\(\int \frac {(f+g x) (a+b \log (c (d+e x^2)^p))}{(h x)^{3/2}} \, dx\) [607]

Optimal result

Integrand size = 29, antiderivative size = 603 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {2 a g \sqrt {h x}}{h^2}-\frac {8 b g p \sqrt {h x}}{h^2}-\frac {2 \sqrt {2} b \sqrt [4]{e} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {\sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} h^{3/2}} \]

[Out]

-2*b*e^(1/4)*f*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(1/4)/h^(3/2)-2*b*d^(1/4)*g*p
*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/e^(1/4)/h^(3/2)+2*b*e^(1/4)*f*p*arctan(1+e^(1/4
)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(1/4)/h^(3/2)+2*b*d^(1/4)*g*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^
(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/e^(1/4)/h^(3/2)+b*e^(1/4)*f*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)-d^(1/4)*e^(1
/4)*2^(1/2)*(h*x)^(1/2))*2^(1/2)/d^(1/4)/h^(3/2)-b*d^(1/4)*g*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)-d^(1/4)*e^
(1/4)*2^(1/2)*(h*x)^(1/2))*2^(1/2)/e^(1/4)/h^(3/2)-b*e^(1/4)*f*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)+d^(1/4)*
e^(1/4)*2^(1/2)*(h*x)^(1/2))*2^(1/2)/d^(1/4)/h^(3/2)+b*d^(1/4)*g*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)+d^(1/4
)*e^(1/4)*2^(1/2)*(h*x)^(1/2))*2^(1/2)/e^(1/4)/h^(3/2)-2*f*(a+b*ln(c*(e*x^2+d)^p))/h/(h*x)^(1/2)+2*a*g*(h*x)^(
1/2)/h^2-8*b*g*p*(h*x)^(1/2)/h^2+2*b*g*ln(c*(e*x^2+d)^p)*(h*x)^(1/2)/h^2

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {2517, 2526, 2498, 327, 217, 1179, 642, 1176, 631, 210, 2505, 303} \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 a g \sqrt {h x}}{h^2}-\frac {2 \sqrt {2} b \sqrt [4]{e} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{d} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}+\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} g p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{e} h^{3/2}}+\frac {\sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{e} h^{3/2}}-\frac {8 b g p \sqrt {h x}}{h^2} \]

[In]

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

[Out]

(2*a*g*Sqrt[h*x])/h^2 - (8*b*g*p*Sqrt[h*x])/h^2 - (2*Sqrt[2]*b*e^(1/4)*f*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*
x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(3/2)) - (2*Sqrt[2]*b*d^(1/4)*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d
^(1/4)*Sqrt[h])])/(e^(1/4)*h^(3/2)) + (2*Sqrt[2]*b*e^(1/4)*f*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)
*Sqrt[h])])/(d^(1/4)*h^(3/2)) + (2*Sqrt[2]*b*d^(1/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[
h])])/(e^(1/4)*h^(3/2)) + (2*b*g*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/h^2 - (2*f*(a + b*Log[c*(d + e*x^2)^p]))/(h*S
qrt[h*x]) + (Sqrt[2]*b*e^(1/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]
])/(d^(1/4)*h^(3/2)) - (Sqrt[2]*b*d^(1/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4
)*Sqrt[h*x]])/(e^(1/4)*h^(3/2)) - (Sqrt[2]*b*e^(1/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(
1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(3/2)) + (Sqrt[2]*b*d^(1/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x +
Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*h^(3/2))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 2517

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))^(p_.)]*(b_.))^(q_.)*((h_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(r
_.), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/h, Subst[Int[x^(k*(m + 1) - 1)*(f + g*(x^k/h))^r*(a + b*Lo
g[c*(d + e*(x^(k*n)/h^n))^p])^q, x], x, (h*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && Fract
ionQ[m] && IntegerQ[n] && IntegerQ[r]

