### 3.280 $$\int \frac{x^2 (d+e x+f x^2+g x^3)}{\sqrt{a+b x+c x^2}} \, dx$$

Optimal. Leaf size=346 $-\frac{\sqrt{a+b x+c x^2} \left (-2 c x \left (-40 c^2 (9 a f+10 b e)+14 b c (46 a g+25 b f)-315 b^3 g+480 c^3 d\right )-60 b^2 c (20 c e-49 a g)+40 b c^2 (36 c d-55 a f)+256 a c^2 (5 c e-4 a g)+1050 b^3 c f-945 b^4 g\right )}{1920 c^5}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (48 b^2 c^2 (2 c d-5 a f)-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)-32 a c^3 (4 c d-3 a f)+70 b^4 c f-63 b^5 g\right )}{256 c^{11/2}}+\frac{x^2 \sqrt{a+b x+c x^2} \left (-64 a c g+63 b^2 g-70 b c f+80 c^2 e\right )}{240 c^3}+\frac{x^3 \sqrt{a+b x+c x^2} (10 c f-9 b g)}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}$

[Out]

((80*c^2*e - 70*b*c*f + 63*b^2*g - 64*a*c*g)*x^2*Sqrt[a + b*x + c*x^2])/(240*c^3) + ((10*c*f - 9*b*g)*x^3*Sqrt
[a + b*x + c*x^2])/(40*c^2) + (g*x^4*Sqrt[a + b*x + c*x^2])/(5*c) - ((1050*b^3*c*f + 40*b*c^2*(36*c*d - 55*a*f
) - 945*b^4*g - 60*b^2*c*(20*c*e - 49*a*g) + 256*a*c^2*(5*c*e - 4*a*g) - 2*c*(480*c^3*d - 40*c^2*(10*b*e + 9*a
*f) - 315*b^3*g + 14*b*c*(25*b*f + 46*a*g))*x)*Sqrt[a + b*x + c*x^2])/(1920*c^5) + ((70*b^4*c*f + 48*b^2*c^2*(
2*c*d - 5*a*f) - 32*a*c^3*(4*c*d - 3*a*f) - 63*b^5*g - 40*b^3*c*(2*c*e - 7*a*g) + 48*a*b*c^2*(4*c*e - 5*a*g))*
ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(11/2))

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Rubi [A]  time = 0.811677, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.152, Rules used = {1653, 832, 779, 621, 206} $-\frac{\sqrt{a+b x+c x^2} \left (-2 c x \left (-40 c^2 (9 a f+10 b e)+14 b c (46 a g+25 b f)-315 b^3 g+480 c^3 d\right )-60 b^2 c (20 c e-49 a g)+40 b c^2 (36 c d-55 a f)+256 a c^2 (5 c e-4 a g)+1050 b^3 c f-945 b^4 g\right )}{1920 c^5}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (48 b^2 c^2 (2 c d-5 a f)-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)-32 a c^3 (4 c d-3 a f)+70 b^4 c f-63 b^5 g\right )}{256 c^{11/2}}+\frac{x^2 \sqrt{a+b x+c x^2} \left (-64 a c g+63 b^2 g-70 b c f+80 c^2 e\right )}{240 c^3}+\frac{x^3 \sqrt{a+b x+c x^2} (10 c f-9 b g)}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((80*c^2*e - 70*b*c*f + 63*b^2*g - 64*a*c*g)*x^2*Sqrt[a + b*x + c*x^2])/(240*c^3) + ((10*c*f - 9*b*g)*x^3*Sqrt
[a + b*x + c*x^2])/(40*c^2) + (g*x^4*Sqrt[a + b*x + c*x^2])/(5*c) - ((1050*b^3*c*f + 40*b*c^2*(36*c*d - 55*a*f
) - 945*b^4*g - 60*b^2*c*(20*c*e - 49*a*g) + 256*a*c^2*(5*c*e - 4*a*g) - 2*c*(480*c^3*d - 40*c^2*(10*b*e + 9*a
*f) - 315*b^3*g + 14*b*c*(25*b*f + 46*a*g))*x)*Sqrt[a + b*x + c*x^2])/(1920*c^5) + ((70*b^4*c*f + 48*b^2*c^2*(
2*c*d - 5*a*f) - 32*a*c^3*(4*c*d - 3*a*f) - 63*b^5*g - 40*b^3*c*(2*c*e - 7*a*g) + 48*a*b*c^2*(4*c*e - 5*a*g))*
ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(11/2))

