### 3.237 $$\int \frac{d+e x+f x^2}{(g+h x) (a+b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=225 $\frac{2 \left (-x \left (-c (-2 a e h+2 a f g+b d h+b e g)+b f (b g-a h)+2 c^2 d g\right )-b (a e h+a f g+c d g)+2 a (a f h-c d h+c e g)+b^2 d h\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a h^2-b g h+c g^2\right )}+\frac{\left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right )}{\left (a h^2-b g h+c g^2\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.265658, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {1646, 12, 724, 206} $\frac{2 \left (-x \left (-c (-2 a e h+2 a f g+b d h+b e g)+b f (b g-a h)+2 c^2 d g\right )-b (a e h+a f g+c d g)+2 a (a f h-c d h+c e g)+b^2 d h\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a h^2-b g h+c g^2\right )}+\frac{\left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right )}{\left (a h^2-b g h+c g^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 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])

Rubi steps

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

Mathematica [A]  time = 0.584231, size = 271, normalized size = 1.2 $\frac{\frac{-b^2 \left (a f h^2+2 c d h^2+c f g (g-2 h x)\right )-2 b c h (-a e h+a f (g+h x)+c (-d g+d h x+e g x))+4 c^2 (a h (d h-e g+e h x)+a f g (g-h x)+c d g h x)+b^3 f g h}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (h (b g-a h)-c g^2\right )}-\frac{c h \left (h (d h-e g)+f g^2\right ) \tanh ^{-1}\left (\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right )}{\left (h (a h-b g)+c g^2\right )^{3/2}}-\frac{f}{\sqrt{a+x (b+c x)}}}{c h}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.293, size = 2079, normalized size = 9.2 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 131.668, size = 3934, normalized size = 17.48 \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)/(h*x+g)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.20519, size = 971, normalized size = 4.32 \begin{align*} -\frac{2 \,{\left (\frac{{\left (2 \, c^{3} d g^{3} + b^{2} c f g^{3} - 2 \, a c^{2} f g^{3} - 3 \, b c^{2} d g^{2} h - b^{3} f g^{2} h + a b c f g^{2} h + b^{2} c d g h^{2} + 2 \, a c^{2} d g h^{2} + 2 \, a b^{2} f g h^{2} - 2 \, a^{2} c f g h^{2} - a b c d h^{3} - a^{2} b f h^{3} - b c^{2} g^{3} e + b^{2} c g^{2} h e + 2 \, a c^{2} g^{2} h e - 3 \, a b c g h^{2} e + 2 \, a^{2} c h^{3} e\right )} x}{b^{2} c^{2} g^{4} - 4 \, a c^{3} g^{4} - 2 \, b^{3} c g^{3} h + 8 \, a b c^{2} g^{3} h + b^{4} g^{2} h^{2} - 2 \, a b^{2} c g^{2} h^{2} - 8 \, a^{2} c^{2} g^{2} h^{2} - 2 \, a b^{3} g h^{3} + 8 \, a^{2} b c g h^{3} + a^{2} b^{2} h^{4} - 4 \, a^{3} c h^{4}} + \frac{b c^{2} d g^{3} + a b c f g^{3} - 2 \, b^{2} c d g^{2} h + 2 \, a c^{2} d g^{2} h - a b^{2} f g^{2} h - 2 \, a^{2} c f g^{2} h + b^{3} d g h^{2} - a b c d g h^{2} + 3 \, a^{2} b f g h^{2} - a b^{2} d h^{3} + 2 \, a^{2} c d h^{3} - 2 \, a^{3} f h^{3} - 2 \, a c^{2} g^{3} e + 3 \, a b c g^{2} h e - a b^{2} g h^{2} e - 2 \, a^{2} c g h^{2} e + a^{2} b h^{3} e}{b^{2} c^{2} g^{4} - 4 \, a c^{3} g^{4} - 2 \, b^{3} c g^{3} h + 8 \, a b c^{2} g^{3} h + b^{4} g^{2} h^{2} - 2 \, a b^{2} c g^{2} h^{2} - 8 \, a^{2} c^{2} g^{2} h^{2} - 2 \, a b^{3} g h^{3} + 8 \, a^{2} b c g h^{3} + a^{2} b^{2} h^{4} - 4 \, a^{3} c h^{4}}\right )}}{\sqrt{c x^{2} + b x + a}} + \frac{2 \,{\left (f g^{2} + d h^{2} - g h e\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} h + \sqrt{c} g}{\sqrt{-c g^{2} + b g h - a h^{2}}}\right )}{{\left (c g^{2} - b g h + a h^{2}\right )} \sqrt{-c g^{2} + b g h - a h^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((2*c^3*d*g^3 + b^2*c*f*g^3 - 2*a*c^2*f*g^3 - 3*b*c^2*d*g^2*h - b^3*f*g^2*h + a*b*c*f*g^2*h + b^2*c*d*g*h^2
+ 2*a*c^2*d*g*h^2 + 2*a*b^2*f*g*h^2 - 2*a^2*c*f*g*h^2 - a*b*c*d*h^3 - a^2*b*f*h^3 - b*c^2*g^3*e + b^2*c*g^2*h
*e + 2*a*c^2*g^2*h*e - 3*a*b*c*g*h^2*e + 2*a^2*c*h^3*e)*x/(b^2*c^2*g^4 - 4*a*c^3*g^4 - 2*b^3*c*g^3*h + 8*a*b*c
^2*g^3*h + b^4*g^2*h^2 - 2*a*b^2*c*g^2*h^2 - 8*a^2*c^2*g^2*h^2 - 2*a*b^3*g*h^3 + 8*a^2*b*c*g*h^3 + a^2*b^2*h^4
- 4*a^3*c*h^4) + (b*c^2*d*g^3 + a*b*c*f*g^3 - 2*b^2*c*d*g^2*h + 2*a*c^2*d*g^2*h - a*b^2*f*g^2*h - 2*a^2*c*f*g
^2*h + b^3*d*g*h^2 - a*b*c*d*g*h^2 + 3*a^2*b*f*g*h^2 - a*b^2*d*h^3 + 2*a^2*c*d*h^3 - 2*a^3*f*h^3 - 2*a*c^2*g^3
*e + 3*a*b*c*g^2*h*e - a*b^2*g*h^2*e - 2*a^2*c*g*h^2*e + a^2*b*h^3*e)/(b^2*c^2*g^4 - 4*a*c^3*g^4 - 2*b^3*c*g^3
*h + 8*a*b*c^2*g^3*h + b^4*g^2*h^2 - 2*a*b^2*c*g^2*h^2 - 8*a^2*c^2*g^2*h^2 - 2*a*b^3*g*h^3 + 8*a^2*b*c*g*h^3 +
a^2*b^2*h^4 - 4*a^3*c*h^4))/sqrt(c*x^2 + b*x + a) + 2*(f*g^2 + d*h^2 - g*h*e)*arctan(-((sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*h + sqrt(c)*g)/sqrt(-c*g^2 + b*g*h - a*h^2))/((c*g^2 - b*g*h + a*h^2)*sqrt(-c*g^2 + b*g*h - a*h^
2))