\(\int \frac {d+e x+f x^2}{(g+h x)^2 (a+c x^2)^{3/2}} \, dx\) [113]

Optimal result

Integrand size = 29, antiderivative size = 239 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=-\frac {a (c g (e g-2 d h)+a h (2 f g-e h))-\left (c^2 d g^2+a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^2 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{\left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (a h^2 (2 f g-e h)-c g \left (f g^2-h (2 e g-3 d h)\right )\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{\left (c g^2+a h^2\right )^{5/2}} \]

[Out]

(a*h^2*(-e*h+2*f*g)-c*g*(f*g^2-h*(-3*d*h+2*e*g)))*arctanh((-c*g*x+a*h)/(a*h^2+c*g^2)^(1/2)/(c*x^2+a)^(1/2))/(a
*h^2+c*g^2)^(5/2)+(-a*(c*g*(-2*d*h+e*g)+a*h*(-e*h+2*f*g))+(c^2*d*g^2+a^2*f*h^2-a*c*(f*g^2-h*(-d*h+2*e*g)))*x)/
a/(a*h^2+c*g^2)^2/(c*x^2+a)^(1/2)-h*(d*h^2-e*g*h+f*g^2)*(c*x^2+a)^(1/2)/(a*h^2+c*g^2)^2/(h*x+g)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1661, 821, 739, 212} \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=-\frac {a (a h (2 f g-e h)+c g (e g-2 d h))-x \left (a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )+c^2 d g^2\right )}{a \sqrt {a+c x^2} \left (a h^2+c g^2\right )^2}-\frac {\text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (-a h^2 (2 f g-e h)-c g h (2 e g-3 d h)+c f g^3\right )}{\left (a h^2+c g^2\right )^{5/2}}-\frac {h \sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{(g+h x) \left (a h^2+c g^2\right )^2} \]

[In]

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

[Out]

-((a*(c*g*(e*g - 2*d*h) + a*h*(2*f*g - e*h)) - (c^2*d*g^2 + a^2*f*h^2 - a*c*(f*g^2 - h*(2*e*g - d*h)))*x)/(a*(
c*g^2 + a*h^2)^2*Sqrt[a + c*x^2])) - (h*(f*g^2 - e*g*h + d*h^2)*Sqrt[a + c*x^2])/((c*g^2 + a*h^2)^2*(g + h*x))
- ((c*f*g^3 - c*g*h*(2*e*g - 3*d*h) - a*h^2*(2*f*g - e*h))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a
+ c*x^2])])/(c*g^2 + a*h^2)^(5/2)