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 832

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

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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{x^2 \left (d+e x+f x^2+g x^3\right )}{\sqrt{a+b x+c x^2}} \, dx &=\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x^2 \left (5 c d+(5 c e-4 a g) x+\frac{1}{2} (10 c f-9 b g) x^2\right )}{\sqrt{a+b x+c x^2}} \, dx}{5 c}\\ &=\frac{(10 c f-9 b g) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x^2 \left (\frac{1}{2} \left (40 c^2 d-30 a c f+27 a b g\right )+\frac{1}{4} \left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{20 c^2}\\ &=\frac{\left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}+\frac{(10 c f-9 b g) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x \left (-\frac{1}{2} a \left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right )+\frac{1}{8} \left (480 c^3 d-40 c^2 (10 b e+9 a f)-315 b^3 g+14 b c (25 b f+46 a g)\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{60 c^3}\\ &=\frac{\left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}+\frac{(10 c f-9 b g) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}-\frac{\left (1050 b^3 c f+40 b c^2 (36 c d-55 a f)-945 b^4 g-60 b^2 c (20 c e-49 a g)+256 a c^2 (5 c e-4 a g)-2 c \left (480 c^3 d-40 c^2 (10 b e+9 a f)-315 b^3 g+14 b c (25 b f+46 a g)\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}+\frac{\left (70 b^4 c f+48 b^2 c^2 (2 c d-5 a f)-32 a c^3 (4 c d-3 a f)-63 b^5 g-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{256 c^5}\\ &=\frac{\left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}+\frac{(10 c f-9 b g) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}-\frac{\left (1050 b^3 c f+40 b c^2 (36 c d-55 a f)-945 b^4 g-60 b^2 c (20 c e-49 a g)+256 a c^2 (5 c e-4 a g)-2 c \left (480 c^3 d-40 c^2 (10 b e+9 a f)-315 b^3 g+14 b c (25 b f+46 a g)\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}+\frac{\left (70 b^4 c f+48 b^2 c^2 (2 c d-5 a f)-32 a c^3 (4 c d-3 a f)-63 b^5 g-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{128 c^5}\\ &=\frac{\left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}+\frac{(10 c f-9 b g) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}-\frac{\left (1050 b^3 c f+40 b c^2 (36 c d-55 a f)-945 b^4 g-60 b^2 c (20 c e-49 a g)+256 a c^2 (5 c e-4 a g)-2 c \left (480 c^3 d-40 c^2 (10 b e+9 a f)-315 b^3 g+14 b c (25 b f+46 a g)\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}+\frac{\left (70 b^4 c f+48 b^2 c^2 (2 c d-5 a f)-32 a c^3 (4 c d-3 a f)-63 b^5 g-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.72651, size = 282, normalized size = 0.82 $\frac{\sqrt{a+x (b+c x)} \left (16 c^2 \left (64 a^2 g-a c (80 e+x (45 f+32 g x))+2 c^2 x (30 d+x (20 e+3 x (5 f+4 g x)))\right )+4 b^2 c (-735 a g+300 c e+7 c x (25 f+18 g x))-8 b c^2 \left (2 c \left (90 d+x \left (50 e+35 f x+27 g x^2\right )\right )-a (275 f+161 g x)\right )-210 b^3 c (5 f+3 g x)+945 b^4 g\right )}{1920 c^5}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (-48 b^2 c^2 (2 c d-5 a f)+40 b^3 c (2 c e-7 a g)+48 a b c^2 (5 a g-4 c e)+32 a c^3 (4 c d-3 a f)-70 b^4 c f+63 b^5 g\right )}{256 c^{11/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[a + x*(b + c*x)]*(945*b^4*g - 210*b^3*c*(5*f + 3*g*x) + 4*b^2*c*(300*c*e - 735*a*g + 7*c*x*(25*f + 18*g*
x)) - 8*b*c^2*(-(a*(275*f + 161*g*x)) + 2*c*(90*d + x*(50*e + 35*f*x + 27*g*x^2))) + 16*c^2*(64*a^2*g - a*c*(8
0*e + x*(45*f + 32*g*x)) + 2*c^2*x*(30*d + x*(20*e + 3*x*(5*f + 4*g*x))))))/(1920*c^5) - ((-70*b^4*c*f - 48*b^
2*c^2*(2*c*d - 5*a*f) + 32*a*c^3*(4*c*d - 3*a*f) + 63*b^5*g + 40*b^3*c*(2*c*e - 7*a*g) + 48*a*b*c^2*(-4*c*e +
5*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(256*c^(11/2))

