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

Optimal. Leaf size=220 \[ \frac{2 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^2 n^{3/2}}+\frac{2 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^2 n^{3/2}}-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]

[Out]

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

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Rubi [A]  time = 0.402761, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310} \[ \frac{2 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^2 n^{3/2}}+\frac{2 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^2 n^{3/2}}-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2400

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

Rule 2401

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]

Rule 2389

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 2300

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 2180

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] &&  !$UseGamma === True

Rule 2204

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 2390

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 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps

\begin{align*} \int \frac{f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx &=-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{4 \int \frac{f+g x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b n}-\frac{(2 (e f-d g)) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e n}\\ &=-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{4 \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}{b n}-\frac{(2 (e f-d g)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^2 n}\\ &=-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{(4 g) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e n}+\frac{(4 (e f-d g)) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e n}-\frac{\left (2 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^2 n^2}\\ &=-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{(4 g) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^2 n}+\frac{(4 (e f-d g)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^2 n}-\frac{\left (4 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{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^2 e^2 n^2}\\ &=-\frac{2 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 )}{b^{3/2} e^2 n^{3/2}}-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (4 g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^2 n^2}+\frac{\left (4 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^2 n^2}\\ &=-\frac{2 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 )}{b^{3/2} e^2 n^{3/2}}-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (8 g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{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^2 e^2 n^2}+\frac{\left (8 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{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^2 e^2 n^2}\\ &=\frac{2 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 )}{b^{3/2} e^2 n^{3/2}}+\frac{2 e^{-\frac{2 a}{b n}} g \sqrt{2 \pi } (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 )}{b^{3/2} e^2 n^{3/2}}-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}\\ \end{align*}

Mathematica [A]  time = 0.807175, size = 338, normalized size = 1.54 \[ \frac{2 e^{-\frac{2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (\sqrt{b} \sqrt{n} e^{\frac{a}{b n}} \left (c (d+e x)^n\right )^{\frac{1}{n}} \left ((d g+e f) \sqrt{-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )-e e^{\frac{a}{b n}} (f+g x) \left (c (d+e x)^n\right )^{\frac{1}{n}}\right )-2 \sqrt{\pi } d g e^{\frac{a}{b n}} \left (c (d+e x)^n\right )^{\frac{1}{n}} \sqrt{a+b \log \left (c (d+e x)^n\right )} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+\sqrt{2 \pi } g (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )\right )}{b^{3/2} e^2 n^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)*(-2*d*E^(a/(b*n))*g*Sqrt[Pi]*(c*(d + e*x)^n)^n^(-1)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*
Sqrt[n])]*Sqrt[a + b*Log[c*(d + e*x)^n]] + g*Sqrt[2*Pi]*(d + e*x)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]]
)/(Sqrt[b]*Sqrt[n])]*Sqrt[a + b*Log[c*(d + e*x)^n]] + Sqrt[b]*E^(a/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^n^(-1)*(-(e*
E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(f + g*x)) + (e*f + d*g)*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*Sq
rt[-((a + b*Log[c*(d + e*x)^n])/(b*n))])))/(b^(3/2)*e^2*E^((2*a)/(b*n))*n^(3/2)*(c*(d + e*x)^n)^(2/n)*Sqrt[a +
 b*Log[c*(d + e*x)^n]])

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Maple [F]  time = 0.466, size = 0, normalized size = 0. \begin{align*} \int{(gx+f) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{g x + f}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{g x + f}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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