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

Optimal. Leaf size=239 \[ -\frac {a (a h (2 f g-e h)+c g (e g-2 d h))-x \left (a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )+c^2 d g^2\right )}{a \sqrt {a+c x^2} \left (a h^2+c g^2\right )^2}-\frac {h \sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{(g+h x) \left (a h^2+c g^2\right )^2}+\frac {\tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (a h^2 (2 f g-e h)-c g \left (f g^2-h (2 e g-3 d h)\right )\right )}{\left (a h^2+c g^2\right )^{5/2}} \]

[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)

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Rubi [A]  time = 0.42, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1647, 807, 725, 206} \[ -\frac {a (a h (2 f g-e h)+c g (e g-2 d h))-x \left (a^2 f h^2-a c \left (f g^2-h (2 e g-d h)\right )+c^2 d g^2\right )}{a \sqrt {a+c x^2} \left (a h^2+c g^2\right )^2}-\frac {h \sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{(g+h x) \left (a h^2+c g^2\right )^2}-\frac {\tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (-a h^2 (2 f g-e h)-c g h (2 e g-3 d h)+c f g^3\right )}{\left (a h^2+c g^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[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 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 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 807

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 1647

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*} \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 {\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 ) \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 )}{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*}

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Mathematica [A]  time = 0.61, size = 285, normalized size = 1.19 \[ \frac {\left (a h^2+c g^2\right )^{3/2} \left (-a^2 f h^2+a c (2 h (d h-e g+e h x)+f g (g-2 h x))+2 c^2 d g h x\right )+2 h \left (h \left (a+c x^2\right ) \sqrt {a h^2+c g^2} \left (a^2 f h^2+a c \left (h (3 e g-2 d h)-2 f g^2\right )+c^2 d g^2\right )-a c \sqrt {a+c x^2} (g+h x) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (a h^2 (e h-2 f g)+c g h (3 d h-2 e g)+c f g^3\right )\right )-a f \left (a h^2+c g^2\right )^{5/2}}{2 a c h \sqrt {a+c x^2} (g+h x) \left (a h^2+c g^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 7.18, size = 1573, normalized size = 6.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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

[Out]

Timed out

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maple [B]  time = 0.02, size = 1663, normalized size = 6.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.77, size = 1085, normalized size = 4.54 \[ \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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {f\,x^2+e\,x+d}{{\left (g+h\,x\right )}^2\,{\left (c\,x^2+a\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [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)/(h*x+g)**2/(c*x**2+a)**(3/2),x)

[Out]

Timed out

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