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\left (f+\frac {g x^2}{h}\right ) \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{x^2} \, dx,x,\sqrt {h x}\right )}{h} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {g \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h}+\frac {f \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{x^2}\right ) \, dx,x,\sqrt {h x}\right )}{h} \\ & = \frac {(2 g) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right ) \, dx,x,\sqrt {h x}\right )}{h^2}+\frac {(2 f) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )}{x^2} \, dx,x,\sqrt {h x}\right )}{h} \\ & = \frac {2 a g \sqrt {h x}}{h^2}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {(2 b g) \text {Subst}\left (\int \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right ) \, dx,x,\sqrt {h x}\right )}{h^2}+\frac {(8 b e f p) \text {Subst}\left (\int \frac {x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^3} \\ & = \frac {2 a g \sqrt {h x}}{h^2}+\frac {2 b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}-\frac {(8 b e g p) \text {Subst}\left (\int \frac {x^4}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^4}-\frac {\left (4 b \sqrt {e} f p\right ) \text {Subst}\left (\int \frac {\sqrt {d} h-\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^3}+\frac {\left (4 b \sqrt {e} f p\right ) \text {Subst}\left (\int \frac {\sqrt {d} h+\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^3} \\ & = \frac {2 a g \sqrt {h x}}{h^2}-\frac {8 b g p \sqrt {h x}}{h^2}+\frac {2 b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {(8 b d g p) \text {Subst}\left (\int \frac {1}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^2}+\frac {\left (\sqrt {2} b \sqrt [4]{e} f p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}+2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {\left (\sqrt {2} b \sqrt [4]{e} f p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}-2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {(2 b f p) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{h}+\frac {(2 b f p) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{h} \\ & = \frac {2 a g \sqrt {h x}}{h^2}-\frac {8 b g p \sqrt {h x}}{h^2}+\frac {2 b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {\left (4 b \sqrt {d} g p\right ) \text {Subst}\left (\int \frac {\sqrt {d} h-\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^3}+\frac {\left (4 b \sqrt {d} g p\right ) \text {Subst}\left (\int \frac {\sqrt {d} h+\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^3}+\frac {\left (2 \sqrt {2} b \sqrt [4]{e} f p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{e} f p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}} \\ & = \frac {2 a g \sqrt {h x}}{h^2}-\frac {8 b g p \sqrt {h x}}{h^2}-\frac {2 \sqrt {2} b \sqrt [4]{e} f p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {\left (\sqrt {2} b \sqrt [4]{d} g p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}+2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt [4]{e} h^{3/2}}-\frac {\left (\sqrt {2} b \sqrt [4]{d} g p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}-2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {\left (2 b \sqrt {d} g p\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt {e} h}+\frac {\left (2 b \sqrt {d} g p\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt {e} h} \\ & = \frac {2 a g \sqrt {h x}}{h^2}-\frac {8 b g p \sqrt {h x}}{h^2}-\frac {2 \sqrt {2} b \sqrt [4]{e} f p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {\sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {\left (2 \sqrt {2} b \sqrt [4]{d} g p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{d} g p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}} \\ & = \frac {2 a g \sqrt {h x}}{h^2}-\frac {8 b g p \sqrt {h x}}{h^2}-\frac {2 \sqrt {2} b \sqrt [4]{e} f p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 b g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {\sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} h^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.55 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {2 x^{3/2} \left (a g \sqrt {x}-4 b g p \sqrt {x}-\frac {\sqrt {2} b \sqrt [4]{d} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {2 b \sqrt [4]{e} f p \left (\arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\text {arctanh}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {b \sqrt [4]{d} g p \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+\frac {b \sqrt [4]{d} g p \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+b g \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )-\frac {f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}\right )}{(h x)^{3/2}} \]

[In]

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

[Out]

(2*x^(3/2)*(a*g*Sqrt[x] - 4*b*g*p*Sqrt[x] - (Sqrt[2]*b*d^(1/4)*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4
)])/e^(1/4) + (Sqrt[2]*b*d^(1/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (2*b*e^(1/4)*f*p
*(ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)] + ArcTanh[(d*e^(1/4)*Sqrt[x])/(-d)^(5/4)]))/(-d)^(1/4) - (b*d^(1/4)*g*p
*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) + (b*d^(1/4)*g*p*Log[Sqrt[d] +
Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/(Sqrt[2]*e^(1/4)) + b*g*Sqrt[x]*Log[c*(d + e*x^2)^p] - (f*(a + b
*Log[c*(d + e*x^2)^p]))/Sqrt[x]))/(h*x)^(3/2)

Maple [F]

\[\int \frac {\left (g x +f \right ) \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {3}{2}}}d x\]

[In]

int((g*x+f)*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(3/2),x)

[Out]

int((g*x+f)*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(3/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1162 vs. \(2 (427) = 854\).

