\(\int \frac {d+e x+f x^2}{(g+h x)^3 \sqrt {a+c x^2}} \, dx\) [107]

Optimal result

Integrand size = 29, antiderivative size = 225 \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \sqrt {a+c x^2}} \, dx=-\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac {\left (2 a h^2 (2 f g-e h)+c g \left (f g^2+h (e g-3 d h)\right )\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right )^2 (g+h x)}-\frac {\left (2 c^2 d g^2+2 a^2 f h^2-a c \left (f g^2-h (3 e g-d h)\right )\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{2 \left (c g^2+a h^2\right )^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 224, 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 = {1665, 821, 739, 212} \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \sqrt {a+c x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (2 a^2 f h^2-a c \left (f g^2-h (3 e g-d h)\right )+2 c^2 d g^2\right )}{2 \left (a h^2+c g^2\right )^{5/2}}-\frac {\sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{2 h (g+h x)^2 \left (a h^2+c g^2\right )}+\frac {\sqrt {a+c x^2} \left (2 a h^2 (2 f g-e h)+c g h (e g-3 d h)+c f g^3\right )}{2 h (g+h x) \left (a h^2+c g^2\right )^2} \]

[In]

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

[Out]

-1/2*((f*g^2 - e*g*h + d*h^2)*Sqrt[a + c*x^2])/(h*(c*g^2 + a*h^2)*(g + h*x)^2) + ((c*f*g^3 + c*g*h*(e*g - 3*d*
h) + 2*a*h^2*(2*f*g - e*h))*Sqrt[a + c*x^2])/(2*h*(c*g^2 + a*h^2)^2*(g + h*x)) - ((2*c^2*d*g^2 + 2*a^2*f*h^2 -
a*c*(f*g^2 - h*(3*e*g - d*h)))*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)^(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 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 {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\int \frac {-2 (c d g-a f g+a e h)-\left (2 a f h+c \left (e g+\frac {f g^2}{h}-d h\right )\right ) x}{(g+h x)^2 \sqrt {a+c x^2}} \, dx}{2 \left (c g^2+a h^2\right )} \\ & = -\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac {\left (c f g^3+c g h (e g-3 d h)+2 a h^2 (2 f g-e h)\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (2 c^2 d g^2+2 a^2 f h^2-a c \left (f g^2-h (3 e g-d h)\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{2 \left (c g^2+a h^2\right )^2} \\ & = -\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac {\left (c f g^3+c g h (e g-3 d h)+2 a h^2 (2 f g-e h)\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right )^2 (g+h x)}-\frac {\left (2 c^2 d g^2+2 a^2 f h^2-a c \left (f g^2-h (3 e g-d h)\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 )^2} \\ & = -\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac {\left (c f g^3+c g h (e g-3 d h)+2 a h^2 (2 f g-e h)\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right )^2 (g+h x)}-\frac {\left (2 c^2 d g^2+2 a^2 f h^2-a c \left (f g^2-h (3 e g-d h)\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 )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.90 \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \sqrt {a+c x^2}} \, dx=\frac {\sqrt {a+c x^2} \left (c g \left (f g^2 x+e g (2 g+h x)-d h (4 g+3 h x)\right )-a h (-f g (3 g+4 h x)+h (d h+e (g+2 h x)))\right )}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac {\left (2 c^2 d g^2+2 a^2 f h^2-a c \left (f g^2+h (-3 e g+d h)\right )\right ) \arctan \left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{\left (-c g^2-a h^2\right )^{5/2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(809\) vs. \(2(209)=418\).

Time = 0.64 (sec) , antiderivative size = 810, normalized size of antiderivative = 3.60

method	result	size
default	\(-\frac {f \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 )}{h^{3} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {\left (e h -2 f g \right ) \left (-\frac {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}}}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )}-\frac {c g h \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^{4}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (-\frac {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}}}}{2 \left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )^{2}}+\frac {3 c g h \left (-\frac {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}}}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )}-\frac {c g h \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 )}{2 \left (a \,h^{2}+c \,g^{2}\right )}+\frac {c \,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 )}{2 \left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{5}}\)	\(810\)

[In]

int((f*x^2+e*x+d)/(h*x+g)^3/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-f/h^3/((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)-c*g*h/(a*h^2+c*g^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)))+(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)+3/2*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)-c*g*h/(a*h^2+c*g^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)))+1/2*c/(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. 531 vs. \(2 (210) = 420\).

