\(\int \frac {\sqrt {h x} (a+b \log (c (d+e x^2)^p))}{f+g x} \, dx\) [616]

Optimal result

Integrand size = 31, antiderivative size = 1680 \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\frac {2 a \sqrt {h x}}{g}-\frac {8 b p \sqrt {h x}}{g}-\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 b \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{g^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} 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} g}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} 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} g}-\frac {8 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {4 i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 2.28 (sec) , antiderivative size = 1680, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.645, Rules used = {2517, 2526, 2498, 327, 217, 1179, 642, 1176, 631, 210, 211, 2520, 12, 266, 6857, 5048, 4966, 2449, 2352, 2497} \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\frac {2 \sqrt {h x} a}{g}-\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e} g}+\frac {2 b \sqrt {h x} \log \left (c \left (e x^2+d\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{g^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}-\frac {8 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {4 i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}+1\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}-\frac {8 b p \sqrt {h x}}{g} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 211

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

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 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

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 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 2520

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[u*(x^(n - 1)/(d + e*x^n)
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

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]

Rule 4966

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x]))*(Log[2/(1
- I*c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((
d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], x] + Simp[(a + b*ArcTan[c*x])*(Log[2*c*((d + e*x)/((c*
d + I*e)*(1 - I*c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 5048

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 1506, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\frac {\sqrt {h x} \left (2 a \sqrt {g} \sqrt {x}-8 b \sqrt {g} p \sqrt {x}-\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {g} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {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} \sqrt {g} p \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {g} p \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt [4]{e}}+2 b \sqrt {g} \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )+\sqrt {-f} \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-\sqrt {-f} \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-b \sqrt {-f} p \left (\log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-\sqrt [4]{e} \sqrt {x}\right )}{-\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )+\log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+i \sqrt [4]{e} \sqrt {x}\right )}{i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )+\log \left (\frac {\sqrt {g} \left (i \sqrt [4]{-d}+\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )+\log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-\sqrt [4]{-d} \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-i \sqrt [4]{-d} \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right )\right )+b \sqrt {-f} p \left (\log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+\log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-i \sqrt [4]{e} \sqrt {x}\right )}{i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+\log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+i \sqrt [4]{e} \sqrt {x}\right )}{-i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+\log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+\sqrt [4]{e} \sqrt {x}\right )}{-\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-\sqrt [4]{-d} \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-i \sqrt [4]{-d} \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right )\right )\right )}{g^{3/2} \sqrt {x}} \]

[In]

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

[Out]

(Sqrt[h*x]*(2*a*Sqrt[g]*Sqrt[x] - 8*b*Sqrt[g]*p*Sqrt[x] - (2*Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*ArcTan[1 - (Sqrt[2]*e
^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (2*Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4
)])/e^(1/4) - (Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/e^(1/4)
+ (Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/e^(1/4) + 2*b*Sqrt
[g]*Sqrt[x]*Log[c*(d + e*x^2)^p] + Sqrt[-f]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]]*(a + b*Log[c*(d + e*x^2)^p]) - Sqr
t[-f]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]]*(a + b*Log[c*(d + e*x^2)^p]) - b*Sqrt[-f]*p*(Log[(Sqrt[g]*((-d)^(1/4) -
e^(1/4)*Sqrt[x]))/(-(e^(1/4)*Sqrt[-f]) + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] + Log[(Sqrt[g]*(
(-d)^(1/4) + I*e^(1/4)*Sqrt[x]))/(I*e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] +
Log[(Sqrt[g]*(I*(-d)^(1/4) + e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[
g]*Sqrt[x]] + Log[(Sqrt[g]*((-d)^(1/4) + e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-
f] - Sqrt[g]*Sqrt[x]] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] - (-d)^(1/4)*Sqrt[
g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] - I*(-d)^(1/4)*Sqrt[g])] + PolyLog[
2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sq
rt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]) + b*Sqrt[-f]*p*(Log[(Sqrt[g]*((-d)^(1/4)
- e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]] + Log[(Sqrt[g]*((
-d)^(1/4) - I*e^(1/4)*Sqrt[x]))/(I*e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]] + L
og[(Sqrt[g]*((-d)^(1/4) + I*e^(1/4)*Sqrt[x]))/((-I)*e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] + Sqr
t[g]*Sqrt[x]] + Log[(Sqrt[g]*((-d)^(1/4) + e^(1/4)*Sqrt[x]))/(-(e^(1/4)*Sqrt[-f]) + (-d)^(1/4)*Sqrt[g])]*Log[S
qrt[-f] + Sqrt[g]*Sqrt[x]] + PolyLog[2, (e^(1/4)*(Sqrt[-f] + Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] - (-d)^(1/4)*
Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] + Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] - I*(-d)^(1/4)*Sqrt[g])] + Pol
yLog[2, (e^(1/4)*(Sqrt[-f] + Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4
)*(Sqrt[-f] + Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])])))/(g^(3/2)*Sqrt[x])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

b*integrate((sqrt(h)*sqrt(x)*log((e*x^2 + d)^p) + sqrt(h)*sqrt(x)*log(c))/(g*x + f), x) - 2*(f*h^2*arctan(sqrt
(h*x)*g/sqrt(f*g*h))/(sqrt(f*g*h)*g) - sqrt(h*x)*h/g)*a/h

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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