Time = 0.37 (sec) , antiderivative size = 1162, normalized size of antiderivative = 1.93 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\frac {2 \, {\left (h^{2} x \sqrt {-\frac {2 \, b^{2} f g p^{2} + h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}} \log \left (-32 \, {\left (b^{3} e^{2} f^{4} - b^{3} d^{2} g^{4}\right )} \sqrt {h x} p^{3} + 32 \, {\left (d e f h^{5} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}} - {\left (b^{2} d e f^{2} g - b^{2} d^{2} g^{3}\right )} h^{2} p^{2}\right )} \sqrt {-\frac {2 \, b^{2} f g p^{2} + h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}}\right ) - h^{2} x \sqrt {-\frac {2 \, b^{2} f g p^{2} + h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}} \log \left (-32 \, {\left (b^{3} e^{2} f^{4} - b^{3} d^{2} g^{4}\right )} \sqrt {h x} p^{3} - 32 \, {\left (d e f h^{5} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}} - {\left (b^{2} d e f^{2} g - b^{2} d^{2} g^{3}\right )} h^{2} p^{2}\right )} \sqrt {-\frac {2 \, b^{2} f g p^{2} + h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}}\right ) - h^{2} x \sqrt {-\frac {2 \, b^{2} f g p^{2} - h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}} \log \left (-32 \, {\left (b^{3} e^{2} f^{4} - b^{3} d^{2} g^{4}\right )} \sqrt {h x} p^{3} + 32 \, {\left (d e f h^{5} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}} + {\left (b^{2} d e f^{2} g - b^{2} d^{2} g^{3}\right )} h^{2} p^{2}\right )} \sqrt {-\frac {2 \, b^{2} f g p^{2} - h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}}\right ) + h^{2} x \sqrt {-\frac {2 \, b^{2} f g p^{2} - h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}} \log \left (-32 \, {\left (b^{3} e^{2} f^{4} - b^{3} d^{2} g^{4}\right )} \sqrt {h x} p^{3} - 32 \, {\left (d e f h^{5} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}} + {\left (b^{2} d e f^{2} g - b^{2} d^{2} g^{3}\right )} h^{2} p^{2}\right )} \sqrt {-\frac {2 \, b^{2} f g p^{2} - h^{3} \sqrt {-\frac {{\left (b^{4} e^{2} f^{4} - 2 \, b^{4} d e f^{2} g^{2} + b^{4} d^{2} g^{4}\right )} p^{4}}{d e h^{6}}}}{h^{3}}}\right ) - {\left (a f + {\left (4 \, b g p - a g\right )} x - {\left (b g p x - b f p\right )} \log \left (e x^{2} + d\right ) - {\left (b g x - b f\right )} \log \left (c\right )\right )} \sqrt {h x}\right )}}{h^{2} x} \]

[In]

integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(3/2),x, algorithm="fricas")

[Out]

2*(h^2*x*sqrt(-(2*b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)
*log(-32*(b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 + 32*(d*e*f*h^5*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^
4*d^2*g^4)*p^4/(d*e*h^6)) - (b^2*d*e*f^2*g - b^2*d^2*g^3)*h^2*p^2)*sqrt(-(2*b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f
^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)) - h^2*x*sqrt(-(2*b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*
f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)*log(-32*(b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3
- 32*(d*e*f*h^5*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)) - (b^2*d*e*f^2*g - b^2*d^
2*g^3)*h^2*p^2)*sqrt(-(2*b^2*f*g*p^2 + h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)
))/h^3)) - h^2*x*sqrt(-(2*b^2*f*g*p^2 - h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6
)))/h^3)*log(-32*(b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 + 32*(d*e*f*h^5*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*
g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)) + (b^2*d*e*f^2*g - b^2*d^2*g^3)*h^2*p^2)*sqrt(-(2*b^2*f*g*p^2 - h^3*sqrt(-(b
^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)) + h^2*x*sqrt(-(2*b^2*f*g*p^2 - h^3*sqrt(-(
b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)))/h^3)*log(-32*(b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h
*x)*p^3 - 32*(d*e*f*h^5*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(d*e*h^6)) + (b^2*d*e*f^2*g
- b^2*d^2*g^3)*h^2*p^2)*sqrt(-(2*b^2*f*g*p^2 - h^3*sqrt(-(b^4*e^2*f^4 - 2*b^4*d*e*f^2*g^2 + b^4*d^2*g^4)*p^4/(
d*e*h^6)))/h^3)) - (a*f + (4*b*g*p - a*g)*x - (b*g*p*x - b*f*p)*log(e*x^2 + d) - (b*g*x - b*f)*log(c))*sqrt(h*
x))/(h^2*x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((g*x+f)*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(3/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.21 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=-\frac {b e f p {\left (\frac {\sqrt {2} \log \left (\sqrt {e} h x + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {e} h x - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {e}} - \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {e}}\right )}}{h} + \frac {2 \, b g x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{\left (h x\right )^{\frac {3}{2}}} + \frac {2 \, a g x^{2}}{\left (h x\right )^{\frac {3}{2}}} - \frac {2 \, b f \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{\sqrt {h x} h} - \frac {{\left (\frac {8 \, \sqrt {h x} h^{2}}{e} - \frac {{\left (\frac {\sqrt {2} h^{4} \log \left (\sqrt {e} h x + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {3}{4}} e^{\frac {1}{4}}} - \frac {\sqrt {2} h^{4} \log \left (\sqrt {e} h x - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {3}{4}} e^{\frac {1}{4}}} + \frac {\sqrt {2} h^{3} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {d}} + \frac {\sqrt {2} h^{3} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {d}}\right )} d}{e}\right )} b e g p}{h^{4}} - \frac {2 \, a f}{\sqrt {h x} h} \]

