3.2.25 \(\int \frac {f+g x}{\sqrt {a+b \log (c (d+e x)^n)}} \, dx\) [125]

Optimal. Leaf size=181 \[ \frac {e^{-\frac {a}{b n}} (e f-d g) \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^2 \sqrt {n}}+\frac {e^{-\frac {2 a}{b n}} g \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^2 \sqrt {n}} \]

[Out]

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

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Rubi [A]
time = 0.21, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2448, 2436, 2337, 2211, 2235, 2437, 2347} \begin {gather*} \frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g) \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 )}{\sqrt {b} e^2 \sqrt {n}}+\frac {\sqrt {\frac {\pi }{2}} g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^2 \sqrt {n}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 2347

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

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 164, normalized size = 0.91 \begin {gather*} \frac {e^{-\frac {2 a}{b n}} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (2 e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+\sqrt {2} g (d+e x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )}{2 \sqrt {b} e^2 \sqrt {n}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Pi]*(d + e*x)*(2*E^(a/(b*n))*(e*f - d*g)*(c*(d + e*x)^n)^n^(-1)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqr
t[b]*Sqrt[n])] + Sqrt[2]*g*(d + e*x)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]))/(2*Sqr
t[b]*e^2*E^((2*a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^(2/n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((g*x+f)/(a+b*ln(c*(e*x+d)^n))^(1/2),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((g*x+f)/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="maxima")

[Out]

integrate((g*x + f)/sqrt(b*log((x*e + d)^n*c) + a), 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((g*x+f)/(a+b*log(c*(e*x+d)^n))^(1/2),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 \frac {f + g x}{\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(a+b*ln(c*(e*x+d)**n))**(1/2),x)

[Out]

Integral((f + g*x)/sqrt(a + b*log(c*(d + e*x)**n)), 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((g*x+f)/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="giac")

[Out]

integrate((g*x + f)/sqrt(b*log((x*e + d)^n*c) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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