\(\int \frac {d+e x+f x^2}{(g+h x)^3 (a+c x^2)^{3/2}} \, dx\) [114]

Optimal result

Integrand size = 29, antiderivative size = 374 \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\frac {a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}+\frac {h \left (2 a h^2 (2 f g-e h)-c g \left (3 f g^2-h (5 e g-7 d h)\right )\right ) \sqrt {a+c x^2}}{2 \left (c g^2+a h^2\right )^3 (g+h x)}-\frac {\left (2 a^2 f h^4-a c h^2 \left (11 f g^2-9 e g h+3 d h^2\right )+2 c^2 g^2 \left (f g^2-3 e g h+6 d h^2\right )\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{2 \left (c g^2+a h^2\right )^{7/2}} \]

[Out]

-1/2*(2*a^2*f*h^4-a*c*h^2*(3*d*h^2-9*e*g*h+11*f*g^2)+2*c^2*g^2*(6*d*h^2-3*e*g*h+f*g^2))*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)^(7/2)+(a*(a^2*f*h^3-c^2*g^2*(-3*d*h+e*g)-a*c*h*(3*f*g^2-h*(-
d*h+3*e*g)))+c*(c^2*d*g^3+a^2*h^2*(-e*h+3*f*g)-a*c*g*(f*g^2-3*h*(-d*h+e*g)))*x)/a/(a*h^2+c*g^2)^3/(c*x^2+a)^(1
/2)-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)^2+1/2*h*(2*a*h^2*(-e*h+2*f*g)-c*g*(3*f*g
^2-h*(-7*d*h+5*e*g)))*(c*x^2+a)^(1/2)/(a*h^2+c*g^2)^3/(h*x+g)

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1661, 1665, 821, 739, 212} \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \left (a+c x^2\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (2 a^2 f h^4-a c h^2 \left (3 d h^2-9 e g h+11 f g^2\right )+2 c^2 g^2 \left (6 d h^2-3 e g h+f g^2\right )\right )}{2 \left (a h^2+c g^2\right )^{7/2}}+\frac {a \left (a^2 f h^3-a c h \left (3 f g^2-h (3 e g-d h)\right )-c^2 g^2 (e g-3 d h)\right )+c x \left (a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )+c^2 d g^3\right )}{a \sqrt {a+c x^2} \left (a h^2+c g^2\right )^3}-\frac {h \sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{2 (g+h x)^2 \left (a h^2+c g^2\right )^2}-\frac {h \sqrt {a+c x^2} \left (-2 a h^2 (2 f g-e h)-c g h (5 e g-7 d h)+3 c f g^3\right )}{2 (g+h x) \left (a h^2+c g^2\right )^3} \]

[In]

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

[Out]

(a*(a^2*f*h^3 - c^2*g^2*(e*g - 3*d*h) - a*c*h*(3*f*g^2 - h*(3*e*g - d*h))) + c*(c^2*d*g^3 + a^2*h^2*(3*f*g - e
*h) - a*c*g*(f*g^2 - 3*h*(e*g - d*h)))*x)/(a*(c*g^2 + a*h^2)^3*Sqrt[a + c*x^2]) - (h*(f*g^2 - e*g*h + d*h^2)*S
qrt[a + c*x^2])/(2*(c*g^2 + a*h^2)^2*(g + h*x)^2) - (h*(3*c*f*g^3 - c*g*h*(5*e*g - 7*d*h) - 2*a*h^2*(2*f*g - e
*h))*Sqrt[a + c*x^2])/(2*(c*g^2 + a*h^2)^3*(g + h*x)) - ((2*a^2*f*h^4 - a*c*h^2*(11*f*g^2 - 9*e*g*h + 3*d*h^2)
+ 2*c^2*g^2*(f*g^2 - 3*e*g*h + 6*d*h^2))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(2*(c*
g^2 + a*h^2)^(7/2))

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]