[In]

integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(3/2),x, algorithm="maxima")

[Out]

-b*e*f*p*(sqrt(2)*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4
)) - sqrt(2)*log(sqrt(e)*h*x - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) -
sqrt(2)*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)
*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sq
rt(e)) - sqrt(2)*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))
/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt
(e)*h)*sqrt(e)))/h + 2*b*g*x^2*log((e*x^2 + d)^p*c)/(h*x)^(3/2) + 2*a*g*x^2/(h*x)^(3/2) - 2*b*f*log((e*x^2 + d
)^p*c)/(sqrt(h*x)*h) - (8*sqrt(h*x)*h^2/e - (sqrt(2)*h^4*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(
1/4) + sqrt(d)*h)/((d*h^2)^(3/4)*e^(1/4)) - sqrt(2)*h^4*log(sqrt(e)*h*x - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1
/4) + sqrt(d)*h)/((d*h^2)^(3/4)*e^(1/4)) + sqrt(2)*h^3*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2
)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*s
qrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(d)) + sqrt(2)*h^3*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - s
qrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*
e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(d)))*d/e)*b*e*g*p/h^4 - 2*a*f/(sqrt(h*x)*h)

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.74 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {b f p}{\sqrt {h x}} - \frac {\sqrt {h x} b g p}{h}\right )} \log \left (e h^{2} x^{2} + d h^{2}\right ) - \frac {2 \, {\left (b f p \log \left (h^{2}\right ) - b f \log \left (c\right ) - a f\right )}}{\sqrt {h x}} + \frac {2 \, {\left (b g p \log \left (h^{2}\right ) + 4 \, b g p - b g \log \left (c\right ) - a g\right )} \sqrt {h x}}{h} - \frac {2 \, {\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p + \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} + 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d e^{2} h^{2}} - \frac {2 \, {\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p + \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} - 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d e^{2} h^{2}} - \frac {{\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p - \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \log \left (h x + \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d e^{2} h^{2}} + \frac {{\left (\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g h p - \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f p\right )} \log \left (h x - \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d e^{2} h^{2}}}{h} \]

[In]

integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(3/2),x, algorithm="giac")

[Out]

-(2*(b*f*p/sqrt(h*x) - sqrt(h*x)*b*g*p/h)*log(e*h^2*x^2 + d*h^2) - 2*(b*f*p*log(h^2) - b*f*log(c) - a*f)/sqrt(
h*x) + 2*(b*g*p*log(h^2) + 4*b*g*p - b*g*log(c) - a*g)*sqrt(h*x)/h - 2*(sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e*g*h*p
+ sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*p)*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4) + 2*sqrt(h*x))/(d*h^2/e)^(1/4))
/(d*e^2*h^2) - 2*(sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e*g*h*p + sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*p)*arctan(-1/2*sqrt(2)
*(sqrt(2)*(d*h^2/e)^(1/4) - 2*sqrt(h*x))/(d*h^2/e)^(1/4))/(d*e^2*h^2) - (sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e*g*h*p
- sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*p)*log(h*x + sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/(d*e^2*h^2) +
(sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e*g*h*p - sqrt(2)*(d*e^3*h^2)^(3/4)*b*f*p)*log(h*x - sqrt(2)*(d*h^2/e)^(1/4)*s
qrt(h*x) + sqrt(d*h^2/e))/(d*e^2*h^2))/h

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{3/2}} \,d x \]

[In]

int(((f + g*x)*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(3/2),x)

[Out]

int(((f + g*x)*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(3/2), x)