3.1.25 \(\int (a+b \log (c (d+e x)^n))^{3/2} \, dx\) [25]

Optimal. Leaf size=143 \[ \frac {3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e}-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2436, 2333, 2337, 2211, 2235} \begin {gather*} \frac {3 \sqrt {\pi } b^{3/2} n^{3/2} e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2337

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 127, normalized size = 0.89 \begin {gather*} \frac {(d+e x) \left (3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \sqrt {\pi } \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (2 a-3 b n+2 b \log \left (c (d+e x)^n\right )\right )\right )}{4 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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