Rule 1665

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

Rubi steps \begin{align*} \text {integral}& = \frac {a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {a c \left (a^2 d h^6+c^2 g^4 \left (f g^2-3 e g h+6 d h^2\right )-a c g^2 h^2 \left (3 f g^2-h (e g+3 d h)\right )\right )}{\left (c g^2+a h^2\right )^3}-\frac {a c h^2 \left (a^2 e h^4-c^2 g^3 (3 e g-8 d h)-2 a c g^2 h (4 f g-3 e h)\right ) x}{\left (c g^2+a h^2\right )^3}-\frac {a c h^3 \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right ) x^2}{\left (c g^2+a h^2\right )^3}}{(g+h x)^3 \sqrt {a+c x^2}} \, dx}{a c} \\ & = \frac {a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}+\frac {\int \frac {-\frac {2 a c \left (a^2 h^4 (f g-e h)-c^2 g^3 \left (f g^2-3 e g h+6 d h^2\right )+2 a c g h^2 \left (2 f g^2-h (e g+d h)\right )\right )}{\left (c g^2+a h^2\right )^2}+\frac {a c h \left (2 a^2 f h^4-c^2 \left (f g^4+g^2 h (e g-5 d h)\right )-a c h^2 \left (7 f g^2-h (7 e g-3 d h)\right )\right ) x}{\left (c g^2+a h^2\right )^2}}{(g+h x)^2 \sqrt {a+c x^2}} \, dx}{2 a c \left (c g^2+a h^2\right )} \\ & = \frac {a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac {h \left (3 c f g^3-c g h (5 e g-7 d h)-2 a h^2 (2 f g-e h)\right ) \sqrt {a+c x^2}}{2 \left (c g^2+a h^2\right )^3 (g+h x)}+\frac {\left (2 a^2 f h^4-a c h^2 \left (11 f g^2-9 e g h+3 d h^2\right )+2 c^2 g^2 \left (f g^2-3 e g h+6 d h^2\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{2 \left (c g^2+a h^2\right )^3} \\ & = \frac {a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac {h \left (3 c f g^3-c g h (5 e g-7 d h)-2 a h^2 (2 f g-e h)\right ) \sqrt {a+c x^2}}{2 \left (c g^2+a h^2\right )^3 (g+h x)}-\frac {\left (2 a^2 f h^4-a c h^2 \left (11 f g^2-9 e g h+3 d h^2\right )+2 c^2 g^2 \left (f g^2-3 e g h+6 d h^2\right )\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 )}{2 \left (c g^2+a h^2\right )^3} \\ & = \frac {a \left (a^2 f h^3-c^2 g^2 (e g-3 d h)-a c h \left (3 f g^2-h (3 e g-d h)\right )\right )+c \left (c^2 d g^3+a^2 h^2 (3 f g-e h)-a c g \left (f g^2-3 h (e g-d h)\right )\right ) x}{a \left (c g^2+a h^2\right )^3 \sqrt {a+c x^2}}-\frac {h \left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac {h \left (3 c f g^3-c g h (5 e g-7 d h)-2 a h^2 (2 f g-e h)\right ) \sqrt {a+c x^2}}{2 \left (c g^2+a h^2\right )^3 (g+h x)}-\frac {\left (2 a^2 f h^4-a c h^2 \left (11 f g^2-9 e g h+3 d h^2\right )+2 c^2 g^2 \left (f g^2-3 e g h+6 d h^2\right )\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{2 \left (c g^2+a h^2\right )^{7/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1537\) vs. \(2(374)=748\).

