3.105 \(\int (f+g x)^2 \sqrt{a+b \log (c (d+e x)^n)} \, dx\)

Optimal. Leaf size=404 \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} g \sqrt{n} e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 e^3}-\frac{\sqrt{\pi } \sqrt{b} \sqrt{n} e^{-\frac{a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 e^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} g^2 \sqrt{n} e^{-\frac{3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{6 e^3}+\frac{g (d+e x)^2 (e f-d g) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{(d+e x) (e f-d g)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{g^2 (d+e x)^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{3 e^3} \]

[Out]

-(Sqrt[b]*(e*f - d*g)^2*Sqrt[n]*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(2*
e^3*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) - (Sqrt[b]*g*(e*f - d*g)*Sqrt[n]*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*
Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(2*e^3*E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) - (Sqrt[b]*g
^2*Sqrt[n]*Sqrt[Pi/3]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(6*e^3*E^(
(3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)) + ((e*f - d*g)^2*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/e^3 + (g*(e*f -
d*g)*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/e^3 + (g^2*(d + e*x)^3*Sqrt[a + b*Log[c*(d + e*x)^n]])/(3*e^
3)

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Rubi [A]  time = 0.701143, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {2401, 2389, 2296, 2300, 2180, 2204, 2390, 2305, 2310} \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} g \sqrt{n} e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 e^3}-\frac{\sqrt{\pi } \sqrt{b} \sqrt{n} e^{-\frac{a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{2 e^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} g^2 \sqrt{n} e^{-\frac{3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{6 e^3}+\frac{g (d+e x)^2 (e f-d g) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{(d+e x) (e f-d g)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{g^2 (d+e x)^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{3 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[b]*(e*f - d*g)^2*Sqrt[n]*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(2*
e^3*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) - (Sqrt[b]*g*(e*f - d*g)*Sqrt[n]*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*
Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(2*e^3*E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) - (Sqrt[b]*g
^2*Sqrt[n]*Sqrt[Pi/3]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(6*e^3*E^(
(3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)) + ((e*f - d*g)^2*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/e^3 + (g*(e*f -
d*g)*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/e^3 + (g^2*(d + e*x)^3*Sqrt[a + b*Log[c*(d + e*x)^n]])/(3*e^
3)

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 2296

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 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 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 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 (f+g x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx &=\int \left (\frac{(e f-d g)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac{2 g (e f-d g) (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac{g^2 (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^2}\right ) \, dx\\ &=\frac{g^2 \int (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^2}+\frac{(2 g (e f-d g)) \int (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^2}+\frac{(e f-d g)^2 \int \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^2}\\ &=\frac{g^2 \operatorname{Subst}\left (\int x^2 \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^3}+\frac{(2 g (e f-d g)) \operatorname{Subst}\left (\int x \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^3}+\frac{(e f-d g)^2 \operatorname{Subst}\left (\int \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^3}\\ &=\frac{(e f-d g)^2 (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{g (e f-d g) (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{g^2 (d+e x)^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{3 e^3}-\frac{\left (b g^2 n\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{6 e^3}-\frac{(b g (e f-d g) n) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{2 e^3}-\frac{\left (b (e f-d g)^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{2 e^3}\\ &=\frac{(e f-d g)^2 (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{g (e f-d g) (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{g^2 (d+e x)^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{3 e^3}-\frac{\left (b g^2 (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 )}{6 e^3}-\frac{\left (b g (e f-d 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 )}{2 e^3}-\frac{\left (b (e f-d g)^2 (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 )}{2 e^3}\\ &=\frac{(e f-d g)^2 (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{g (e f-d g) (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{g^2 (d+e x)^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{3 e^3}-\frac{\left (g^2 (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 )}{3 e^3}-\frac{\left (g (e f-d 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 )}{e^3}-\frac{\left ((e f-d g)^2 (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 )}{e^3}\\ &=-\frac{\sqrt{b} e^{-\frac{a}{b n}} (e f-d g)^2 \sqrt{n} \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 )}{2 e^3}-\frac{\sqrt{b} e^{-\frac{2 a}{b n}} g (e f-d g) \sqrt{n} \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 )}{2 e^3}-\frac{\sqrt{b} e^{-\frac{3 a}{b n}} g^2 \sqrt{n} \sqrt{\frac{\pi }{3}} (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 )}{6 e^3}+\frac{(e f-d g)^2 (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{g (e f-d g) (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac{g^2 (d+e x)^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.513818, size = 374, normalized size = 0.93 \[ \frac{(d+e x) \left (9 \sqrt{2 \pi } \sqrt{b} g \sqrt{n} e^{-\frac{2 a}{b n}} (d+e x) (d g-e f) \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )-18 \sqrt{\pi } \sqrt{b} \sqrt{n} e^{-\frac{a}{b n}} (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )-2 \sqrt{3 \pi } \sqrt{b} g^2 \sqrt{n} e^{-\frac{3 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+36 (e f-d g)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}+36 g (d+e x) (e f-d g) \sqrt{a+b \log \left (c (d+e x)^n\right )}+12 g^2 (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{36 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*((-18*Sqrt[b]*(e*f - d*g)^2*Sqrt[n]*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)) + (9*Sqrt[b]*g*(-(e*f) + d*g)*Sqrt[n]*Sqrt[2*Pi]*(d + e*x)*Erfi[(Sqrt[2
]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) - (2*Sqrt[b]*g^2
*Sqrt[n]*Sqrt[3*Pi]*(d + e*x)^2*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b
*n))*(c*(d + e*x)^n)^(3/n)) + 36*(e*f - d*g)^2*Sqrt[a + b*Log[c*(d + e*x)^n]] + 36*g*(e*f - d*g)*(d + e*x)*Sqr
t[a + b*Log[c*(d + e*x)^n]] + 12*g^2*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]]))/(36*e^3)

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Maple [F]  time = 0.796, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) ^{2}\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)^2*(a+b*ln(c*(e*x+d)^n))^(1/2),x)

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x + f\right )}^{2} \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)^2*(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="giac")

[Out]

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