Rule 212

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

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 1661

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[(a*g - c*f*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {a (c g (e g-2 d h)+a h (2 f g-e h))-\left (c^2 d g^2+a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^2 \sqrt {a+c x^2}}-\frac {\int \frac {\frac {a c \left (a h^2 \left (f g^2-d h^2\right )-c \left (f g^4-g^2 h (2 e g-3 d h)\right )\right )}{\left (c g^2+a h^2\right )^2}+\frac {a c h^2 (c g (e g-2 d h)+a h (2 f g-e h)) x}{\left (c g^2+a h^2\right )^2}}{(g+h x)^2 \sqrt {a+c x^2}} \, dx}{a c} \\ & = -\frac {a (c g (e g-2 d h)+a h (2 f g-e h))-\left (c^2 d g^2+a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^2 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{\left (c g^2+a h^2\right )^2 (g+h x)}-\frac {\left (\frac {a^2 c h^3 (c g (e g-2 d h)+a h (2 f g-e h))}{\left (c g^2+a h^2\right )^2}+\frac {a c^2 g \left (a h^2 \left (f g^2-d h^2\right )-c \left (f g^4-g^2 h (2 e g-3 d h)\right )\right )}{\left (c g^2+a h^2\right )^2}\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{a c \left (c g^2+a h^2\right )} \\ & = -\frac {a (c g (e g-2 d h)+a h (2 f g-e h))-\left (c^2 d g^2+a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^2 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{\left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (\frac {a^2 c h^3 (c g (e g-2 d h)+a h (2 f g-e h))}{\left (c g^2+a h^2\right )^2}+\frac {a c^2 g \left (a h^2 \left (f g^2-d h^2\right )-c \left (f g^4-g^2 h (2 e g-3 d h)\right )\right )}{\left (c g^2+a h^2\right )^2}\right ) \text {Subst}\left (\int \frac {1}{c g^2+a h^2-x^2} \, dx,x,\frac {a h-c g x}{\sqrt {a+c x^2}}\right )}{a c \left (c g^2+a h^2\right )} \\ & = -\frac {a (c g (e g-2 d h)+a h (2 f g-e h))-\left (c^2 d g^2+a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^2 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{\left (c g^2+a h^2\right )^2 (g+h x)}-\frac {\left (c f g^3-c g h (2 e g-3 d h)-a h^2 (2 f g-e h)\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{\left (c g^2+a h^2\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.36 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.04 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\frac {c^2 d g^2 x (g+h x)+a^2 h \left (h (2 e g-d h+e h x)+f \left (-3 g^2-g h x+h^2 x^2\right )\right )+a c \left (-f g^2 x (g+2 h x)+d h \left (2 g^2+g h x-2 h^2 x^2\right )+e g \left (-g^2+g h x+3 h^2 x^2\right )\right )}{a \left (c g^2+a h^2\right )^2 (g+h x) \sqrt {a+c x^2}}-\frac {2 \left (c f g^3+c g h (-2 e g+3 d h)+a h^2 (-2 f g+e h)\right ) \arctan \left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{\left (-c g^2-a h^2\right )^{5/2}} \]

[In]

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

[Out]

(c^2*d*g^2*x*(g + h*x) + a^2*h*(h*(2*e*g - d*h + e*h*x) + f*(-3*g^2 - g*h*x + h^2*x^2)) + a*c*(-(f*g^2*x*(g +
2*h*x)) + d*h*(2*g^2 + g*h*x - 2*h^2*x^2) + e*g*(-g^2 + g*h*x + 3*h^2*x^2)))/(a*(c*g^2 + a*h^2)^2*(g + h*x)*Sq
rt[a + c*x^2]) - (2*(c*f*g^3 + c*g*h*(-2*e*g + 3*d*h) + a*h^2*(-2*f*g + e*h))*ArcTan[(Sqrt[c]*(g + h*x) - h*Sq
rt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2]])/(-(c*g^2) - a*h^2)^(5/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(871\) vs. \(2(228)=456\).

Time = 0.53 (sec) , antiderivative size = 872, normalized size of antiderivative = 3.65

method	result	size
default	\(\frac {f x}{h^{2} a \sqrt {c \,x^{2}+a}}+\frac {\left (e h -2 f g \right ) \left (\frac {h^{2}}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {2 c g h \left (2 c \left (x +\frac {g}{h}\right )-\frac {2 c g}{h}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \left (\frac {4 c \left (a \,h^{2}+c \,g^{2}\right )}{h^{2}}-\frac {4 c^{2} g^{2}}{h^{2}}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}-\frac {h^{2} \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{3}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (-\frac {h^{2}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {3 c g h \left (\frac {h^{2}}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {2 c g h \left (2 c \left (x +\frac {g}{h}\right )-\frac {2 c g}{h}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \left (\frac {4 c \left (a \,h^{2}+c \,g^{2}\right )}{h^{2}}-\frac {4 c^{2} g^{2}}{h^{2}}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}-\frac {h^{2} \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{a \,h^{2}+c \,g^{2}}-\frac {4 c \,h^{2} \left (2 c \left (x +\frac {g}{h}\right )-\frac {2 c g}{h}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \left (\frac {4 c \left (a \,h^{2}+c \,g^{2}\right )}{h^{2}}-\frac {4 c^{2} g^{2}}{h^{2}}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{4}}\)	\(872\)

[In]

int((f*x^2+e*x+d)/(h*x+g)^2/(c*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (230) = 460\).