Time = 11.11 (sec) , antiderivative size = 1537, normalized size of antiderivative = 4.11 \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\frac {a^4 h^3 \left (f \left (5 g^2+8 g h x+2 h^2 x^2\right )-h (d h+e (g+2 h x))\right )-4 c^{7/2} g^2 x^2 \left (-\sqrt {c} x+\sqrt {a+c x^2}\right ) \left (d \left (-g^3+4 g^2 h x+18 g h^2 x^2+12 h^3 x^3\right )+g x \left (f g x (3 g+2 h x)-e \left (2 g^2+9 g h x+6 h^2 x^2\right )\right )\right )-a^3 \left (\sqrt {c} h^2 \sqrt {a+c x^2} \left (-3 d h^3 x+e h \left (4 g^2+5 g h x-2 h^2 x^2\right )+f \left (-10 g^3-5 g^2 h x+14 g h^2 x^2+6 h^3 x^3\right )\right )+c h \left (f \left (10 g^4+39 g^3 h x+26 g^2 h^2 x^2-20 g h^3 x^3-10 h^4 x^4\right )+h \left (d h \left (10 g^2+11 g h x+8 h^2 x^2\right )-e \left (12 g^3+27 g^2 h x+20 g h^2 x^2-2 h^3 x^3\right )\right )\right )\right )+a^2 \left (c^2 \left (d h \left (6 g^4+45 g^3 h x+14 g^2 h^2 x^2-29 g h^3 x^3-19 h^4 x^4\right )+e g \left (-2 g^4-31 g^3 h x+10 g^2 h^2 x^2+81 g h^3 x^3+57 h^4 x^4\right )+f x \left (13 g^5-28 g^4 h x-105 g^3 h^2 x^2-75 g^2 h^3 x^3+12 g h^4 x^4+8 h^5 x^5\right )\right )+c^{3/2} \sqrt {a+c x^2} \left (f \left (-5 g^5+20 g^4 h x+72 g^3 h^2 x^2+53 g^2 h^3 x^3-12 g h^4 x^4-8 h^5 x^5\right )+h \left (e g \left (11 g^3-14 g^2 h x-54 g h^2 x^2-39 h^3 x^3\right )+d h \left (-13 g^3+4 g^2 h x+20 g h^2 x^2+13 h^3 x^3\right )\right )\right )\right )+a \left (c^{5/2} \sqrt {a+c x^2} \left (d \left (2 g^5-14 g^4 h x-81 g^3 h^2 x^2-48 g^2 h^3 x^3+18 g h^4 x^4+12 h^5 x^5\right )+g x \left (f g x \left (-19 g^3+14 g^2 h x+66 g h^2 x^2+44 h^3 x^3\right )+e \left (6 g^4+49 g^3 h x+14 g^2 h^2 x^2-54 g h^3 x^3-36 h^4 x^4\right )\right )\right )-c^3 x \left (d \left (4 g^5-22 g^4 h x-117 g^3 h^2 x^2-72 g^2 h^3 x^3+18 g h^4 x^4+12 h^5 x^5\right )+g x \left (f g x \left (-25 g^3+10 g^2 h x+66 g h^2 x^2+44 h^3 x^3\right )+e \left (10 g^4+67 g^3 h x+26 g^2 h^2 x^2-54 g h^3 x^3-36 h^4 x^4\right )\right )\right )\right )}{2 \left (c g^2+a h^2\right )^3 (g+h x)^2 \left (a+c x^2\right ) \left (a \left (-3 \sqrt {c} x+\sqrt {a+c x^2}\right )+4 c x^2 \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )\right )}-\frac {15 a f h^2 \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}}+\frac {21 a e h^3 \arctan \left (\frac {-\sqrt {c} (g+h x)+h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{g \left (-c g^2-a h^2\right )^{5/2}}-\frac {2 f \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 )^{3/2}}+\frac {6 e h \arctan \left (\frac {-\sqrt {c} (g+h x)+h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{g \left (-c g^2-a h^2\right )^{3/2}}-\frac {12 d h^2 \arctan \left (\frac {-\sqrt {c} (g+h x)+h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{g^2 \left (-c g^2-a h^2\right )^{3/2}}-\frac {27 a d h^4 \arctan \left (\frac {-\sqrt {c} (g+h x)+h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{\sqrt {-c g^2-a h^2} \left (c g^3+a g h^2\right )^2}-\frac {15 a^2 h^4 \left (f g^2+h (-e g+d h)\right ) \arctan \left (\frac {-\sqrt {c} (g+h x)+h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{g^2 \left (-c g^2-a h^2\right )^{7/2}} \]

[In]

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

[Out]

