^{2} n x + 6 \, b d^{4} \log \relax (c) + 6 \, a d^{4} - 6 \, {\left (b e^{4} n x^{2} - b d^{4} n\right )} \log \left (e \sqrt {x} + d\right ) + 2 \, {\left (3 \, b d e^{3} n x + b d^{3} e n\right )} \sqrt {x}}{12 \, d^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.19, size = 366, normalized size = 3.36 \[ \frac {{\left (6 \, {\left (\sqrt {x} e + d\right )}^{4} b n e^{5} \log \left (\sqrt {x} e + d\right ) - 24 \, {\left (\sqrt {x} e + d\right )}^{3} b d n e^{5} \log \left (\sqrt {x} e + d\right ) + 36 \, {\left (\sqrt {x} e + d\right )}^{2} b d^{2} n e^{5} \log \left (\sqrt {x} e + d\right ) - 24 \, {\left (\sqrt {x} e + d\right )} b d^{3} n e^{5} \log \left (\sqrt {x} e + d\right ) - 6 \, {\left (\sqrt {x} e + d\right )}^{4} b n e^{5} \log \left (\sqrt {x} e\right ) + 24 \, {\left (\sqrt {x} e + d\right )}^{3} b d n e^{5} \log \left (\sqrt {x} e\right ) - 36 \, {\left (\sqrt {x} e + d\right )}^{2} b d^{2} n e^{5} \log \left (\sqrt {x} e\right ) + 24 \, {\left (\sqrt {x} e + d\right )} b d^{3} n e^{5} \log \left (\sqrt {x} e\right ) - 6 \, b d^{4} n e^{5} \log \left (\sqrt {x} e\right ) - 6 \, {\left (\sqrt {x} e + d\right )}^{3} b d n e^{5} + 21 \, {\left (\sqrt {x} e + d\right )}^{2} b d^{2} n e^{5} - 26 \, {\left (\sqrt {x} e + d\right )} b d^{3} n e^{5} + 11 \, b d^{4} n e^{5} - 6 \, b d^{4} e^{5} \log \relax (c) - 6 \, a d^{4} e^{5}\right )} e^{\left (-1\right )}}{12 \, {\left ({\left (\sqrt {x} e + d\right )}^{4} d^{4} - 4 \, {\left (\sqrt {x} e + d\right )}^{3} d^{5} + 6 \, {\left (\sqrt {x} e + d\right )}^{2} d^{6} - 4 \, {\left (\sqrt {x} e + d\right )} d^{7} + d^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^3,x, algorithm="giac")

[Out]

1/12*(6*(sqrt(x)*e + d)^4*b*n*e^5*log(sqrt(x)*e + d) - 24*(sqrt(x)*e + d)^3*b*d*n*e^5*log(sqrt(x)*e + d) + 36*
(sqrt(x)*e + d)^2*b*d^2*n*e^5*log(sqrt(x)*e + d) - 24*(sqrt(x)*e + d)*b*d^3*n*e^5*log(sqrt(x)*e + d) - 6*(sqrt
(x)*e + d)^4*b*n*e^5*log(sqrt(x)*e) + 24*(sqrt(x)*e + d)^3*b*d*n*e^5*log(sqrt(x)*e) - 36*(sqrt(x)*e + d)^2*b*d
^2*n*e^5*log(sqrt(x)*e) + 24*(sqrt(x)*e + d)*b*d^3*n*e^5*log(sqrt(x)*e) - 6*b*d^4*n*e^5*log(sqrt(x)*e) - 6*(sq
rt(x)*e + d)^3*b*d*n*e^5 + 21*(sqrt(x)*e + d)^2*b*d^2*n*e^5 - 26*(sqrt(x)*e + d)*b*d^3*n*e^5 + 11*b*d^4*n*e^5
- 6*b*d^4*e^5*log(c) - 6*a*d^4*e^5)*e^(-1)/((sqrt(x)*e + d)^4*d^4 - 4*(sqrt(x)*e + d)^3*d^5 + 6*(sqrt(x)*e + d
)^2*d^6 - 4*(sqrt(x)*e + d)*d^7 + d^8)

________________________________________________________________________________________

maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*(e*x^(1/2)+d)^n)+a)/x^3,x)

[Out]

int((b*ln(c*(e*x^(1/2)+d)^n)+a)/x^3,x)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 84, normalized size = 0.77 \[ \frac {1}{12} \, b e n {\left (\frac {6 \, e^{3} \log \left (e \sqrt {x} + d\right )}{d^{4}} - \frac {3 \, e^{3} \log \relax (x)}{d^{4}} - \frac {6 \, e^{2} x - 3 \, d e \sqrt {x} + 2 \, d^{2}}{d^{3} x^{\frac {3}{2}}}\right )} - \frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.63, size = 83, normalized size = 0.76 \[ \frac {b\,e^4\,n\,\mathrm {atanh}\left (\frac {2\,e\,\sqrt {x}}{d}+1\right )}{d^4}-\frac {\frac {b\,e\,n}{3\,d}+\frac {b\,e^3\,n\,x}{d^3}-\frac {b\,e^2\,n\,\sqrt {x}}{2\,d^2}}{2\,x^{3/2}}-\frac {b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{2\,x^2}-\frac {a}{2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x^(1/2))^n))/x^3,x)

[Out]

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

________________________________________________________________________________________

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**3,x)

[Out]

Timed out

________________________________________________________________________________________