### 3.84 $$\int \frac{\sqrt{a+c x^2} (d+e x+f x^2)}{(g+h x)^3} \, dx$$

Optimal. Leaf size=296 $-\frac{\tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (2 a^2 f h^4+a c h^2 \left (9 f g^2-h (3 e g-d h)\right )+2 c^2 g^3 (3 f g-e h)\right )}{2 h^4 \left (a h^2+c g^2\right )^{3/2}}-\frac{\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{2 h (g+h x)^2 \left (a h^2+c g^2\right )}+\frac{\sqrt{a+c x^2} \left (h x \left (2 a f h^2+c \left (3 f g^2-h (e g-d h)\right )\right )+2 \left (a h^2+c g^2\right ) (3 f g-e h)\right )}{2 h^3 (g+h x) \left (a h^2+c g^2\right )}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (3 f g-e h)}{h^4}$

[Out]

((2*(3*f*g - e*h)*(c*g^2 + a*h^2) + h*(2*a*f*h^2 + c*(3*f*g^2 - h*(e*g - d*h)))*x)*Sqrt[a + c*x^2])/(2*h^3*(c*
g^2 + a*h^2)*(g + h*x)) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(3/2))/(2*h*(c*g^2 + a*h^2)*(g + h*x)^2) - (Sqr
t[c]*(3*f*g - e*h)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/h^4 - ((2*a^2*f*h^4 + 2*c^2*g^3*(3*f*g - e*h) + a*c*h
^2*(9*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*h^4*(c*g^2 +
a*h^2)^(3/2))

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Rubi [A]  time = 0.545689, antiderivative size = 295, normalized size of antiderivative = 1., 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, 813, 844, 217, 206, 725} $-\frac{\tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (2 a^2 f h^4+a c h^2 \left (9 f g^2-h (3 e g-d h)\right )+2 c^2 g^3 (3 f g-e h)\right )}{2 h^4 \left (a h^2+c g^2\right )^{3/2}}-\frac{\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{2 h (g+h x)^2 \left (a h^2+c g^2\right )}+\frac{\sqrt{a+c x^2} \left (h x \left (2 a f h^2-c h (e g-d h)+3 c f g^2\right )+2 \left (a h^2+c g^2\right ) (3 f g-e h)\right )}{2 h^3 (g+h x) \left (a h^2+c g^2\right )}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (3 f g-e h)}{h^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((2*(3*f*g - e*h)*(c*g^2 + a*h^2) + h*(3*c*f*g^2 + 2*a*f*h^2 - c*h*(e*g - d*h))*x)*Sqrt[a + c*x^2])/(2*h^3*(c*
g^2 + a*h^2)*(g + h*x)) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(3/2))/(2*h*(c*g^2 + a*h^2)*(g + h*x)^2) - (Sqr
t[c]*(3*f*g - e*h)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/h^4 - ((2*a^2*f*h^4 + 2*c^2*g^3*(3*f*g - e*h) + a*c*h
^2*(9*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*h^4*(c*g^2 +
a*h^2)^(3/2))