(a^4*h^3*(f*(5*g^2 + 8*g*h*x + 2*h^2*x^2) - h*(d*h + e*(g + 2*h*x))) - 4*c^(7/2)*g^2*x^2*(-(Sqrt[c]*x) + Sqrt[
a + c*x^2])*(d*(-g^3 + 4*g^2*h*x + 18*g*h^2*x^2 + 12*h^3*x^3) + g*x*(f*g*x*(3*g + 2*h*x) - e*(2*g^2 + 9*g*h*x
+ 6*h^2*x^2))) - a^3*(Sqrt[c]*h^2*Sqrt[a + c*x^2]*(-3*d*h^3*x + e*h*(4*g^2 + 5*g*h*x - 2*h^2*x^2) + f*(-10*g^3
- 5*g^2*h*x + 14*g*h^2*x^2 + 6*h^3*x^3)) + c*h*(f*(10*g^4 + 39*g^3*h*x + 26*g^2*h^2*x^2 - 20*g*h^3*x^3 - 10*h
^4*x^4) + h*(d*h*(10*g^2 + 11*g*h*x + 8*h^2*x^2) - e*(12*g^3 + 27*g^2*h*x + 20*g*h^2*x^2 - 2*h^3*x^3)))) + a^2
*(c^2*(d*h*(6*g^4 + 45*g^3*h*x + 14*g^2*h^2*x^2 - 29*g*h^3*x^3 - 19*h^4*x^4) + e*g*(-2*g^4 - 31*g^3*h*x + 10*g
^2*h^2*x^2 + 81*g*h^3*x^3 + 57*h^4*x^4) + f*x*(13*g^5 - 28*g^4*h*x - 105*g^3*h^2*x^2 - 75*g^2*h^3*x^3 + 12*g*h
^4*x^4 + 8*h^5*x^5)) + c^(3/2)*Sqrt[a + c*x^2]*(f*(-5*g^5 + 20*g^4*h*x + 72*g^3*h^2*x^2 + 53*g^2*h^3*x^3 - 12*
g*h^4*x^4 - 8*h^5*x^5) + h*(e*g*(11*g^3 - 14*g^2*h*x - 54*g*h^2*x^2 - 39*h^3*x^3) + d*h*(-13*g^3 + 4*g^2*h*x +
20*g*h^2*x^2 + 13*h^3*x^3)))) + a*(c^(5/2)*Sqrt[a + c*x^2]*(d*(2*g^5 - 14*g^4*h*x - 81*g^3*h^2*x^2 - 48*g^2*h
^3*x^3 + 18*g*h^4*x^4 + 12*h^5*x^5) + g*x*(f*g*x*(-19*g^3 + 14*g^2*h*x + 66*g*h^2*x^2 + 44*h^3*x^3) + e*(6*g^4
+ 49*g^3*h*x + 14*g^2*h^2*x^2 - 54*g*h^3*x^3 - 36*h^4*x^4))) - c^3*x*(d*(4*g^5 - 22*g^4*h*x - 117*g^3*h^2*x^2
- 72*g^2*h^3*x^3 + 18*g*h^4*x^4 + 12*h^5*x^5) + g*x*(f*g*x*(-25*g^3 + 10*g^2*h*x + 66*g*h^2*x^2 + 44*h^3*x^3)
+ e*(10*g^4 + 67*g^3*h*x + 26*g^2*h^2*x^2 - 54*g*h^3*x^3 - 36*h^4*x^4)))))/(2*(c*g^2 + a*h^2)^3*(g + h*x)^2*(
a + c*x^2)*(a*(-3*Sqrt[c]*x + Sqrt[a + c*x^2]) + 4*c*x^2*(-(Sqrt[c]*x) + Sqrt[a + c*x^2]))) - (15*a*f*h^2*ArcT
an[(-(Sqrt[c]*(g + h*x)) + h*Sqrt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2]])/(-(c*g^2) - a*h^2)^(5/2) + (21*a*e*h^3*
ArcTan[(-(Sqrt[c]*(g + h*x)) + h*Sqrt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2]])/(g*(-(c*g^2) - a*h^2)^(5/2)) - (2*f
*ArcTan[(-(Sqrt[c]*(g + h*x)) + h*Sqrt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2]])/(-(c*g^2) - a*h^2)^(3/2) + (6*e*h*
ArcTan[(-(Sqrt[c]*(g + h*x)) + h*Sqrt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2]])/(g*(-(c*g^2) - a*h^2)^(3/2)) - (12*
d*h^2*ArcTan[(-(Sqrt[c]*(g + h*x)) + h*Sqrt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2]])/(g^2*(-(c*g^2) - a*h^2)^(3/2)
) - (27*a*d*h^4*ArcTan[(-(Sqrt[c]*(g + h*x)) + h*Sqrt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2]])/(Sqrt[-(c*g^2) - a*
h^2]*(c*g^3 + a*g*h^2)^2) - (15*a^2*h^4*(f*g^2 + h*(-(e*g) + d*h))*ArcTan[(-(Sqrt[c]*(g + h*x)) + h*Sqrt[a + c
*x^2])/Sqrt[-(c*g^2) - a*h^2]])/(g^2*(-(c*g^2) - a*h^2)^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1786\) vs. \(2(356)=712\).

Time = 0.66 (sec) , antiderivative size = 1787, normalized size of antiderivative = 4.78

method	result	size
default	\(\text {Expression too large to display}\)	\(1787\)

[In]

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

[Out]

f/h^3*(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)))+(e*h-2*f*g)/h^4*(-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))+(d*h^2-
e*g*h+f*g^2)/h^5*(-1/2/(a*h^2+c*g^2)*h^2/(x+1/h*g)^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)
+5/2*c*g*h/(a*h^2+c*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))-3/2*c/(a*h^2+c*g^2)*h^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))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1413 vs. \(2 (357) = 714\).

