3.2.41 \(\int \frac {a+b \log (c (d+e x)^n)}{(f+g x)^{3/2}} \, dx\) [141]

Optimal. Leaf size=81 \[ -\frac {4 b \sqrt {e} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f+g x}} \]

[Out]

-4*b*n*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))*e^(1/2)/g/(-d*g+e*f)^(1/2)-2*(a+b*ln(c*(e*x+d)^n))/g/(g
*x+f)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2442, 65, 214} \begin {gather*} -\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f+g x}}-\frac {4 b \sqrt {e} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])/(f + g*x)^(3/2),x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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]

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 80, normalized size = 0.99 \begin {gather*} \frac {2 \left (-\frac {2 b \sqrt {e} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}}\right )}{g} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])/(f + g*x)^(3/2),x]

[Out]

(2*((-2*b*Sqrt[e]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/Sqrt[e*f - d*g] - (a + b*Log[c*(d + e*x)
^n])/Sqrt[f + g*x]))/g

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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{\left (g x +f \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*%e^2*f-4*%e*d*g>0)', see `as
sume?` for m

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Fricas [A]
time = 0.39, size = 216, normalized size = 2.67 \begin {gather*} \left [\frac {2 \, {\left ({\left (b g n x + b f n\right )} \sqrt {-\frac {e}{d g - f e}} \log \left (-\frac {d g - 2 \, {\left (d g - f e\right )} \sqrt {g x + f} \sqrt {-\frac {e}{d g - f e}} - {\left (g x + 2 \, f\right )} e}{x e + d}\right ) - {\left (b n \log \left (x e + d\right ) + b \log \left (c\right ) + a\right )} \sqrt {g x + f}\right )}}{g^{2} x + f g}, \frac {2 \, {\left (\frac {2 \, {\left (b g n x + b f n\right )} \arctan \left (-\frac {\sqrt {d g - f e} e^{\left (-\frac {1}{2}\right )}}{\sqrt {g x + f}}\right ) e^{\frac {1}{2}}}{\sqrt {d g - f e}} - {\left (b n \log \left (x e + d\right ) + b \log \left (c\right ) + a\right )} \sqrt {g x + f}\right )}}{g^{2} x + f g}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2*((b*g*n*x + b*f*n)*sqrt(-e/(d*g - f*e))*log(-(d*g - 2*(d*g - f*e)*sqrt(g*x + f)*sqrt(-e/(d*g - f*e)) - (g*x
+ 2*f)*e)/(x*e + d)) - (b*n*log(x*e + d) + b*log(c) + a)*sqrt(g*x + f))/(g^2*x + f*g), 2*(2*(b*g*n*x + b*f*n)
*arctan(-sqrt(d*g - f*e)*e^(-1/2)/sqrt(g*x + f))*e^(1/2)/sqrt(d*g - f*e) - (b*n*log(x*e + d) + b*log(c) + a)*s
qrt(g*x + f))/(g^2*x + f*g)]

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Sympy [A]
time = 7.32, size = 85, normalized size = 1.05 \begin {gather*} \frac {- \frac {2 a}{\sqrt {f + g x}} + 2 b \left (\frac {2 n \operatorname {atan}{\left (\frac {\sqrt {f + g x}}{\sqrt {\frac {g \left (d - \frac {e f}{g}\right )}{e}}} \right )}}{\sqrt {\frac {g \left (d - \frac {e f}{g}\right )}{e}}} - \frac {\log {\left (c \left (d - \frac {e f}{g} + \frac {e \left (f + g x\right )}{g}\right )^{n} \right )}}{\sqrt {f + g x}}\right )}{g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))/(g*x+f)**(3/2),x)

[Out]

(-2*a/sqrt(f + g*x) + 2*b*(2*n*atan(sqrt(f + g*x)/sqrt(g*(d - e*f/g)/e))/sqrt(g*(d - e*f/g)/e) - log(c*(d - e*
f/g + e*(f + g*x)/g)**n)/sqrt(f + g*x)))/g

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Giac [A]
time = 2.80, size = 92, normalized size = 1.14 \begin {gather*} \frac {4 \, b n \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right ) e}{\sqrt {d g e - f e^{2}} g} - \frac {2 \, {\left (b n \log \left (d g + {\left (g x + f\right )} e - f e\right ) - b n \log \left (g\right ) + b \log \left (c\right ) + a\right )}}{\sqrt {g x + f} g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b*n*arctan(sqrt(g*x + f)*e/sqrt(d*g*e - f*e^2))*e/(sqrt(d*g*e - f*e^2)*g) - 2*(b*n*log(d*g + (g*x + f)*e - f
*e) - b*n*log(g) + b*log(c) + a)/(sqrt(g*x + f)*g)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{{\left (f+g\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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