Time = 4.71 (sec) , antiderivative size = 1088, normalized size of antiderivative = 4.84 \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \sqrt {a+c x^2}} \, dx=\left [\frac {{\left (3 \, a c e g^{3} h + {\left (2 \, c^{2} d - a c f\right )} g^{4} - {\left (a c d - 2 \, a^{2} f\right )} g^{2} h^{2} + {\left (3 \, a c e g h^{3} + {\left (2 \, c^{2} d - a c f\right )} g^{2} h^{2} - {\left (a c d - 2 \, a^{2} f\right )} h^{4}\right )} x^{2} + 2 \, {\left (3 \, a c e g^{2} h^{2} + {\left (2 \, c^{2} d - a c f\right )} g^{3} h - {\left (a c d - 2 \, a^{2} f\right )} g h^{3}\right )} x\right )} \sqrt {c g^{2} + a h^{2}} \log \left (\frac {2 \, a c g h x - a c g^{2} - 2 \, a^{2} h^{2} - {\left (2 \, c^{2} g^{2} + a c h^{2}\right )} x^{2} - 2 \, \sqrt {c g^{2} + a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{h^{2} x^{2} + 2 \, g h x + g^{2}}\right ) + 2 \, {\left (2 \, c^{2} e g^{5} + a c e g^{3} h^{2} - a^{2} e g h^{4} - a^{2} d h^{5} - {\left (4 \, c^{2} d - 3 \, a c f\right )} g^{4} h - {\left (5 \, a c d - 3 \, a^{2} f\right )} g^{2} h^{3} + {\left (c^{2} f g^{5} + c^{2} e g^{4} h - a c e g^{2} h^{3} - 2 \, a^{2} e h^{5} - {\left (3 \, c^{2} d - 5 \, a c f\right )} g^{3} h^{2} - {\left (3 \, a c d - 4 \, a^{2} f\right )} g h^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{3} g^{8} + 3 \, a c^{2} g^{6} h^{2} + 3 \, a^{2} c g^{4} h^{4} + a^{3} g^{2} h^{6} + {\left (c^{3} g^{6} h^{2} + 3 \, a c^{2} g^{4} h^{4} + 3 \, a^{2} c g^{2} h^{6} + a^{3} h^{8}\right )} x^{2} + 2 \, {\left (c^{3} g^{7} h + 3 \, a c^{2} g^{5} h^{3} + 3 \, a^{2} c g^{3} h^{5} + a^{3} g h^{7}\right )} x\right )}}, -\frac {{\left (3 \, a c e g^{3} h + {\left (2 \, c^{2} d - a c f\right )} g^{4} - {\left (a c d - 2 \, a^{2} f\right )} g^{2} h^{2} + {\left (3 \, a c e g h^{3} + {\left (2 \, c^{2} d - a c f\right )} g^{2} h^{2} - {\left (a c d - 2 \, a^{2} f\right )} h^{4}\right )} x^{2} + 2 \, {\left (3 \, a c e g^{2} h^{2} + {\left (2 \, c^{2} d - a c f\right )} g^{3} h - {\left (a c d - 2 \, a^{2} f\right )} g h^{3}\right )} x\right )} \sqrt {-c g^{2} - a h^{2}} \arctan \left (\frac {\sqrt {-c g^{2} - a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{a c g^{2} + a^{2} h^{2} + {\left (c^{2} g^{2} + a c h^{2}\right )} x^{2}}\right ) - {\left (2 \, c^{2} e g^{5} + a c e g^{3} h^{2} - a^{2} e g h^{4} - a^{2} d h^{5} - {\left (4 \, c^{2} d - 3 \, a c f\right )} g^{4} h - {\left (5 \, a c d - 3 \, a^{2} f\right )} g^{2} h^{3} + {\left (c^{2} f g^{5} + c^{2} e g^{4} h - a c e g^{2} h^{3} - 2 \, a^{2} e h^{5} - {\left (3 \, c^{2} d - 5 \, a c f\right )} g^{3} h^{2} - {\left (3 \, a c d - 4 \, a^{2} f\right )} g h^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} g^{8} + 3 \, a c^{2} g^{6} h^{2} + 3 \, a^{2} c g^{4} h^{4} + a^{3} g^{2} h^{6} + {\left (c^{3} g^{6} h^{2} + 3 \, a c^{2} g^{4} h^{4} + 3 \, a^{2} c g^{2} h^{6} + a^{3} h^{8}\right )} x^{2} + 2 \, {\left (c^{3} g^{7} h + 3 \, a c^{2} g^{5} h^{3} + 3 \, a^{2} c g^{3} h^{5} + a^{3} g h^{7}\right )} x\right )}}\right ] \]