Time = 8.79 (sec) , antiderivative size = 2853, normalized size of antiderivative = 7.63 \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/4*((2*a^2*c^2*f*g^6 - 6*a^2*c^2*e*g^5*h + 9*a^3*c*e*g^3*h^3 + (12*a^2*c^2*d - 11*a^3*c*f)*g^4*h^2 - (3*a^3*
c*d - 2*a^4*f)*g^2*h^4 + (2*a*c^3*f*g^4*h^2 - 6*a*c^3*e*g^3*h^3 + 9*a^2*c^2*e*g*h^5 + (12*a*c^3*d - 11*a^2*c^2
*f)*g^2*h^4 - (3*a^2*c^2*d - 2*a^3*c*f)*h^6)*x^4 + 2*(2*a*c^3*f*g^5*h - 6*a*c^3*e*g^4*h^2 + 9*a^2*c^2*e*g^2*h^
4 + (12*a*c^3*d - 11*a^2*c^2*f)*g^3*h^3 - (3*a^2*c^2*d - 2*a^3*c*f)*g*h^5)*x^3 + (2*a*c^3*f*g^6 - 6*a*c^3*e*g^
5*h + 3*a^2*c^2*e*g^3*h^3 + 9*a^3*c*e*g*h^5 + 3*(4*a*c^3*d - 3*a^2*c^2*f)*g^4*h^2 + 9*(a^2*c^2*d - a^3*c*f)*g^
2*h^4 - (3*a^3*c*d - 2*a^4*f)*h^6)*x^2 + 2*(2*a^2*c^2*f*g^5*h - 6*a^2*c^2*e*g^4*h^2 + 9*a^3*c*e*g^2*h^4 + (12*
a^2*c^2*d - 11*a^3*c*f)*g^3*h^3 - (3*a^3*c*d - 2*a^4*f)*g*h^5)*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*(2*a*c^3*e*g^7 - 10*a^2*c^2*e*g^5*h^2 - 11*a^3*c*e*g^3*h^4 + a^4*e*g*h^6 + a^4*d*h^7 - 2*(3
*a*c^3*d - 5*a^2*c^2*f)*g^6*h + (4*a^2*c^2*d + 5*a^3*c*f)*g^4*h^3 + (11*a^3*c*d - 5*a^4*f)*g^2*h^5 - (11*a*c^3
*e*g^4*h^3 + 7*a^2*c^2*e*g^2*h^5 - 4*a^3*c*e*h^7 + (2*c^4*d - 5*a*c^3*f)*g^5*h^2 - (11*a*c^3*d - 5*a^2*c^2*f)*
g^3*h^4 - (13*a^2*c^2*d - 10*a^3*c*f)*g*h^6)*x^3 - (16*a*c^3*e*g^5*h^2 + 17*a^2*c^2*e*g^3*h^4 + a^3*c*e*g*h^6
+ 4*(c^4*d - 2*a*c^3*f)*g^6*h - (10*a*c^3*d - a^2*c^2*f)*g^4*h^3 - (17*a^2*c^2*d - 11*a^3*c*f)*g^2*h^5 - (3*a^
3*c*d - 2*a^4*f)*h^7)*x^2 - (2*a*c^3*e*g^6*h + 17*a^2*c^2*e*g^4*h^3 + 13*a^3*c*e*g^2*h^5 - 2*a^4*e*h^7 + 2*(c^
4*d - a*c^3*f)*g^7 + (8*a*c^3*d - 11*a^2*c^2*f)*g^5*h^2 - (5*a^2*c^2*d + a^3*c*f)*g^3*h^4 - (11*a^3*c*d - 8*a^
4*f)*g*h^6)*x)*sqrt(c*x^2 + a))/(a^2*c^4*g^10 + 4*a^3*c^3*g^8*h^2 + 6*a^4*c^2*g^6*h^4 + 4*a^5*c*g^4*h^6 + a^6*
g^2*h^8 + (a*c^5*g^8*h^2 + 4*a^2*c^4*g^6*h^4 + 6*a^3*c^3*g^4*h^6 + 4*a^4*c^2*g^2*h^8 + a^5*c*h^10)*x^4 + 2*(a*
c^5*g^9*h + 4*a^2*c^4*g^7*h^3 + 6*a^3*c^3*g^5*h^5 + 4*a^4*c^2*g^3*h^7 + a^5*c*g*h^9)*x^3 + (a*c^5*g^10 + 5*a^2
*c^4*g^8*h^2 + 10*a^3*c^3*g^6*h^4 + 10*a^4*c^2*g^4*h^6 + 5*a^5*c*g^2*h^8 + a^6*h^10)*x^2 + 2*(a^2*c^4*g^9*h +
