\(\int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx\) [106]

Optimal result

Integrand size = 29, antiderivative size = 168 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=-\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac {f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} h^2}+\frac {\left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{h^2 \left (c g^2+a h^2\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1665, 858, 223, 212, 739} \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=\frac {\text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right )}{h^2 \left (a h^2+c g^2\right )^{3/2}}+\frac {f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} h^2}-\frac {\sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )} \]

[In]

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

[Out]

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

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[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 {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{h \left (c g^2+a h^2\right ) (g+h x)}-\frac {\int \frac {-c d g+a f g-a e h-f \left (\frac {c g^2}{h}+a h\right ) x}{(g+h x) \sqrt {a+c x^2}} \, dx}{c g^2+a h^2} \\ & = -\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac {f \int \frac {1}{\sqrt {a+c x^2}} \, dx}{h^2}+\frac {\left (c d g-2 a f g-\frac {c f g^3}{h^2}+a e h\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{c g^2+a h^2} \\ & = -\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac {f \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{h^2}-\frac {\left (c d g-2 a f g-\frac {c f g^3}{h^2}+a e h\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 )}{c g^2+a h^2} \\ & = -\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac {f \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} h^2}-\frac {\left (c d g-2 a f g-\frac {c f g^3}{h^2}+a 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 )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.05 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=-\frac {\frac {h \left (f g^2+h (-e g+d h)\right ) \sqrt {a+c x^2}}{\left (c g^2+a h^2\right ) (g+h x)}+\frac {2 \left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right ) \arctan \left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{\left (-c g^2-a h^2\right )^{3/2}}+\frac {f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{h^2} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(389\) vs. \(2(154)=308\).

Time = 0.60 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.32

method	result	size
default	\(\frac {f \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{h^{2} \sqrt {c}}-\frac {\left (e h -2 f g \right ) \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{h^{3} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (-\frac {h^{2} \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )}-\frac {c g h \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{4}}\)	\(390\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (155) = 310\).

Time = 0.23 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.49 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=-\frac {\sqrt {c x^{2} + a} f g^{2}}{c g^{2} h^{2} x + a h^{4} x + c g^{3} h + a g h^{3}} + \frac {\sqrt {c x^{2} + a} e g}{c g^{2} h x + a h^{3} x + c g^{3} + a g h^{2}} - \frac {\sqrt {c x^{2} + a} d}{c g^{2} x + a h^{2} x + \frac {c g^{3}}{h} + a g h} + \frac {f \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c} h^{2}} + \frac {c f g^{3} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{5}} - \frac {c e g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{4}} + \frac {c d g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{3}} - \frac {2 \, f g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{3}} + \frac {e \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{2}} \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

Mupad [F(-1)]

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

[In]

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

[Out]

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