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Maple [B]  time = 0.058, size = 783, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x)

[Out]

35/96*f*b^2/c^3*x*(c*x^2+b*x+a)^(1/2)+35/32*g*b^3/c^(9/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-15/16*
f*b^2/c^(7/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+55/48*f*b/c^3*a*(c*x^2+b*x+a)^(1/2)-3/8*f*a/c^2*x*
(c*x^2+b*x+a)^(1/2)-7/24*f*b/c^2*x^2*(c*x^2+b*x+a)^(1/2)-9/40*g*b/c^2*x^3*(c*x^2+b*x+a)^(1/2)+21/80*g*b^2/c^3*
x^2*(c*x^2+b*x+a)^(1/2)-21/64*g*b^3/c^4*x*(c*x^2+b*x+a)^(1/2)-5/12*e*b/c^2*x*(c*x^2+b*x+a)^(1/2)+3/4*e*b/c^(5/
2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-49/32*g*b^2/c^4*a*(c*x^2+b*x+a)^(1/2)-15/16*g*b/c^(7/2)*a^2*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-4/15*g*a/c^2*x^2*(c*x^2+b*x+a)^(1/2)+3/8*f*a^2/c^(5/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/2*d*x/c*(c*x^2+b*x+a)^(1/2)-3/4*d*b/c^2*(c*x^2+b*x+a)^(1/2)+3/8*d*b^2/c^(5/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/2*d*a/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/3*
e*x^2/c*(c*x^2+b*x+a)^(1/2)+5/8*e*b^2/c^3*(c*x^2+b*x+a)^(1/2)-5/16*e*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))-2/3*e*a/c^2*(c*x^2+b*x+a)^(1/2)+8/15*g*a^2/c^3*(c*x^2+b*x+a)^(1/2)+63/128*g*b^4/c^5*(c*x^2+b*x+
a)^(1/2)-63/256*g*b^5/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/4*f*x^3/c*(c*x^2+b*x+a)^(1/2)-35/
64*f*b^3/c^4*(c*x^2+b*x+a)^(1/2)+35/128*f*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+161/240*g*b/
c^3*a*x*(c*x^2+b*x+a)^(1/2)+1/5*g*x^4*(c*x^2+b*x+a)^(1/2)/c