4*a^3*c^3*g^7*h^3 + 6*a^4*c^2*g^5*h^5 + 4*a^5*c*g^3*h^7 + a^6*g*h^9)*x), -1/2*((2*a^2*c^2*f*g^6 - 6*a^2*c^2*e*
g^5*h + 9*a^3*c*e*g^3*h^3 + (12*a^2*c^2*d - 11*a^3*c*f)*g^4*h^2 - (3*a^3*c*d - 2*a^4*f)*g^2*h^4 + (2*a*c^3*f*g
^4*h^2 - 6*a*c^3*e*g^3*h^3 + 9*a^2*c^2*e*g*h^5 + (12*a*c^3*d - 11*a^2*c^2*f)*g^2*h^4 - (3*a^2*c^2*d - 2*a^3*c*
f)*h^6)*x^4 + 2*(2*a*c^3*f*g^5*h - 6*a*c^3*e*g^4*h^2 + 9*a^2*c^2*e*g^2*h^4 + (12*a*c^3*d - 11*a^2*c^2*f)*g^3*h
^3 - (3*a^2*c^2*d - 2*a^3*c*f)*g*h^5)*x^3 + (2*a*c^3*f*g^6 - 6*a*c^3*e*g^5*h + 3*a^2*c^2*e*g^3*h^3 + 9*a^3*c*e
*g*h^5 + 3*(4*a*c^3*d - 3*a^2*c^2*f)*g^4*h^2 + 9*(a^2*c^2*d - a^3*c*f)*g^2*h^4 - (3*a^3*c*d - 2*a^4*f)*h^6)*x^
2 + 2*(2*a^2*c^2*f*g^5*h - 6*a^2*c^2*e*g^4*h^2 + 9*a^3*c*e*g^2*h^4 + (12*a^2*c^2*d - 11*a^3*c*f)*g^3*h^3 - (3*
a^3*c*d - 2*a^4*f)*g*h^5)*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)) + (2*a*c^3*e*g^7 - 10*a^2*c^2*e*g^5*h^2 - 11*a^3*c*e*g^3*h^4 + a^
4*e*g*h^6 + a^4*d*h^7 - 2*(3*a*c^3*d - 5*a^2*c^2*f)*g^6*h + (4*a^2*c^2*d + 5*a^3*c*f)*g^4*h^3 + (11*a^3*c*d -
5*a^4*f)*g^2*h^5 - (11*a*c^3*e*g^4*h^3 + 7*a^2*c^2*e*g^2*h^5 - 4*a^3*c*e*h^7 + (2*c^4*d - 5*a*c^3*f)*g^5*h^2 -
(11*a*c^3*d - 5*a^2*c^2*f)*g^3*h^4 - (13*a^2*c^2*d - 10*a^3*c*f)*g*h^6)*x^3 - (16*a*c^3*e*g^5*h^2 + 17*a^2*c^
2*e*g^3*h^4 + a^3*c*e*g*h^6 + 4*(c^4*d - 2*a*c^3*f)*g^6*h - (10*a*c^3*d - a^2*c^2*f)*g^4*h^3 - (17*a^2*c^2*d -
11*a^3*c*f)*g^2*h^5 - (3*a^3*c*d - 2*a^4*f)*h^7)*x^2 - (2*a*c^3*e*g^6*h + 17*a^2*c^2*e*g^4*h^3 + 13*a^3*c*e*g
^2*h^5 - 2*a^4*e*h^7 + 2*(c^4*d - a*c^3*f)*g^7 + (8*a*c^3*d - 11*a^2*c^2*f)*g^5*h^2 - (5*a^2*c^2*d + a^3*c*f)*
g^3*h^4 - (11*a^3*c*d - 8*a^4*f)*g*h^6)*x)*sqrt(c*x^2 + a))/(a^2*c^4*g^10 + 4*a^3*c^3*g^8*h^2 + 6*a^4*c^2*g^6*
h^4 + 4*a^5*c*g^4*h^6 + a^6*g^2*h^8 + (a*c^5*g^8*h^2 + 4*a^2*c^4*g^6*h^4 + 6*a^3*c^3*g^4*h^6 + 4*a^4*c^2*g^2*h
^8 + a^5*c*h^10)*x^4 + 2*(a*c^5*g^9*h + 4*a^2*c^4*g^7*h^3 + 6*a^3*c^3*g^5*h^5 + 4*a^4*c^2*g^3*h^7 + a^5*c*g*h^
9)*x^3 + (a*c^5*g^10 + 5*a^2*c^4*g^8*h^2 + 10*a^3*c^3*g^6*h^4 + 10*a^4*c^2*g^4*h^6 + 5*a^5*c*g^2*h^8 + a^6*h^1
0)*x^2 + 2*(a^2*c^4*g^9*h + 4*a^3*c^3*g^7*h^3 + 6*a^4*c^2*g^5*h^5 + 4*a^5*c*g^3*h^7 + a^6*g*h^9)*x)]

