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

Optimal. Leaf size=298 \[ -\frac{1}{2} \sqrt{\pi } \sqrt{b} d^2 \sqrt{n} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+d^2 x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\frac{\pi }{2}} \sqrt{b} d e \sqrt{n} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+d e x^2 \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{6} \sqrt{\frac{\pi }{3}} \sqrt{b} e^2 \sqrt{n} x^3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+\frac{1}{3} e^2 x^3 \sqrt{a+b \log \left (c x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.466567, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {2330, 2296, 2300, 2180, 2204, 2305, 2310} \[ -\frac{1}{2} \sqrt{\pi } \sqrt{b} d^2 \sqrt{n} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+d^2 x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\frac{\pi }{2}} \sqrt{b} d e \sqrt{n} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+d e x^2 \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{6} \sqrt{\frac{\pi }{3}} \sqrt{b} e^2 \sqrt{n} x^3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+\frac{1}{3} e^2 x^3 \sqrt{a+b \log \left (c x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

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

Mathematica [A]  time = 0.336213, size = 287, normalized size = 0.96 \[ \frac{1}{36} x \left (-18 \sqrt{\pi } \sqrt{b} d^2 \sqrt{n} e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+36 d^2 \sqrt{a+b \log \left (c x^n\right )}-9 \sqrt{2 \pi } \sqrt{b} d e \sqrt{n} x e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+36 d e x \sqrt{a+b \log \left (c x^n\right )}-2 \sqrt{3 \pi } \sqrt{b} e^2 \sqrt{n} x^2 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+12 e^2 x^2 \sqrt{a+b \log \left (c x^n\right )}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*((-18*Sqrt[b]*d^2*Sqrt[n]*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*x^n)^n^(
-1)) - (9*Sqrt[b]*d*e*Sqrt[n]*Sqrt[2*Pi]*x*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)
/(b*n))*(c*x^n)^(2/n)) - (2*Sqrt[b]*e^2*Sqrt[n]*Sqrt[3*Pi]*x^2*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*
Sqrt[n])])/(E^((3*a)/(b*n))*(c*x^n)^(3/n)) + 36*d^2*Sqrt[a + b*Log[c*x^n]] + 36*d*e*x*Sqrt[a + b*Log[c*x^n]] +
12*e^2*x^2*Sqrt[a + b*Log[c*x^n]]))/36

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Maple [F]  time = 0.392, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{2}\sqrt{a+b\ln \left ( c{x}^{n} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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