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

Optimal. Leaf size=308 \[ -\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a f h^2+2 c \left (3 f g^2-h (2 e g-d h)\right )\right )}{2 \sqrt {c} h^4}+\frac {\tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (a h^2 (2 f g-e h)+c g \left (3 f g^2-h (2 e g-d h)\right )\right )}{h^4 \sqrt {a h^2+c g^2}}-\frac {\sqrt {a+c x^2} \left (2 \left (a h^2 (2 f g-e h)+c g \left (3 f g^2-h (2 e g-d h)\right )\right )-h x \left (a f h^2+c \left (3 f g^2-2 h (e g-d h)\right )\right )\right )}{2 h^3 \left (a h^2+c g^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.51, antiderivative size = 303, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1651, 815, 844, 217, 206, 725} \[ -\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}-\frac {\sqrt {a+c x^2} \left (2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )-h x \left (a f h^2-2 c h (e g-d h)+3 c f g^2\right )\right )}{2 h^3 \left (a h^2+c g^2\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a f h^2-2 c h (2 e g-d h)+6 c f g^2\right )}{2 \sqrt {c} h^4}+\frac {\tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )}{h^4 \sqrt {a h^2+c g^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*(3*c*f*g^3 - c*g*h*(2*e*g - d*h) + a*h^2*(2*f*g - e*h)) - h*(3*c*f*g^2 + a*f*h^2 - 2*c*h*(e*g - d*h))*x)*
Sqrt[a + c*x^2])/(2*h^3*(c*g^2 + a*h^2)) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(3/2))/(h*(c*g^2 + a*h^2)*(g +
h*x)) + ((6*c*f*g^2 + a*f*h^2 - 2*c*h*(2*e*g - d*h))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*Sqrt[c]*h^4) +
((3*c*f*g^3 - c*g*h*(2*e*g - 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])])/(h^4*Sqrt[c*g^2 + a*h^2])

Rule 206

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

Rule 217

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 725

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 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 844

