3.5.6 \(\int \frac {a+b \log (c (d+e \sqrt {x})^n)}{x^3} \, dx\) [406]

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

[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)

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Rubi [A]
time = 0.05, 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 = {2504, 2442, 46} \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(b*e*n)/(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 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^3} \, dx &=2 \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.03, size = 104, normalized size = 0.95 \begin {gather*} -\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 {1}{3 d x^{3/2}}+\frac {e}{2 d^2 x}-\frac {e^2}{d^3 \sqrt {x}}+\frac {e^3 \log \left (d+e \sqrt {x}\right )}{d^4}-\frac {e^3 \log (x)}{2 d^4}\right ) \end {gather*}

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 84, normalized size = 0.77 \begin {gather*} \frac {1}{12} \, b n {\left (\frac {6 \, e^{3} \log \left (\sqrt {x} e + d\right )}{d^{4}} - \frac {3 \, e^{3} \log \left (x\right )}{d^{4}} + \frac {3 \, d \sqrt {x} e - 2 \, d^{2} - 6 \, x e^{2}}{d^{3} x^{\frac {3}{2}}}\right )} e - \frac {b \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 95, normalized size = 0.87 \begin {gather*} \frac {3 \, b d^{2} n x e^{2} - 6 \, b d^{4} \log \left (c\right ) - 6 \, b n x^{2} e^{4} \log \left (\sqrt {x}\right ) - 6 \, a d^{4} - 6 \, {\left (b d^{4} n - b n x^{2} e^{4}\right )} \log \left (\sqrt {x} e + d\right ) - 2 \, {\left (b d^{3} n e + 3 \, b d n x e^{3}\right )} \sqrt {x}}{12 \, d^{4} x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (105) = 210\).
time = 107.56, size = 493, normalized size = 4.52 \begin {gather*} \begin {cases} - \frac {6 a d^{5} \sqrt {x}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {6 a d^{4} e x}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {6 b d^{5} \sqrt {x} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {2 b d^{4} e n x}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {6 b d^{4} e x \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} + \frac {b d^{3} e^{2} n x^{\frac {3}{2}}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {3 b d^{2} e^{3} n x^{2}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {3 b d e^{4} n x^{\frac {5}{2}} \log {\left (x \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {6 b d e^{4} n x^{\frac {5}{2}}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} + \frac {6 b d e^{4} x^{\frac {5}{2}} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {3 b e^{5} n x^{3} \log {\left (x \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} + \frac {6 b e^{5} x^{3} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} & \text {for}\: d \neq 0 \\- \frac {a}{2 x^{2}} - \frac {b n}{8 x^{2}} - \frac {b \log {\left (c \left (e \sqrt {x}\right )^{n} \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \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**3,x)

[Out]

Piecewise((-6*a*d**5*sqrt(x)/(12*d**5*x**(5/2) + 12*d**4*e*x**3) - 6*a*d**4*e*x/(12*d**5*x**(5/2) + 12*d**4*e*
x**3) - 6*b*d**5*sqrt(x)*log(c*(d + e*sqrt(x))**n)/(12*d**5*x**(5/2) + 12*d**4*e*x**3) - 2*b*d**4*e*n*x/(12*d*
*5*x**(5/2) + 12*d**4*e*x**3) - 6*b*d**4*e*x*log(c*(d + e*sqrt(x))**n)/(12*d**5*x**(5/2) + 12*d**4*e*x**3) + b
*d**3*e**2*n*x**(3/2)/(12*d**5*x**(5/2) + 12*d**4*e*x**3) - 3*b*d**2*e**3*n*x**2/(12*d**5*x**(5/2) + 12*d**4*e
*x**3) - 3*b*d*e**4*n*x**(5/2)*log(x)/(12*d**5*x**(5/2) + 12*d**4*e*x**3) - 6*b*d*e**4*n*x**(5/2)/(12*d**5*x**
(5/2) + 12*d**4*e*x**3) + 6*b*d*e**4*x**(5/2)*log(c*(d + e*sqrt(x))**n)/(12*d**5*x**(5/2) + 12*d**4*e*x**3) -
3*b*e**5*n*x**3*log(x)/(12*d**5*x**(5/2) + 12*d**4*e*x**3) + 6*b*e**5*x**3*log(c*(d + e*sqrt(x))**n)/(12*d**5*
x**(5/2) + 12*d**4*e*x**3), Ne(d, 0)), (-a/(2*x**2) - b*n/(8*x**2) - b*log(c*(e*sqrt(x))**n)/(2*x**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (88) = 176\).
time = 4.84, size = 366, normalized size = 3.36 \begin {gather*} \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 \left (c\right ) - 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 )}} \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^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)

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Mupad [B]
time = 0.63, size = 83, normalized size = 0.76 \begin {gather*} \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} \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^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)

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