[In]

integrate((f*x^2+e*x+d)/(h*x+g)^3/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[1/4*((3*a*c*e*g^3*h + (2*c^2*d - a*c*f)*g^4 - (a*c*d - 2*a^2*f)*g^2*h^2 + (3*a*c*e*g*h^3 + (2*c^2*d - a*c*f)*
g^2*h^2 - (a*c*d - 2*a^2*f)*h^4)*x^2 + 2*(3*a*c*e*g^2*h^2 + (2*c^2*d - a*c*f)*g^3*h - (a*c*d - 2*a^2*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*(2*c^2*e*g^5 + a*c*e*g^3*h^2 - a^2*e*g*h^4
- a^2*d*h^5 - (4*c^2*d - 3*a*c*f)*g^4*h - (5*a*c*d - 3*a^2*f)*g^2*h^3 + (c^2*f*g^5 + c^2*e*g^4*h - a*c*e*g^2*h
^3 - 2*a^2*e*h^5 - (3*c^2*d - 5*a*c*f)*g^3*h^2 - (3*a*c*d - 4*a^2*f)*g*h^4)*x)*sqrt(c*x^2 + a))/(c^3*g^8 + 3*a
*c^2*g^6*h^2 + 3*a^2*c*g^4*h^4 + a^3*g^2*h^6 + (c^3*g^6*h^2 + 3*a*c^2*g^4*h^4 + 3*a^2*c*g^2*h^6 + a^3*h^8)*x^2
+ 2*(c^3*g^7*h + 3*a*c^2*g^5*h^3 + 3*a^2*c*g^3*h^5 + a^3*g*h^7)*x), -1/2*((3*a*c*e*g^3*h + (2*c^2*d - a*c*f)*
g^4 - (a*c*d - 2*a^2*f)*g^2*h^2 + (3*a*c*e*g*h^3 + (2*c^2*d - a*c*f)*g^2*h^2 - (a*c*d - 2*a^2*f)*h^4)*x^2 + 2*
(3*a*c*e*g^2*h^2 + (2*c^2*d - a*c*f)*g^3*h - (a*c*d - 2*a^2*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)) - (2*c^2*e*g^5 + a*c*
e*g^3*h^2 - a^2*e*g*h^4 - a^2*d*h^5 - (4*c^2*d - 3*a*c*f)*g^4*h - (5*a*c*d - 3*a^2*f)*g^2*h^3 + (c^2*f*g^5 + c
^2*e*g^4*h - a*c*e*g^2*h^3 - 2*a^2*e*h^5 - (3*c^2*d - 5*a*c*f)*g^3*h^2 - (3*a*c*d - 4*a^2*f)*g*h^4)*x)*sqrt(c*
x^2 + a))/(c^3*g^8 + 3*a*c^2*g^6*h^2 + 3*a^2*c*g^4*h^4 + a^3*g^2*h^6 + (c^3*g^6*h^2 + 3*a*c^2*g^4*h^4 + 3*a^2*
c*g^2*h^6 + a^3*h^8)*x^2 + 2*(c^3*g^7*h + 3*a*c^2*g^5*h^3 + 3*a^2*c*g^3*h^5 + a^3*g*h^7)*x)]

Sympy [F]

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

[In]

integrate((f*x**2+e*x+d)/(h*x+g)**3/(c*x**2+a)**(1/2),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 896 vs. \(2 (210) = 420\).

