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

Optimal. Leaf size=383 \[ \frac{\sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{3 \sqrt{\frac{\pi }{2}} g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{\sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{\sqrt{\pi } g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 \sqrt{b} e^4 \sqrt{n}} \]

[Out]

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

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Rubi [A]  time = 0.727602, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {2401, 2389, 2300, 2180, 2204, 2390, 2310} \[ \frac{\sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{3 \sqrt{\frac{\pi }{2}} g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{\sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}+\frac{\sqrt{\pi } g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 \sqrt{b} e^4 \sqrt{n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^3}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx &=\int \left (\frac{(e f-d g)^3}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{3 g (e f-d g)^2 (d+e x)}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{3 g^2 (e f-d g) (d+e x)^2}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g^3 (d+e x)^3}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx\\ &=\frac{g^3 \int \frac{(d+e x)^3}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac{\left (3 g^2 (e f-d g)\right ) \int \frac{(d+e x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac{\left (3 g (e f-d g)^2\right ) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac{(e f-d g)^3 \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}\\ &=\frac{g^3 \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 g^2 (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac{(e f-d g)^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}\\ &=\frac{\left (g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac{\left (3 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac{\left (3 g (e f-d g)^2 (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 )}{e^4 n}+\frac{\left ((e f-d g)^3 (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 )}{e^4 n}\\ &=\frac{\left (2 g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{4 a}{b n}+\frac{4 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b e^4 n}+\frac{\left (6 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b n}+\frac{3 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b e^4 n}+\frac{\left (6 g (e f-d g)^2 (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 e^4 n}+\frac{\left (2 (e f-d g)^3 (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 e^4 n}\\ &=\frac{e^{-\frac{a}{b n}} (e f-d g)^3 \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^4 \sqrt{n}}+\frac{e^{-\frac{4 a}{b n}} g^3 \sqrt{\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 \sqrt{b} e^4 \sqrt{n}}+\frac{3 e^{-\frac{2 a}{b n}} g (e f-d g)^2 \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^4 \sqrt{n}}+\frac{e^{-\frac{3 a}{b n}} g^2 (e f-d g) \sqrt{3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{\sqrt{b} e^4 \sqrt{n}}\\ \end{align*}

Mathematica [A]  time = 0.417212, size = 331, normalized size = 0.86 \[ \frac{\sqrt{\pi } e^{-\frac{4 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-4/n} \left (2 \sqrt{3} g^2 e^{\frac{a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+3 \sqrt{2} g e^{\frac{2 a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+2 e^{\frac{3 a}{b n}} (e f-d g)^3 \left (c (d+e x)^n\right )^{3/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+g^3 (d+e x)^3 \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )\right )}{2 \sqrt{b} e^4 \sqrt{n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.741, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{3}{\frac{1}{\sqrt{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3/(a+b*ln(c*(e*x+d)^n))^(1/2),x)

[Out]

int((g*x+f)^3/(a+b*ln(c*(e*x+d)^n))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="maxima")

[Out]

integrate((g*x + f)^3/sqrt(b*log((e*x + d)^n*c) + a), 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)^3/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**3/(a+b*ln(c*(e*x+d)**n))**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="giac")

[Out]

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