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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(x^2*(g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.15884, size = 1666, normalized size = 4.82 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/7680*(15*(32*(3*b^2*c^3 - 4*a*c^4)*d - 16*(5*b^3*c^2 - 12*a*b*c^3)*e + 2*(35*b^4*c - 120*a*b^2*c^2 + 48*a^
2*c^3)*f - (63*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*g)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b
*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(384*c^5*g*x^4 - 1440*b*c^4*d + 48*(10*c^5*f - 9*b*c^4*g)*x^3 + 8*(80
*c^5*e - 70*b*c^4*f + (63*b^2*c^3 - 64*a*c^4)*g)*x^2 + 80*(15*b^2*c^3 - 16*a*c^4)*e - 50*(21*b^3*c^2 - 44*a*b*
c^3)*f + (945*b^4*c - 2940*a*b^2*c^2 + 1024*a^2*c^3)*g + 2*(480*c^5*d - 400*b*c^4*e + 10*(35*b^2*c^3 - 36*a*c^
4)*f - 7*(45*b^3*c^2 - 92*a*b*c^3)*g)*x)*sqrt(c*x^2 + b*x + a))/c^6, -1/3840*(15*(32*(3*b^2*c^3 - 4*a*c^4)*d -
16*(5*b^3*c^2 - 12*a*b*c^3)*e + 2*(35*b^4*c - 120*a*b^2*c^2 + 48*a^2*c^3)*f - (63*b^5 - 280*a*b^3*c + 240*a^2
*b*c^2)*g)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(384*c^
5*g*x^4 - 1440*b*c^4*d + 48*(10*c^5*f - 9*b*c^4*g)*x^3 + 8*(80*c^5*e - 70*b*c^4*f + (63*b^2*c^3 - 64*a*c^4)*g)
*x^2 + 80*(15*b^2*c^3 - 16*a*c^4)*e - 50*(21*b^3*c^2 - 44*a*b*c^3)*f + (945*b^4*c - 2940*a*b^2*c^2 + 1024*a^2*
c^3)*g + 2*(480*c^5*d - 400*b*c^4*e + 10*(35*b^2*c^3 - 36*a*c^4)*f - 7*(45*b^3*c^2 - 92*a*b*c^3)*g)*x)*sqrt(c*
x^2 + b*x + a))/c^6]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(g*x**3+f*x**2+e*x+d)/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.23068, size = 446, normalized size = 1.29 \begin{align*} \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (\frac{8 \, g x}{c} + \frac{10 \, c^{4} f - 9 \, b c^{3} g}{c^{5}}\right )} x - \frac{70 \, b c^{3} f - 63 \, b^{2} c^{2} g + 64 \, a c^{3} g - 80 \, c^{4} e}{c^{5}}\right )} x + \frac{480 \, c^{4} d + 350 \, b^{2} c^{2} f - 360 \, a c^{3} f - 315 \, b^{3} c g + 644 \, a b c^{2} g - 400 \, b c^{3} e}{c^{5}}\right )} x - \frac{1440 \, b c^{3} d + 1050 \, b^{3} c f - 2200 \, a b c^{2} f - 945 \, b^{4} g + 2940 \, a b^{2} c g - 1024 \, a^{2} c^{2} g - 1200 \, b^{2} c^{2} e + 1280 \, a c^{3} e}{c^{5}}\right )} - \frac{{\left (96 \, b^{2} c^{3} d - 128 \, a c^{4} d + 70 \, b^{4} c f - 240 \, a b^{2} c^{2} f + 96 \, a^{2} c^{3} f - 63 \, b^{5} g + 280 \, a b^{3} c g - 240 \, a^{2} b c^{2} g - 80 \, b^{3} c^{2} e + 192 \, a b c^{3} e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{11}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*g*x/c + (10*c^4*f - 9*b*c^3*g)/c^5)*x - (70*b*c^3*f - 63*b^2*c^2*g +
64*a*c^3*g - 80*c^4*e)/c^5)*x + (480*c^4*d + 350*b^2*c^2*f - 360*a*c^3*f - 315*b^3*c*g + 644*a*b*c^2*g - 400*b
*c^3*e)/c^5)*x - (1440*b*c^3*d + 1050*b^3*c*f - 2200*a*b*c^2*f - 945*b^4*g + 2940*a*b^2*c*g - 1024*a^2*c^2*g -
1200*b^2*c^2*e + 1280*a*c^3*e)/c^5) - 1/256*(96*b^2*c^3*d - 128*a*c^4*d + 70*b^4*c*f - 240*a*b^2*c^2*f + 96*a
^2*c^3*f - 63*b^5*g + 280*a*b^3*c*g - 240*a^2*b*c^2*g - 80*b^3*c^2*e + 192*a*b*c^3*e)*log(abs(-2*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)