Time = 0.24 (sec) , antiderivative size = 896, normalized size of antiderivative = 3.98 \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \sqrt {a+c x^2}} \, dx=-\frac {3 \, \sqrt {c x^{2} + a} c f g^{3}}{2 \, {\left (c^{2} g^{4} h^{2} x + 2 \, a c g^{2} h^{4} x + a^{2} h^{6} x + c^{2} g^{5} h + 2 \, a c g^{3} h^{3} + a^{2} g h^{5}\right )}} + \frac {3 \, \sqrt {c x^{2} + a} c e g^{2}}{2 \, {\left (c^{2} g^{4} h x + 2 \, a c g^{2} h^{3} x + a^{2} h^{5} x + c^{2} g^{5} + 2 \, a c g^{3} h^{2} + a^{2} g h^{4}\right )}} - \frac {3 \, \sqrt {c x^{2} + a} c d g}{2 \, {\left (c^{2} g^{4} x + 2 \, a c g^{2} h^{2} x + a^{2} h^{4} x + \frac {c^{2} g^{5}}{h} + 2 \, a c g^{3} h + a^{2} g h^{3}\right )}} - \frac {\sqrt {c x^{2} + a} f g^{2}}{2 \, {\left (c g^{2} h^{3} x^{2} + a h^{5} x^{2} + 2 \, c g^{3} h^{2} x + 2 \, a g h^{4} x + c g^{4} h + a g^{2} h^{3}\right )}} + \frac {\sqrt {c x^{2} + a} e g}{2 \, {\left (c g^{2} h^{2} x^{2} + a h^{4} x^{2} + 2 \, c g^{3} h x + 2 \, a g h^{3} x + c g^{4} + a g^{2} h^{2}\right )}} + \frac {2 \, \sqrt {c x^{2} + a} f g}{c g^{2} h^{2} x + a h^{4} x + c g^{3} h + a g h^{3}} - \frac {\sqrt {c x^{2} + a} d}{2 \, {\left (c g^{2} h x^{2} + a h^{3} x^{2} + 2 \, c g^{3} x + 2 \, a g h^{2} x + \frac {c g^{4}}{h} + a g^{2} h\right )}} - \frac {\sqrt {c x^{2} + a} e}{c g^{2} h x + a h^{3} x + c g^{3} + a g h^{2}} + \frac {3 \, c^{2} f g^{4} \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 )}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{7}} - \frac {3 \, c^{2} e 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 )}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{6}} + \frac {3 \, c^{2} d 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 )}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{5}} - \frac {5 \, c f 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 )}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{5}} + \frac {3 \, c e 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 )}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{4}} - \frac {c d \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 )}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{3}} + \frac {f \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 )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{3}} \]

[In]

integrate((f*x^2+e*x+d)/(h*x+g)^3/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

-3/2*sqrt(c*x^2 + a)*c*f*g^3/(c^2*g^4*h^2*x + 2*a*c*g^2*h^4*x + a^2*h^6*x + c^2*g^5*h + 2*a*c*g^3*h^3 + a^2*g*
h^5) + 3/2*sqrt(c*x^2 + a)*c*e*g^2/(c^2*g^4*h*x + 2*a*c*g^2*h^3*x + a^2*h^5*x + c^2*g^5 + 2*a*c*g^3*h^2 + a^2*
g*h^4) - 3/2*sqrt(c*x^2 + a)*c*d*g/(c^2*g^4*x + 2*a*c*g^2*h^2*x + a^2*h^4*x + c^2*g^5/h + 2*a*c*g^3*h + a^2*g*
h^3) - 1/2*sqrt(c*x^2 + a)*f*g^2/(c*g^2*h^3*x^2 + a*h^5*x^2 + 2*c*g^3*h^2*x + 2*a*g*h^4*x + c*g^4*h + a*g^2*h^
3) + 1/2*sqrt(c*x^2 + a)*e*g/(c*g^2*h^2*x^2 + a*h^4*x^2 + 2*c*g^3*h*x + 2*a*g*h^3*x + c*g^4 + a*g^2*h^2) + 2*s
qrt(c*x^2 + a)*f*g/(c*g^2*h^2*x + a*h^4*x + c*g^3*h + a*g*h^3) - 1/2*sqrt(c*x^2 + a)*d/(c*g^2*h*x^2 + a*h^3*x^
2 + 2*c*g^3*x + 2*a*g*h^2*x + c*g^4/h + a*g^2*h) - sqrt(c*x^2 + a)*e/(c*g^2*h*x + a*h^3*x + c*g^3 + a*g*h^2) +
3/2*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)^(5/2)*h
^7) - 3/2*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)^(5
/2)*h^6) + 3/2*c^2*d*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) - 5/2*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)^(3/2)*h^5) + 3/2*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)^(3/2)*h^4) - 1/2*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)^(3/2)*h^3) + f*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/(sqrt(a + c*g^2/h^
2)*h^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (210) = 420\).