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 1651

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*} \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx &=-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{h \left (c g^2+a h^2\right ) (g+h x)}-\frac {\int \frac {\left (-c d g+a f g-a e h-\left (a f h-c \left (2 e g-\frac {3 f g^2}{h}-2 d h\right )\right ) x\right ) \sqrt {a+c x^2}}{g+h x} \, dx}{c g^2+a h^2}\\ &=-\frac {\left (2 \left (3 c f g^3-c g h (2 e g-d h)+a h^2 (2 f g-e h)\right )-h \left (3 c f g^2+a f h^2-2 c h (e g-d h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^3 \left (c g^2+a h^2\right )}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{h \left (c g^2+a h^2\right ) (g+h x)}-\frac {\int \frac {a c (3 f g-2 e h) \left (c g^2+a h^2\right )-\frac {c \left (c g^2+a h^2\right ) \left (6 c f g^2+a f h^2-2 c h (2 e g-d h)\right ) x}{h}}{(g+h x) \sqrt {a+c x^2}} \, dx}{2 c h^2 \left (c g^2+a h^2\right )}\\ &=-\frac {\left (2 \left (3 c f g^3-c g h (2 e g-d h)+a h^2 (2 f g-e h)\right )-h \left (3 c f g^2+a f h^2-2 c h (e g-d h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^3 \left (c g^2+a h^2\right )}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac {\left (6 c f g^2+a f h^2-2 c h (2 e g-d h)\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 h^4}-\frac {\left (3 c f g^3-c g h (2 e g-d h)+a h^2 (2 f g-e h)\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{h^4}\\ &=-\frac {\left (2 \left (3 c f g^3-c g h (2 e g-d h)+a h^2 (2 f g-e h)\right )-h \left (3 c f g^2+a f h^2-2 c h (e g-d h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^3 \left (c g^2+a h^2\right )}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac {\left (6 c f g^2+a f h^2-2 c h (2 e g-d h)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 h^4}+\frac {\left (3 c f g^3-c g h (2 e g-d h)+a h^2 (2 f g-e h)\right ) \operatorname {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 )}{h^4}\\ &=-\frac {\left (2 \left (3 c f g^3-c g h (2 e g-d h)+a h^2 (2 f g-e h)\right )-h \left (3 c f g^2+a f h^2-2 c h (e g-d h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^3 \left (c g^2+a h^2\right )}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac {\left (6 c f g^2+a f h^2-2 c h (2 e g-d h)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} h^4}+\frac {\left (3 c f g^3-c g h (2 e g-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 )}{h^4 \sqrt {c g^2+a h^2}}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 264, normalized size = 0.86 \[ \frac {\frac {h \sqrt {a+c x^2} \left (2 h (-d h+2 e g+e h x)+f \left (-6 g^2-3 g h x+h^2 x^2\right )\right )}{g+h x}+\frac {\log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right ) \left (a f h^2+2 c h (d h-2 e g)+6 c f g^2\right )}{\sqrt {c}}+\frac {2 \log \left (\sqrt {a+c x^2} \sqrt {a h^2+c g^2}+a h-c g x\right ) \left (a h^2 (2 f g-e h)+c g h (d h-2 e g)+3 c f g^3\right )}{\sqrt {a h^2+c g^2}}-\frac {2 \log (g+h x) \left (a h^2 (2 f g-e h)+c g h (d h-2 e g)+3 c f g^3\right )}{\sqrt {a h^2+c g^2}}}{2 h^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((h*Sqrt[a + c*x^2]*(2*h*(2*e*g - d*h + e*h*x) + f*(-6*g^2 - 3*g*h*x + h^2*x^2)))/(g + h*x) - (2*(3*c*f*g^3 +
c*g*h*(-2*e*g + d*h) + a*h^2*(2*f*g - e*h))*Log[g + h*x])/Sqrt[c*g^2 + a*h^2] + ((6*c*f*g^2 + a*f*h^2 + 2*c*h*
(-2*e*g + d*h))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/Sqrt[c] + (2*(3*c*f*g^3 + c*g*h*(-2*e*g + d*h) + a*h^2*(2*
f*g - e*h))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/Sqrt[c*g^2 + a*h^2])/(2*h^4)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.02, size = 2818, normalized size = 9.15 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.65, size = 478, normalized size = 1.55 \[ -\frac {\sqrt {c x^{2} + a} f g^{2}}{h^{4} x + g h^{3}} + \frac {\sqrt {c x^{2} + a} e g}{h^{3} x + g h^{2}} - \frac {\sqrt {c x^{2} + a} d}{h^{2} x + g h} + \frac {\sqrt {c x^{2} + a} f x}{2 \, h^{2}} + \frac {3 \, \sqrt {c} f g^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{4}} - \frac {2 \, \sqrt {c} e g \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{3}} + \frac {\sqrt {c} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{2}} + \frac {a f \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c} h^{2}} - \frac {c f g^{3} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{5}} + \frac {c e g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{4}} - \frac {c d g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{3}} - \frac {2 \, \sqrt {a + \frac {c g^{2}}{h^{2}}} f g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{3}} + \frac {\sqrt {a + \frac {c g^{2}}{h^{2}}} e \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{2}} - \frac {2 \, \sqrt {c x^{2} + a} f g}{h^{3}} + \frac {\sqrt {c x^{2} + a} e}{h^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*x^2 + a)*f*g^2/(h^4*x + g*h^3) + sqrt(c*x^2 + a)*e*g/(h^3*x + g*h^2) - sqrt(c*x^2 + a)*d/(h^2*x + g*h)
+ 1/2*sqrt(c*x^2 + a)*f*x/h^2 + 3*sqrt(c)*f*g^2*arcsinh(c*x/sqrt(a*c))/h^4 - 2*sqrt(c)*e*g*arcsinh(c*x/sqrt(a
*c))/h^3 + sqrt(c)*d*arcsinh(c*x/sqrt(a*c))/h^2 + 1/2*a*f*arcsinh(c*x/sqrt(a*c))/(sqrt(c)*h^2) - c*f*g^3*arcsi
nh(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^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)))/(sqrt(a + c*g^2/h^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)))/(sqrt(a + c*g^2/h^2)*h^3) - 2*sqrt(a + c*g^2/h^2)*f*
g*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^3 + sqrt(a + c*g^2/h^2)*e*arcsinh(c
*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^2 - 2*sqrt(c*x^2 + a)*f*g/h^3 + sqrt(c*x^2 + a
)*e/h^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {c\,x^2+a}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + c x^{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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