3.434 \(\int \frac {(d+e x^r)^{3/2} (a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=284 \[ \frac {2}{3} \left (-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}+\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b d^{3/2} n \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {e x^r+d}}\right )}{r^2}+\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {16 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 r^2}-\frac {4 b d^{3/2} n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}-\frac {16 b d n \sqrt {d+e x^r}}{3 r^2} \]

[Out]

-4/9*b*n*(d+e*x^r)^(3/2)/r^2+16/3*b*d^(3/2)*n*arctanh((d+e*x^r)^(1/2)/d^(1/2))/r^2+2*b*d^(3/2)*n*arctanh((d+e*
x^r)^(1/2)/d^(1/2))^2/r^2-4*b*d^(3/2)*n*arctanh((d+e*x^r)^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)-(d+e*x^r)^(1/2)
))/r^2-2*b*d^(3/2)*n*polylog(2,1-2*d^(1/2)/(d^(1/2)-(d+e*x^r)^(1/2)))/r^2-16/3*b*d*n*(d+e*x^r)^(1/2)/r^2+2/3*(
a+b*ln(c*x^n))*((d+e*x^r)^(3/2)/r-3*d^(3/2)*arctanh((d+e*x^r)^(1/2)/d^(1/2))/r+3*d*(d+e*x^r)^(1/2)/r)

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Rubi [A]  time = 0.39, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {266, 50, 63, 208, 2348, 5984, 5918, 2402, 2315} \[ -\frac {2 b d^{3/2} n \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}+\frac {2}{3} \left (-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}+\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {16 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 r^2}-\frac {4 b d^{3/2} n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}-\frac {16 b d n \sqrt {d+e x^r}}{3 r^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*b*d*n*Sqrt[d + e*x^r])/(3*r^2) - (4*b*n*(d + e*x^r)^(3/2))/(9*r^2) + (16*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x
^r]/Sqrt[d]])/(3*r^2) + (2*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]^2)/r^2 + (2*((3*d*Sqrt[d + e*x^r])/r +
(d + e*x^r)^(3/2)/r - (3*d^(3/2)*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/r)*(a + b*Log[c*x^n]))/3 - (4*b*d^(3/2)*n*
ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/r^2 - (2*b*d^(3/2)*n*PolyLog[2,
1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/r^2

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 208

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2348

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.))/(x_), x_Symbol] :> With[{u = IntHi
de[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b
, c, d, e, n, r}, x] && IntegerQ[q - 1/2]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 5918

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 - c^2
*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 5984

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {2 d \sqrt {d+e x^r}}{r x}+\frac {2 \left (d+e x^r\right )^{3/2}}{3 r x}-\frac {2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r x}\right ) \, dx\\ &=\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(2 b n) \int \frac {\left (d+e x^r\right )^{3/2}}{x} \, dx}{3 r}-\frac {(2 b d n) \int \frac {\sqrt {d+e x^r}}{x} \, dx}{r}+\frac {\left (2 b d^{3/2} n\right ) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{x} \, dx}{r}\\ &=\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {(d+e x)^{3/2}}{x} \, dx,x,x^r\right )}{3 r^2}-\frac {(2 b d n) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^r\right )}{r^2}+\frac {\left (2 b d^{3/2} n\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx,x,x^r\right )}{r^2}\\ &=-\frac {4 b d n \sqrt {d+e x^r}}{r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(2 b d n) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^r\right )}{3 r^2}+\frac {\left (4 b d^{3/2} n\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{-d+x^2} \, dx,x,\sqrt {d+e x^r}\right )}{r^2}-\frac {\left (2 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^r\right )}{r^2}\\ &=-\frac {16 b d n \sqrt {d+e x^r}}{3 r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(4 b d n) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{1-\frac {x}{\sqrt {d}}} \, dx,x,\sqrt {d+e x^r}\right )}{r^2}-\frac {\left (2 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^r\right )}{3 r^2}-\frac {\left (4 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^r}\right )}{e r^2}\\ &=-\frac {16 b d n \sqrt {d+e x^r}}{3 r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r^2}+\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}+\frac {(4 b d n) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {d}}}\right )}{1-\frac {x^2}{d}} \, dx,x,\sqrt {d+e x^r}\right )}{r^2}-\frac {\left (4 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^r}\right )}{3 e r^2}\\ &=-\frac {16 b d n \sqrt {d+e x^r}}{3 r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {16 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 r^2}+\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}-\frac {\left (4 b d^{3/2} n\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x^r}}{\sqrt {d}}}\right )}{r^2}\\ &=-\frac {16 b d n \sqrt {d+e x^r}}{3 r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {16 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 r^2}+\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}-\frac {2 b d^{3/2} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x^r}}{\sqrt {d}}}\right )}{r^2}\\ \end {align*}

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Mathematica [F]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{r} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{r}+d \right )^{\frac {3}{2}} \left (b \ln \left (c \,x^{n}\right )+a \right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, {\left (\frac {3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {e x^{r} + d} - \sqrt {d}}{\sqrt {e x^{r} + d} + \sqrt {d}}\right )}{r} + \frac {2 \, {\left ({\left (e x^{r} + d\right )}^{\frac {3}{2}} + 3 \, \sqrt {e x^{r} + d} d\right )}}{r}\right )} a + b \int \frac {{\left (e x^{r} \log \relax (c) + d \log \relax (c) + {\left (e x^{r} + d\right )} \log \left (x^{n}\right )\right )} \sqrt {e x^{r} + d}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*d^(3/2)*log((sqrt(e*x^r + d) - sqrt(d))/(sqrt(e*x^r + d) + sqrt(d)))/r + 2*((e*x^r + d)^(3/2) + 3*sqrt(
e*x^r + d)*d)/r)*a + b*integrate((e*x^r*log(c) + d*log(c) + (e*x^r + d)*log(x^n))*sqrt(e*x^r + d)/x, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x^r\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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