3.2.26 \(\int (d+e x)^3 \sqrt {a+b \log (c x^n)} \, dx\) [126]

Optimal. Leaf size=402 \[ -\frac {1}{2} \sqrt {b} d^3 e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } x \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {1}{16} \sqrt {b} e^3 e^{-\frac {4 a}{b n}} \sqrt {n} \sqrt {\pi } x^4 \left (c x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {3}{4} \sqrt {b} d^2 e e^{-\frac {2 a}{b n}} \sqrt {n} \sqrt {\frac {\pi }{2}} x^2 \left (c x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {1}{2} \sqrt {b} d e^2 e^{-\frac {3 a}{b n}} \sqrt {n} \sqrt {\frac {\pi }{3}} x^3 \left (c x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+d^3 x \sqrt {a+b \log \left (c x^n\right )}+\frac {3}{2} d^2 e x^2 \sqrt {a+b \log \left (c x^n\right )}+d e^2 x^3 \sqrt {a+b \log \left (c x^n\right )}+\frac {1}{4} e^3 x^4 \sqrt {a+b \log \left (c x^n\right )} \]

[Out]

-1/6*d*e^2*x^3*erfi(3^(1/2)*(a+b*ln(c*x^n))^(1/2)/b^(1/2)/n^(1/2))*b^(1/2)*n^(1/2)*3^(1/2)*Pi^(1/2)/exp(3*a/b/
n)/((c*x^n)^(3/n))-3/8*d^2*e*x^2*erfi(2^(1/2)*(a+b*ln(c*x^n))^(1/2)/b^(1/2)/n^(1/2))*b^(1/2)*n^(1/2)*2^(1/2)*P
i^(1/2)/exp(2*a/b/n)/((c*x^n)^(2/n))-1/2*d^3*x*erfi((a+b*ln(c*x^n))^(1/2)/b^(1/2)/n^(1/2))*b^(1/2)*n^(1/2)*Pi^
(1/2)/exp(a/b/n)/((c*x^n)^(1/n))-1/16*e^3*x^4*erfi(2*(a+b*ln(c*x^n))^(1/2)/b^(1/2)/n^(1/2))*b^(1/2)*n^(1/2)*Pi
^(1/2)/exp(4*a/b/n)/((c*x^n)^(4/n))+d^3*x*(a+b*ln(c*x^n))^(1/2)+3/2*d^2*e*x^2*(a+b*ln(c*x^n))^(1/2)+d*e^2*x^3*
(a+b*ln(c*x^n))^(1/2)+1/4*e^3*x^4*(a+b*ln(c*x^n))^(1/2)

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Rubi [A]
time = 0.45, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2367, 2333, 2337, 2211, 2235, 2342, 2347} \begin {gather*} -\frac {1}{2} \sqrt {\pi } \sqrt {b} d^3 \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^3 x \sqrt {a+b \log \left (c x^n\right )}-\frac {3}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} d^2 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 )+\frac {3}{2} d^2 e x^2 \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{2} \sqrt {\frac {\pi }{3}} \sqrt {b} d 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 )+d e^2 x^3 \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^3 \sqrt {n} x^4 e^{-\frac {4 a}{b n}} \left (c x^n\right )^{-4/n} \text {Erfi}\left (\frac {2 \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+\frac {1}{4} e^3 x^4 \sqrt {a+b \log \left (c x^n\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2333

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 2337

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 2342

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 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2367

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]))

Rubi steps

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

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Mathematica [A]
time = 0.34, size = 366, normalized size = 0.91 \begin {gather*} \frac {1}{48} e^{-\frac {4 a}{b n}} x \left (c x^n\right )^{-4/n} \left (-24 \sqrt {b} d^3 e^{\frac {3 a}{b n}} \sqrt {n} \sqrt {\pi } \left (c x^n\right )^{3/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-3 \sqrt {b} e^3 \sqrt {n} \sqrt {\pi } x^3 \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (-9 \sqrt {b} d^2 e e^{\frac {a}{b n}} \sqrt {n} \sqrt {2 \pi } x \left (c x^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-4 \sqrt {b} d e^2 \sqrt {n} \sqrt {3 \pi } x^2 \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+6 e^{\frac {3 a}{b n}} \left (c x^n\right )^{3/n} \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \sqrt {a+b \log \left (c x^n\right )}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-24*Sqrt[b]*d^3*E^((3*a)/(b*n))*Sqrt[n]*Sqrt[Pi]*(c*x^n)^(3/n)*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt[n
])] - 3*Sqrt[b]*e^3*Sqrt[n]*Sqrt[Pi]*x^3*Erfi[(2*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])] + 2*E^(a/(b*n))*(c
*x^n)^n^(-1)*(-9*Sqrt[b]*d^2*e*E^(a/(b*n))*Sqrt[n]*Sqrt[2*Pi]*x*(c*x^n)^n^(-1)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*
x^n]])/(Sqrt[b]*Sqrt[n])] - 4*Sqrt[b]*d*e^2*Sqrt[n]*Sqrt[3*Pi]*x^2*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*x^n]])/(Sqrt
[b]*Sqrt[n])] + 6*E^((3*a)/(b*n))*(c*x^n)^(3/n)*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*Sqrt[a + b*Log[c*x
^n]])))/(48*E^((4*a)/(b*n))*(c*x^n)^(4/n))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{3} \sqrt {a +b \ln \left (c \,x^{n}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \log {\left (c x^{n} \right )}} \left (d + e x\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {a+b\,\ln \left (c\,x^n\right )}\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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