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

Optimal. Leaf size=330 \[ \frac{3 \sqrt{\pi } b^{3/2} n^{3/2} e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{4 e^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} g n^{3/2} e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{16 e^2}+\frac{(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e^2}-\frac{3 b n (d+e x) (e f-d g) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac{3 b g n (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{8 e^2} \]

[Out]

(3*b^(3/2)*(e*f - d*g)*n^(3/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(4*e
^2*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) + (3*b^(3/2)*g*n^(3/2)*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*
Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(16*e^2*E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) - (3*b*(e*f - d*g)*n*(
d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(2*e^2) - (3*b*g*n*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(8*e^2
) + ((e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/e^2 + (g*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(
3/2))/(2*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.42635, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2401, 2389, 2296, 2300, 2180, 2204, 2390, 2305, 2310} \[ \frac{3 \sqrt{\pi } b^{3/2} n^{3/2} e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{4 e^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} g n^{3/2} e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{16 e^2}+\frac{(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e^2}-\frac{3 b n (d+e x) (e f-d g) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac{3 b g n (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{8 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^(3/2)*(e*f - d*g)*n^(3/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(4*e
^2*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) + (3*b^(3/2)*g*n^(3/2)*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*
Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(16*e^2*E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) - (3*b*(e*f - d*g)*n*(
d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(2*e^2) - (3*b*g*n*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(8*e^2
) + ((e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/e^2 + (g*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(
3/2))/(2*e^2)

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) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx &=\int \left (\frac{(e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}\right ) \, dx\\ &=\frac{g \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx}{e}+\frac{(e f-d g) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx}{e}\\ &=\frac{g \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^{3/2} \, dx,x,d+e x\right )}{e^2}+\frac{(e f-d g) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^{3/2} \, dx,x,d+e x\right )}{e^2}\\ &=\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac{(3 b g n) \operatorname{Subst}\left (\int x \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{4 e^2}-\frac{(3 b (e f-d g) n) \operatorname{Subst}\left (\int \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 e^2}\\ &=-\frac{3 b (e f-d g) n (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac{3 b g n (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{8 e^2}+\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}+\frac{\left (3 b^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{16 e^2}+\frac{\left (3 b^2 (e f-d g) n^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{4 e^2}\\ &=-\frac{3 b (e f-d g) n (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac{3 b g n (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{8 e^2}+\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}+\frac{\left (3 b^2 g n (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 )}{16 e^2}+\frac{\left (3 b^2 (e f-d g) n (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 )}{4 e^2}\\ &=-\frac{3 b (e f-d g) n (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac{3 b g n (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{8 e^2}+\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}+\frac{\left (3 b g n (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 )}{8 e^2}+\frac{\left (3 b (e f-d g) n (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 )}{2 e^2}\\ &=\frac{3 b^{3/2} e^{-\frac{a}{b n}} (e f-d g) n^{3/2} \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 )}{4 e^2}+\frac{3 b^{3/2} e^{-\frac{2 a}{b n}} g n^{3/2} \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 )}{16 e^2}-\frac{3 b (e f-d g) n (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac{3 b g n (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{8 e^2}+\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}\\ \end{align*}

Mathematica [A]  time = 0.433954, size = 282, normalized size = 0.85 \[ \frac{(d+e x) \left (24 b n (e f-d g) \left (\sqrt{\pi } \sqrt{b} \sqrt{n} e^{-\frac{a}{b n}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )-2 \sqrt{a+b \log \left (c (d+e x)^n\right )}\right )+3 b g n (d+e x) \left (\sqrt{2 \pi } \sqrt{b} \sqrt{n} e^{-\frac{2 a}{b n}} \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )-4 \sqrt{a+b \log \left (c (d+e x)^n\right )}\right )+32 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}+16 g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}\right )}{32 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*(32*(e*f - d*g)*(a + b*Log[c*(d + e*x)^n])^(3/2) + 16*g*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2)
+ 3*b*g*n*(d + e*x)*((Sqrt[b]*Sqrt[n]*Sqrt[2*Pi]*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)) - 4*Sqrt[a + b*Log[c*(d + e*x)^n]]) + 24*b*(e*f - d*g)*n*((Sqrt[b
]*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)
) - 2*Sqrt[a + b*Log[c*(d + e*x)^n]])))/(32*e^2)

________________________________________________________________________________________

Maple [F]  time = 0.511, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) \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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x + f\right )}{\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)

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x + f\right )}{\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)