3.461 \(\int (g+h x) \sqrt {a+b \log (c (d (e+f x)^p)^q)} \, dx\)

Optimal. Leaf size=311 \[ -\frac {\sqrt {\pi } \sqrt {b} \sqrt {p} \sqrt {q} (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{2 f^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} h \sqrt {p} \sqrt {q} (e+f x)^2 e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f^2}+\frac {(e+f x) (f g-e h) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac {h (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2} \]

[Out]

-1/8*h*(f*x+e)^2*erfi(2^(1/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/b^(1/2)/p^(1/2)/q^(1/2))*b^(1/2)*p^(1/2)*2^(1/
2)*Pi^(1/2)*q^(1/2)/exp(2*a/b/p/q)/f^2/((c*(d*(f*x+e)^p)^q)^(2/p/q))-1/2*(-e*h+f*g)*(f*x+e)*erfi((a+b*ln(c*(d*
(f*x+e)^p)^q))^(1/2)/b^(1/2)/p^(1/2)/q^(1/2))*b^(1/2)*p^(1/2)*Pi^(1/2)*q^(1/2)/exp(a/b/p/q)/f^2/((c*(d*(f*x+e)
^p)^q)^(1/p/q))+(-e*h+f*g)*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/f^2+1/2*h*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^
p)^q))^(1/2)/f^2

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Rubi [A]  time = 0.81, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2401, 2389, 2296, 2300, 2180, 2204, 2390, 2305, 2310, 2445} \[ -\frac {\sqrt {\pi } \sqrt {b} \sqrt {p} \sqrt {q} (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{2 f^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} h \sqrt {p} \sqrt {q} (e+f x)^2 e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f^2}+\frac {(e+f x) (f g-e h) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac {h (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2} \]

Antiderivative was successfully verified.

[In]

Int[(g + h*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]],x]

[Out]

-(Sqrt[b]*(f*g - e*h)*Sqrt[p]*Sqrt[Pi]*Sqrt[q]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sq
rt[p]*Sqrt[q])])/(2*E^(a/(b*p*q))*f^2*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) - (Sqrt[b]*h*Sqrt[p]*Sqrt[Pi/2]*Sqrt[q]
*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(4*E^((2*a)/(b*p*
q))*f^2*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + ((f*g - e*h)*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/f^2 +
(h*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(2*f^2)

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 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 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]

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 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 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 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 (g+h x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx &=\operatorname {Subst}\left (\int (g+h x) \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {(f g-e h) \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}{f}+\frac {h (e+f x) \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}{f}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\frac {h \int (e+f x) \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(f g-e h) \int \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\frac {h \operatorname {Subst}\left (\int x \sqrt {a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(f g-e h) \operatorname {Subst}\left (\int \sqrt {a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(f g-e h) (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac {h (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\operatorname {Subst}\left (\frac {(b h p q) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{4 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(b (f g-e h) p q) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{2 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(f g-e h) (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac {h (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\operatorname {Subst}\left (\frac {\left (b h (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {2 x}{p q}}}{\sqrt {a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{4 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (b (f g-e h) (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{p q}}}{\sqrt {a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{2 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(f g-e h) (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac {h (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\operatorname {Subst}\left (\frac {\left (h (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \operatorname {Subst}\left (\int e^{-\frac {2 a}{b p q}+\frac {2 x^2}{b p q}} \, dx,x,\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{2 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left ((f g-e h) (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b p q}+\frac {x^2}{b p q}} \, dx,x,\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {\sqrt {b} e^{-\frac {a}{b p q}} (f g-e h) \sqrt {p} \sqrt {\pi } \sqrt {q} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{2 f^2}-\frac {\sqrt {b} e^{-\frac {2 a}{b p q}} h \sqrt {p} \sqrt {\frac {\pi }{2}} \sqrt {q} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f^2}+\frac {(f g-e h) (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac {h (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 298, normalized size = 0.96 \[ -\frac {(e+f x) e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (4 \sqrt {\pi } \sqrt {b} \sqrt {p} \sqrt {q} e^{\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )+\sqrt {2 \pi } \sqrt {b} h \sqrt {p} \sqrt {q} (e+f x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )-4 e^{\frac {2 a}{b p q}} (-e h+2 f g+f h x) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )}{8 f^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]],x]

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^(1/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (h x + g\right )} \sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^(1/2),x, algorithm="giac")

[Out]

integrate((h*x + g)*sqrt(b*log(((f*x + e)^p*d)^q*c) + a), x)

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (h x +g \right ) \sqrt {b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)*(b*ln(c*(d*(f*x+e)^p)^q)+a)^(1/2),x)

[Out]

int((h*x+g)*(b*ln(c*(d*(f*x+e)^p)^q)+a)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (h x + g\right )} \sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^(1/2),x, algorithm="maxima")

[Out]

integrate((h*x + g)*sqrt(b*log(((f*x + e)^p*d)^q*c) + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (g+h\,x\right )\,\sqrt {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g + h*x)*(a + b*log(c*(d*(e + f*x)^p)^q))^(1/2),x)

[Out]

int((g + h*x)*(a + b*log(c*(d*(e + f*x)^p)^q))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}} \left (g + h x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*log(c*(d*(e + f*x)**p)**q))*(g + h*x), x)

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