Sympy [F(-1)]

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

[In]

integrate((f*x**2+e*x+d)/(h*x+g)**3/(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. 2254 vs. \(2 (357) = 714\).

Time = 0.31 (sec) , antiderivative size = 2254, normalized size of antiderivative = 6.03 \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

15/2*c^3*f*g^5*x/(sqrt(c*x^2 + a)*a*c^3*g^6*h^2 + 3*sqrt(c*x^2 + a)*a^2*c^2*g^4*h^4 + 3*sqrt(c*x^2 + a)*a^3*c*
g^2*h^6 + sqrt(c*x^2 + a)*a^4*h^8) - 15/2*c^3*e*g^4*x/(sqrt(c*x^2 + a)*a*c^3*g^6*h + 3*sqrt(c*x^2 + a)*a^2*c^2
*g^4*h^3 + 3*sqrt(c*x^2 + a)*a^3*c*g^2*h^5 + sqrt(c*x^2 + a)*a^4*h^7) + 15/2*c^3*d*g^3*x/(sqrt(c*x^2 + a)*a*c^
3*g^6 + 3*sqrt(c*x^2 + a)*a^2*c^2*g^4*h^2 + 3*sqrt(c*x^2 + a)*a^3*c*g^2*h^4 + sqrt(c*x^2 + a)*a^4*h^6) + 15/2*
c^2*f*g^4/(sqrt(c*x^2 + a)*c^3*g^6*h + 3*sqrt(c*x^2 + a)*a*c^2*g^4*h^3 + 3*sqrt(c*x^2 + a)*a^2*c*g^2*h^5 + sqr
t(c*x^2 + a)*a^3*h^7) - 25/2*c^2*f*g^3*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 + sq
rt(c*x^2 + a)*a^3*h^6) - 15/2*c^2*e*g^3/(sqrt(c*x^2 + a)*c^3*g^6 + 3*sqrt(c*x^2 + a)*a*c^2*g^4*h^2 + 3*sqrt(c*
x^2 + a)*a^2*c*g^2*h^4 + sqrt(c*x^2 + a)*a^3*h^6) + 19/2*c^2*e*g^2*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) + 15/2*c^2*d*g^2/(sqrt(c*x^2 + a)*c^3*g^6/h + 3*sqrt(c*x^2 +
a)*a*c^2*g^4*h + 3*sqrt(c*x^2 + a)*a^2*c*g^2*h^3 + sqrt(c*x^2 + a)*a^3*h^5) - 5/2*c*f*g^3/(sqrt(c*x^2 + a)*c^2
*g^4*h^2*x + 2*sqrt(c*x^2 + a)*a*c*g^2*h^4*x + sqrt(c*x^2 + a)*a^2*h^6*x + sqrt(c*x^2 + a)*c^2*g^5*h + 2*sqrt(
c*x^2 + a)*a*c*g^3*h^3 + sqrt(c*x^2 + a)*a^2*g*h^5) - 13/2*c^2*d*g*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) + 5/2*c*e*g^2/(sqrt(c*x^2 + a)*c^2*g^4*h*x + 2*sqrt(c*x^2 + a)*
a*c*g^2*h^3*x + sqrt(c*x^2 + a)*a^2*h^5*x + sqrt(c*x^2 + a)*c^2*g^5 + 2*sqrt(c*x^2 + a)*a*c*g^3*h^2 + sqrt(c*x
^2 + a)*a^2*g*h^4) - 15/2*c*f*g^2/(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) + 5*c*f*g*x/(sqrt(c*x^2 + a)*a*c*g^2*h^2 + sqrt(c*x^2 + a)*a^2*h^4) - 5/2*c*d*g/(sqrt(c*x^2 + a)*c^2
*g^4*x + 2*sqrt(c*x^2 + a)*a*c*g^2*h^2*x + sqrt(c*x^2 + a)*a^2*h^4*x + sqrt(c*x^2 + a)*c^2*g^5/h + 2*sqrt(c*x^
2 + a)*a*c*g^3*h + sqrt(c*x^2 + a)*a^2*g*h^3) + 9/2*c*e*g/(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) - 1/2*f*g^2/(sqrt(c*x^2 + a)*c*g^2*h^3*x^2 + sqrt(c*x^2 + a)*a*h^5*x^2 + 2*sqr
t(c*x^2 + a)*c*g^3*h^2*x + 2*sqrt(c*x^2 + a)*a*g*h^4*x + sqrt(c*x^2 + a)*c*g^4*h + sqrt(c*x^2 + a)*a*g^2*h^3)
- 2*c*e*x/(sqrt(c*x^2 + a)*a*c*g^2*h + sqrt(c*x^2 + a)*a^2*h^3) - 3/2*c*d/(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) + 1/2*e*g/(sqrt(c*x^2 + a)*c*g^2*h^2*x^2 + sqrt(c*x^2 + a)*a*h
^4*x^2 + 2*sqrt(c*x^2 + a)*c*g^3*h*x + 2*sqrt(c*x^2 + a)*a*g*h^3*x + sqrt(c*x^2 + a)*c*g^4 + sqrt(c*x^2 + a)*a
*g^2*h^2) + 2*f*g/(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) - 1/2*d/(sqrt(c*x^2 + a)*c*g^2*h*x^2 + sqrt(c*x^2 + a)*a*h^3*x^2 + 2*sqrt(c*x^2 + a)*c*g^3*x +
2*sqrt(c*x^2 + a)*a*g*h^2*x + sqrt(c*x^2 + a)*c*g^4/h + sqrt(c*x^2 + a)*a*g^2*h) - e/(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) + f/(sqrt(c*x^2 + a)*c*g^2*h +
sqrt(c*x^2 + a)*a*h^3) + 15/2*c^2*f*g^4*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)^(7/2)*h^7) - 15/2*c^2*e*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)^(7/2)*h^6) + 15/2*c^2*d*g^2*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*a
bs(h*x + g)))/((a + c*g^2/h^2)^(7/2)*h^5) - 15/2*c*f*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^5) + 9/2*c*e*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^4) - 3/2*c*d*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) + f*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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1421 vs. \(2 (357) = 714\).