Time = 0.30 (sec) , antiderivative size = 839, normalized size of antiderivative = 3.73 \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \sqrt {a+c x^2}} \, dx=-\frac {{\left (2 \, c^{2} d g^{2} - a c f g^{2} + 3 \, a c e g h - a c d h^{2} + 2 \, a^{2} f h^{2}\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} h + \sqrt {c} g}{\sqrt {-c g^{2} - a h^{2}}}\right )}{{\left (c^{2} g^{4} + 2 \, a c g^{2} h^{2} + a^{2} h^{4}\right )} \sqrt {-c g^{2} - a h^{2}}} + \frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} f g^{4} h - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} d g^{2} h^{3} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c f g^{2} h^{3} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c e g h^{4} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c d h^{5} + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} f g^{5} + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} e g^{4} h - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} d g^{3} h^{2} + 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} f g^{3} h^{2} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} e g^{2} h^{3} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} d g h^{4} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} \sqrt {c} f g h^{4} + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} \sqrt {c} e h^{5} - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} f g^{4} h - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} e g^{3} h^{2} + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} d g^{2} h^{3} - 11 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c f g^{2} h^{3} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c e g h^{4} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c d h^{5} + a^{2} c^{\frac {3}{2}} f g^{3} h^{2} + a^{2} c^{\frac {3}{2}} e g^{2} h^{3} - 3 \, a^{2} c^{\frac {3}{2}} d g h^{4} + 4 \, a^{3} \sqrt {c} f g h^{4} - 2 \, a^{3} \sqrt {c} e h^{5}}{{\left (c^{2} g^{4} h^{2} + 2 \, a c g^{2} h^{4} + a^{2} h^{6}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} h + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} g - a h\right )}^{2}} \]

[In]

integrate((f*x^2+e*x+d)/(h*x+g)^3/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

-(2*c^2*d*g^2 - a*c*f*g^2 + 3*a*c*e*g*h - a*c*d*h^2 + 2*a^2*f*h^2)*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*h + s
qrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c^2*g^4 + 2*a*c*g^2*h^2 + a^2*h^4)*sqrt(-c*g^2 - a*h^2)) + (2*(sqrt(c)*x - s
qrt(c*x^2 + a))^3*c^2*f*g^4*h - 2*(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 + 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*f*g^5 + 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*e*g^4*h - 6*(s
qrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*d*g^3*h^2 + 7*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*f*g^3*h^2 - 5*(s
qrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*e*g^2*h^3 + 3*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*d*g*h^4 - 4*(s
qrt(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 - 2*(s
qrt(c)*x - sqrt(c*x^2 + a))*a*c^2*f*g^4*h - 4*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c^2*e*g^3*h^2 + 10*(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 + (sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*d*h^5 + a^2*c^(3/2)*f*g^3*h^2 + a^2*c^(3/2)*e*g^2*
h^3 - 3*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^2*g^4*h^2 + 2*a*c*g^2*h^4 + a^2
*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. \[ \int \frac {d+e x+f x^2}{(g+h x)^3 \sqrt {a+c x^2}} \, dx=\int \frac {f\,x^2+e\,x+d}{{\left (g+h\,x\right )}^3\,\sqrt {c\,x^2+a}} \,d x \]

[In]

int((d + e*x + f*x^2)/((g + h*x)^3*(a + c*x^2)^(1/2)),x)

[Out]

int((d + e*x + f*x^2)/((g + h*x)^3*(a + c*x^2)^(1/2)), x)