Time = 1.82 (sec) , antiderivative size = 1573, normalized size of antiderivative = 6.58 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/2*((a^2*c*f*g^4 - 2*a^2*c*e*g^3*h + a^3*e*g*h^3 + (3*a^2*c*d - 2*a^3*f)*g^2*h^2 + (a*c^2*f*g^3*h - 2*a*c^2
*e*g^2*h^2 + a^2*c*e*h^4 + (3*a*c^2*d - 2*a^2*c*f)*g*h^3)*x^3 + (a*c^2*f*g^4 - 2*a*c^2*e*g^3*h + a^2*c*e*g*h^3
+ (3*a*c^2*d - 2*a^2*c*f)*g^2*h^2)*x^2 + (a^2*c*f*g^3*h - 2*a^2*c*e*g^2*h^2 + a^3*e*h^4 + (3*a^2*c*d - 2*a^3*
f)*g*h^3)*x)*sqrt(c*g^2 + a*h^2)*log((2*a*c*g*h*x - a*c*g^2 - 2*a^2*h^2 - (2*c^2*g^2 + a*c*h^2)*x^2 + 2*sqrt(c
*g^2 + a*h^2)*(c*g*x - a*h)*sqrt(c*x^2 + a))/(h^2*x^2 + 2*g*h*x + g^2)) + 2*(a*c^2*e*g^5 - a^2*c*e*g^3*h^2 - 2
*a^3*e*g*h^4 + a^3*d*h^5 - (2*a*c^2*d - 3*a^2*c*f)*g^4*h - (a^2*c*d - 3*a^3*f)*g^2*h^3 - (3*a*c^2*e*g^3*h^2 +
3*a^2*c*e*g*h^4 + (c^3*d - 2*a*c^2*f)*g^4*h - (a*c^2*d + a^2*c*f)*g^2*h^3 - (2*a^2*c*d - a^3*f)*h^5)*x^2 - (a*
c^2*e*g^4*h + 2*a^2*c*e*g^2*h^3 + a^3*e*h^5 + (c^3*d - a*c^2*f)*g^5 + 2*(a*c^2*d - a^2*c*f)*g^3*h^2 + (a^2*c*d
- a^3*f)*g*h^4)*x)*sqrt(c*x^2 + a))/(a^2*c^3*g^7 + 3*a^3*c^2*g^5*h^2 + 3*a^4*c*g^3*h^4 + a^5*g*h^6 + (a*c^4*g
^6*h + 3*a^2*c^3*g^4*h^3 + 3*a^3*c^2*g^2*h^5 + a^4*c*h^7)*x^3 + (a*c^4*g^7 + 3*a^2*c^3*g^5*h^2 + 3*a^3*c^2*g^3
*h^4 + a^4*c*g*h^6)*x^2 + (a^2*c^3*g^6*h + 3*a^3*c^2*g^4*h^3 + 3*a^4*c*g^2*h^5 + a^5*h^7)*x), -((a^2*c*f*g^4 -
2*a^2*c*e*g^3*h + a^3*e*g*h^3 + (3*a^2*c*d - 2*a^3*f)*g^2*h^2 + (a*c^2*f*g^3*h - 2*a*c^2*e*g^2*h^2 + a^2*c*e*
h^4 + (3*a*c^2*d - 2*a^2*c*f)*g*h^3)*x^3 + (a*c^2*f*g^4 - 2*a*c^2*e*g^3*h + a^2*c*e*g*h^3 + (3*a*c^2*d - 2*a^2
*c*f)*g^2*h^2)*x^2 + (a^2*c*f*g^3*h - 2*a^2*c*e*g^2*h^2 + a^3*e*h^4 + (3*a^2*c*d - 2*a^3*f)*g*h^3)*x)*sqrt(-c*
g^2 - a*h^2)*arctan(sqrt(-c*g^2 - a*h^2)*(c*g*x - a*h)*sqrt(c*x^2 + a)/(a*c*g^2 + a^2*h^2 + (c^2*g^2 + a*c*h^2
)*x^2)) + (a*c^2*e*g^5 - a^2*c*e*g^3*h^2 - 2*a^3*e*g*h^4 + a^3*d*h^5 - (2*a*c^2*d - 3*a^2*c*f)*g^4*h - (a^2*c*
d - 3*a^3*f)*g^2*h^3 - (3*a*c^2*e*g^3*h^2 + 3*a^2*c*e*g*h^4 + (c^3*d - 2*a*c^2*f)*g^4*h - (a*c^2*d + a^2*c*f)*
g^2*h^3 - (2*a^2*c*d - a^3*f)*h^5)*x^2 - (a*c^2*e*g^4*h + 2*a^2*c*e*g^2*h^3 + a^3*e*h^5 + (c^3*d - a*c^2*f)*g^
5 + 2*(a*c^2*d - a^2*c*f)*g^3*h^2 + (a^2*c*d - a^3*f)*g*h^4)*x)*sqrt(c*x^2 + a))/(a^2*c^3*g^7 + 3*a^3*c^2*g^5*
h^2 + 3*a^4*c*g^3*h^4 + a^5*g*h^6 + (a*c^4*g^6*h + 3*a^2*c^3*g^4*h^3 + 3*a^3*c^2*g^2*h^5 + a^4*c*h^7)*x^3 + (a
*c^4*g^7 + 3*a^2*c^3*g^5*h^2 + 3*a^3*c^2*g^3*h^4 + a^4*c*g*h^6)*x^2 + (a^2*c^3*g^6*h + 3*a^3*c^2*g^4*h^3 + 3*a
^4*c*g^2*h^5 + a^5*h^7)*x)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (230) = 460\).