Time = 0.31 (sec) , antiderivative size = 1421, normalized size of antiderivative = 3.80 \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((c^6*d*g^9 - a*c^5*f*g^9 + 3*a*c^5*e*g^8*h + 8*a^2*c^4*e*g^6*h^3 - 6*a^2*c^4*d*g^5*h^4 + 6*a^3*c^3*f*g^5*h^4
+ 6*a^3*c^3*e*g^4*h^5 - 8*a^3*c^3*d*g^3*h^6 + 8*a^4*c^2*f*g^3*h^6 - 3*a^4*c^2*d*g*h^8 + 3*a^5*c*f*g*h^8 - a^5*
c*e*h^9)*x/(a*c^6*g^12 + 6*a^2*c^5*g^10*h^2 + 15*a^3*c^4*g^8*h^4 + 20*a^4*c^3*g^6*h^6 + 15*a^5*c^2*g^4*h^8 + 6
*a^6*c*g^2*h^10 + a^7*h^12) - (a*c^5*e*g^9 - 3*a*c^5*d*g^8*h + 3*a^2*c^4*f*g^8*h - 8*a^2*c^4*d*g^6*h^3 + 8*a^3
*c^3*f*g^6*h^3 - 6*a^3*c^3*e*g^5*h^4 - 6*a^3*c^3*d*g^4*h^5 + 6*a^4*c^2*f*g^4*h^5 - 8*a^4*c^2*e*g^3*h^6 - 3*a^5
*c*e*g*h^8 + a^5*c*d*h^9 - a^6*f*h^9)/(a*c^6*g^12 + 6*a^2*c^5*g^10*h^2 + 15*a^3*c^4*g^8*h^4 + 20*a^4*c^3*g^6*h
^6 + 15*a^5*c^2*g^4*h^8 + 6*a^6*c*g^2*h^10 + a^7*h^12))/sqrt(c*x^2 + a) - (2*c^2*f*g^4 - 6*c^2*e*g^3*h + 12*c^
2*d*g^2*h^2 - 11*a*c*f*g^2*h^2 + 9*a*c*e*g*h^3 - 3*a*c*d*h^4 + 2*a^2*f*h^4)*arctan(((sqrt(c)*x - sqrt(c*x^2 +
a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c^3*g^6 + 3*a*c^2*g^4*h^2 + 3*a^2*c*g^2*h^4 + a^3*h^6)*sqrt(-c*g^2
- a*h^2)) - (2*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*f*g^4*h - 4*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*e*g^3*h^2 +
6*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*d*g^2*h^3 - 5*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*f*g^2*h^3 + 3*(sqrt(c
)*x - sqrt(c*x^2 + a))^3*a*c*e*g*h^4 - (sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*d*h^5 + 6*(sqrt(c)*x - sqrt(c*x^2 +
a))^2*c^(5/2)*f*g^5 - 10*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*e*g^4*h + 14*(sqrt(c)*x - sqrt(c*x^2 + a))^2
*c^(5/2)*d*g^3*h^2 - 11*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*f*g^3*h^2 + 9*(sqrt(c)*x - sqrt(c*x^2 + a))^
2*a*c^(3/2)*e*g^2*h^3 - 7*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*d*g*h^4 + 4*(sqrt(c)*x - sqrt(c*x^2 + a))^
2*a^2*sqrt(c)*f*g*h^4 - 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*sqrt(c)*e*h^5 - 10*(sqrt(c)*x - sqrt(c*x^2 + a))
*a*c^2*f*g^4*h + 16*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c^2*e*g^3*h^2 - 22*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c^2*d*g
^2*h^3 + 11*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*f*g^2*h^3 - 5*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*e*g*h^4 - (s
qrt(c)*x - sqrt(c*x^2 + a))*a^2*c*d*h^5 + 3*a^2*c^(3/2)*f*g^3*h^2 - 5*a^2*c^(3/2)*e*g^2*h^3 + 7*a^2*c^(3/2)*d*
g*h^4 - 4*a^3*sqrt(c)*f*g*h^4 + 2*a^3*sqrt(c)*e*h^5)/((c^3*g^6 + 3*a*c^2*g^4*h^2 + 3*a^2*c*g^2*h^4 + a^3*h^6)*
((sqrt(c)*x - sqrt(c*x^2 + a))^2*h + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*g - a*h)^2)

Mupad [F(-1)]

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

[In]

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

[Out]

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