3.411 \(\int (a+b \log (c (d (e+f x)^m)^n))^{5/2} \, dx\)

Optimal. Leaf size=219 \[ -\frac{15 \sqrt{\pi } b^{5/2} m^{5/2} n^{5/2} (e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )}{8 f}+\frac{15 b^2 m^2 n^2 (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{4 f}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}{f}-\frac{5 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}{2 f} \]

[Out]

(-15*b^(5/2)*m^(5/2)*n^(5/2)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqr
t[n])])/(8*E^(a/(b*m*n))*f*(c*(d*(e + f*x)^m)^n)^(1/(m*n))) + (15*b^2*m^2*n^2*(e + f*x)*Sqrt[a + b*Log[c*(d*(e
+ f*x)^m)^n]])/(4*f) - (5*b*m*n*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^(3/2))/(2*f) + ((e + f*x)*(a + b*L
og[c*(d*(e + f*x)^m)^n])^(5/2))/f

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Rubi [A]  time = 0.3677, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2389, 2296, 2300, 2180, 2204, 2445} \[ -\frac{15 \sqrt{\pi } b^{5/2} m^{5/2} n^{5/2} (e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )}{8 f}+\frac{15 b^2 m^2 n^2 (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{4 f}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}{f}-\frac{5 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}{2 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2),x]

[Out]

(-15*b^(5/2)*m^(5/2)*n^(5/2)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqr
t[n])])/(8*E^(a/(b*m*n))*f*(c*(d*(e + f*x)^m)^n)^(1/(m*n))) + (15*b^2*m^2*n^2*(e + f*x)*Sqrt[a + b*Log[c*(d*(e
+ f*x)^m)^n]])/(4*f) - (5*b*m*n*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^(3/2))/(2*f) + ((e + f*x)*(a + b*L
og[c*(d*(e + f*x)^m)^n])^(5/2))/f

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 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin{align*} \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2} \, dx &=\operatorname{Subst}\left (\int \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{5/2} \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right )^{5/2} \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}{f}-\operatorname{Subst}\left (\frac{(5 b m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right )^{3/2} \, dx,x,e+f x\right )}{2 f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-\frac{5 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}{2 f}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}{f}+\operatorname{Subst}\left (\frac{\left (15 b^2 m^2 n^2\right ) \operatorname{Subst}\left (\int \sqrt{a+b \log \left (c d^n x^{m n}\right )} \, dx,x,e+f x\right )}{4 f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{15 b^2 m^2 n^2 (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{4 f}-\frac{5 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}{2 f}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}{f}-\operatorname{Subst}\left (\frac{\left (15 b^3 m^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c d^n x^{m n}\right )}} \, dx,x,e+f x\right )}{8 f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{15 b^2 m^2 n^2 (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{4 f}-\frac{5 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}{2 f}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}{f}-\operatorname{Subst}\left (\frac{\left (15 b^3 m^2 n^2 (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac{1}{m n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{m n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c d^n (e+f x)^{m n}\right )\right )}{8 f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{15 b^2 m^2 n^2 (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{4 f}-\frac{5 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}{2 f}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}{f}-\operatorname{Subst}\left (\frac{\left (15 b^2 m^2 n^2 (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac{1}{m n}}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b m n}+\frac{x^2}{b m n}} \, dx,x,\sqrt{a+b \log \left (c d^n (e+f x)^{m n}\right )}\right )}{4 f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-\frac{15 b^{5/2} e^{-\frac{a}{b m n}} m^{5/2} n^{5/2} \sqrt{\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )}{8 f}+\frac{15 b^2 m^2 n^2 (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{4 f}-\frac{5 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}{2 f}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}{f}\\ \end{align*}

Mathematica [A]  time = 0.31078, size = 190, normalized size = 0.87 \[ \frac{(e+f x) \left (8 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}-5 b m n \left (3 \sqrt{\pi } b^{3/2} m^{3/2} n^{3/2} e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt{b} \sqrt{m} \sqrt{n}}\right )+2 \sqrt{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \left (2 a+2 b \log \left (c \left (d (e+f x)^m\right )^n\right )-3 b m n\right )\right )\right )}{8 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2),x]

[Out]

((e + f*x)*(8*(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2) - 5*b*m*n*((3*b^(3/2)*m^(3/2)*n^(3/2)*Sqrt[Pi]*Erfi[Sqrt[
a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))) +
2*Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]*(2*a - 3*b*m*n + 2*b*Log[c*(d*(e + f*x)^m)^n]))))/(8*f)

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Maple [F]  time = 0.6, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d*(f*x+e)^m)^n))^(5/2),x)

[Out]

int((a+b*ln(c*(d*(f*x+e)^m)^n))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*log(((f*x + e)^m*d)^n*c) + a)^(5/2), 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((a+b*log(c*(d*(f*x+e)^m)^n))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d*(f*x+e)**m)**n))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^(5/2),x, algorithm="giac")

[Out]

integrate((b*log(((f*x + e)^m*d)^n*c) + a)^(5/2), x)