Time = 0.26 (sec) , antiderivative size = 1085, normalized size of antiderivative = 4.54 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\frac {3 \, c^{2} f g^{4} x}{\sqrt {c x^{2} + a} a c^{2} g^{4} h^{2} + 2 \, \sqrt {c x^{2} + a} a^{2} c g^{2} h^{4} + \sqrt {c x^{2} + a} a^{3} h^{6}} - \frac {3 \, c^{2} e g^{3} x}{\sqrt {c x^{2} + a} a c^{2} g^{4} h + 2 \, \sqrt {c x^{2} + a} a^{2} c g^{2} h^{3} + \sqrt {c x^{2} + a} a^{3} h^{5}} + \frac {3 \, c^{2} d g^{2} x}{\sqrt {c x^{2} + a} a c^{2} g^{4} + 2 \, \sqrt {c x^{2} + a} a^{2} c g^{2} h^{2} + \sqrt {c x^{2} + a} a^{3} h^{4}} + \frac {3 \, c f g^{3}}{\sqrt {c x^{2} + a} c^{2} g^{4} h + 2 \, \sqrt {c x^{2} + a} a c g^{2} h^{3} + \sqrt {c x^{2} + a} a^{2} h^{5}} - \frac {4 \, c f g^{2} x}{\sqrt {c x^{2} + a} a c g^{2} h^{2} + \sqrt {c x^{2} + a} a^{2} h^{4}} - \frac {3 \, c e g^{2}}{\sqrt {c x^{2} + a} c^{2} g^{4} + 2 \, \sqrt {c x^{2} + a} a c g^{2} h^{2} + \sqrt {c x^{2} + a} a^{2} h^{4}} + \frac {3 \, c e g x}{\sqrt {c x^{2} + a} a c g^{2} h + \sqrt {c x^{2} + a} a^{2} h^{3}} + \frac {3 \, c d g}{\frac {\sqrt {c x^{2} + a} c^{2} g^{4}}{h} + 2 \, \sqrt {c x^{2} + a} a c g^{2} h + \sqrt {c x^{2} + a} a^{2} h^{3}} - \frac {f g^{2}}{\sqrt {c x^{2} + a} c g^{2} h^{2} x + \sqrt {c x^{2} + a} a h^{4} x + \sqrt {c x^{2} + a} c g^{3} h + \sqrt {c x^{2} + a} a g h^{3}} - \frac {2 \, c d x}{\sqrt {c x^{2} + a} a c g^{2} + \sqrt {c x^{2} + a} a^{2} h^{2}} + \frac {e g}{\sqrt {c x^{2} + a} c g^{2} h x + \sqrt {c x^{2} + a} a h^{3} x + \sqrt {c x^{2} + a} c g^{3} + \sqrt {c x^{2} + a} a g h^{2}} - \frac {2 \, f g}{\sqrt {c x^{2} + a} c g^{2} h + \sqrt {c x^{2} + a} a h^{3}} - \frac {d}{\sqrt {c x^{2} + a} c g^{2} x + \sqrt {c x^{2} + a} a h^{2} x + \frac {\sqrt {c x^{2} + a} c g^{3}}{h} + \sqrt {c x^{2} + a} a g h} + \frac {e}{\sqrt {c x^{2} + a} c g^{2} + \sqrt {c x^{2} + a} a h^{2}} + \frac {f x}{\sqrt {c x^{2} + a} a h^{2}} + \frac {3 \, c f g^{3} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{5}} - \frac {3 \, c e g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{4}} + \frac {3 \, c d g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{3}} - \frac {2 \, f g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{3}} + \frac {e \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{2}} \]