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]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 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 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 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]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^3} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\int \frac{\left (-2 (c d g-a f g+a e h)-\left (2 a f h-c \left (e g-\frac{3 f g^2}{h}-d h\right )\right ) x\right ) \sqrt{a+c x^2}}{(g+h x)^2} \, dx}{2 \left (c g^2+a h^2\right )}\\ &=\frac{\left (2 (3 f g-e h) \left (c g^2+a h^2\right )+h \left (3 c f g^2+2 a f h^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 ) (g+h x)}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac{\int \frac{2 a \left (3 c f g^2+2 a f h^2-c h (e g-d h)\right )-\frac{4 c (3 f g-e h) \left (c g^2+a h^2\right ) x}{h}}{(g+h x) \sqrt{a+c x^2}} \, dx}{4 h^2 \left (c g^2+a h^2\right )}\\ &=\frac{\left (2 (3 f g-e h) \left (c g^2+a h^2\right )+h \left (3 c f g^2+2 a f h^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 ) (g+h x)}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{(c (3 f g-e h)) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{h^4}+\frac{\left (2 a^2 f h^4+2 c^2 g^3 (3 f g-e h)+a c h^2 \left (9 f g^2-h (3 e g-d h)\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{2 h^4 \left (c g^2+a h^2\right )}\\ &=\frac{\left (2 (3 f g-e h) \left (c g^2+a h^2\right )+h \left (3 c f g^2+2 a f h^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 ) (g+h x)}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{(c (3 f g-e h)) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{h^4}-\frac{\left (2 a^2 f h^4+2 c^2 g^3 (3 f g-e h)+a c h^2 \left (9 f g^2-h (3 e g-d h)\right )\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 )}{2 h^4 \left (c g^2+a h^2\right )}\\ &=\frac{\left (2 (3 f g-e h) \left (c g^2+a h^2\right )+h \left (3 c f g^2+2 a f h^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 ) (g+h x)}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\sqrt{c} (3 f g-e h) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{h^4}-\frac{\left (2 a^2 f h^4+2 c^2 g^3 (3 f g-e h)+a c h^2 \left (9 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 h^4 \left (c g^2+a h^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.652985, size = 318, normalized size = 1.07 $\frac{-\frac{\log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (2 a^2 f h^4+a c h^2 \left (h (d h-3 e g)+9 f g^2\right )+2 c^2 g^3 (3 f g-e h)\right )}{\left (a h^2+c g^2\right )^{3/2}}+\frac{\log (g+h x) \left (2 a^2 f h^4+a c h^2 \left (h (d h-3 e g)+9 f g^2\right )+2 c^2 g^3 (3 f g-e h)\right )}{\left (a h^2+c g^2\right )^{3/2}}+h \sqrt{a+c x^2} \left (\frac{-2 a h^2 (e h-2 f g)+c g h (d h-3 e g)+5 c f g^3}{(g+h x) \left (a h^2+c g^2\right )}+\frac{h (e g-d h)-f g^2}{(g+h x)^2}+2 f\right )+2 \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) (e h-3 f g)}{2 h^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(h*Sqrt[a + c*x^2]*(2*f + (-(f*g^2) + h*(e*g - d*h))/(g + h*x)^2 + (5*c*f*g^3 + c*g*h*(-3*e*g + d*h) - 2*a*h^2
*(-2*f*g + e*h))/((c*g^2 + a*h^2)*(g + h*x))) + ((2*a^2*f*h^4 + 2*c^2*g^3*(3*f*g - e*h) + a*c*h^2*(9*f*g^2 + h
*(-3*e*g + d*h)))*Log[g + h*x])/(c*g^2 + a*h^2)^(3/2) + 2*Sqrt[c]*(-3*f*g + e*h)*Log[c*x + Sqrt[c]*Sqrt[a + c*
x^2]] - ((2*a^2*f*h^4 + 2*c^2*g^3*(3*f*g - e*h) + a*c*h^2*(9*f*g^2 + h*(-3*e*g + d*h)))*Log[a*h - c*g*x + Sqrt
[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(3/2))/(2*h^4)

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Maple [B]  time = 0.239, size = 4432, normalized size = 15. \begin{align*} \text{output too large to display} \end{align*}

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)^3,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + c x^{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{3}}\, dx \end{align*}

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)**3,x)

[Out]

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

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Giac [B]  time = 1.42267, size = 1246, normalized size = 4.21 \begin{align*} \text{result too large to display} \end{align*}

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)^3,x, algorithm="giac")

[Out]

-(6*c^2*f*g^4 + 9*a*c*f*g^2*h^2 + a*c*d*h^4 + 2*a^2*f*h^4 - 2*c^2*g^3*h*e - 3*a*c*g*h^3*e)*arctan(((sqrt(c)*x
- sqrt(c*x^2 + a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c*g^2*h^4 + a*h^6)*sqrt(-c*g^2 - a*h^2)) + sqrt(c*x^
2 + a)*f/h^3 + (6*(sqrt(c)*x - sqrt(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 + (sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*d*h^5 - 4*(sqrt(c)*x
- sqrt(c*x^2 + a))^3*c^2*g^3*h^2*e - 3*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c*g*h^4*e + 10*(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)*d*g^3*h^2 + 3*(sqrt(c)*x - sqrt(c*x^2 + a
))^2*a*c^(3/2)*f*g^3*h^2 - (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 - 6*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(5/2)*g^4*h*e - (sqrt(c)*x - sqrt(c*x^2 + a))^2*a
*c^(3/2)*g^2*h^3*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*sqrt(c)*h^5*e - 14*(sqrt(c)*x - sqrt(c*x^2 + a))*a*
c^2*f*g^4*h - 2*(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 + (sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*d*h^5 + 8*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c^2*g^3*h^2*e + 5*(sqrt(c)*
x - sqrt(c*x^2 + a))*a^2*c*g*h^4*e + 5*a^2*c^(3/2)*f*g^3*h^2 + a^2*c^(3/2)*d*g*h^4 + 4*a^3*sqrt(c)*f*g*h^4 - 3
*a^2*c^(3/2)*g^2*h^3*e - 2*a^3*sqrt(c)*h^5*e)/((c*g^2*h^4 + a*h^6)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*h + 2*(sqr
t(c)*x - sqrt(c*x^2 + a))*sqrt(c)*g - a*h)^2) + (3*sqrt(c)*f*g - sqrt(c)*h*e)*log(abs(-sqrt(c)*x + sqrt(c*x^2
+ a)))/h^4