[In]

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

[Out]

3*c^2*f*g^4*x/(sqrt(c*x^2 + a)*a*c^2*g^4*h^2 + 2*sqrt(c*x^2 + a)*a^2*c*g^2*h^4 + sqrt(c*x^2 + a)*a^3*h^6) - 3*
c^2*e*g^3*x/(sqrt(c*x^2 + a)*a*c^2*g^4*h + 2*sqrt(c*x^2 + a)*a^2*c*g^2*h^3 + sqrt(c*x^2 + a)*a^3*h^5) + 3*c^2*
d*g^2*x/(sqrt(c*x^2 + a)*a*c^2*g^4 + 2*sqrt(c*x^2 + a)*a^2*c*g^2*h^2 + sqrt(c*x^2 + a)*a^3*h^4) + 3*c*f*g^3/(s
qrt(c*x^2 + a)*c^2*g^4*h + 2*sqrt(c*x^2 + a)*a*c*g^2*h^3 + sqrt(c*x^2 + a)*a^2*h^5) - 4*c*f*g^2*x/(sqrt(c*x^2
+ a)*a*c*g^2*h^2 + sqrt(c*x^2 + a)*a^2*h^4) - 3*c*e*g^2/(sqrt(c*x^2 + a)*c^2*g^4 + 2*sqrt(c*x^2 + a)*a*c*g^2*h
^2 + sqrt(c*x^2 + a)*a^2*h^4) + 3*c*e*g*x/(sqrt(c*x^2 + a)*a*c*g^2*h + sqrt(c*x^2 + a)*a^2*h^3) + 3*c*d*g/(sqr
t(c*x^2 + a)*c^2*g^4/h + 2*sqrt(c*x^2 + a)*a*c*g^2*h + sqrt(c*x^2 + a)*a^2*h^3) - f*g^2/(sqrt(c*x^2 + a)*c*g^2
*h^2*x + sqrt(c*x^2 + a)*a*h^4*x + sqrt(c*x^2 + a)*c*g^3*h + sqrt(c*x^2 + a)*a*g*h^3) - 2*c*d*x/(sqrt(c*x^2 +
a)*a*c*g^2 + sqrt(c*x^2 + a)*a^2*h^2) + e*g/(sqrt(c*x^2 + a)*c*g^2*h*x + sqrt(c*x^2 + a)*a*h^3*x + sqrt(c*x^2
+ a)*c*g^3 + sqrt(c*x^2 + a)*a*g*h^2) - 2*f*g/(sqrt(c*x^2 + a)*c*g^2*h + sqrt(c*x^2 + a)*a*h^3) - d/(sqrt(c*x^
2 + a)*c*g^2*x + sqrt(c*x^2 + a)*a*h^2*x + sqrt(c*x^2 + a)*c*g^3/h + sqrt(c*x^2 + a)*a*g*h) + e/(sqrt(c*x^2 +
a)*c*g^2 + sqrt(c*x^2 + a)*a*h^2) + f*x/(sqrt(c*x^2 + a)*a*h^2) + 3*c*f*g^3*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x +
g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(5/2)*h^5) - 3*c*e*g^2*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x
+ g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(5/2)*h^4) + 3*c*d*g*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x +
g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(5/2)*h^3) - 2*f*g*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)
) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^3) + e*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h
/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^2)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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