3.288 \(\int \frac{d+e x+f x^2+g x^3}{x^6 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=371 \[ -\frac{\sqrt{a+b x+c x^2} \left (40 a^2 b (55 c e-36 a g)+256 a^2 c (4 c d-5 a f)-60 a b^2 (49 c d-20 a f)-1050 a b^3 e+945 b^4 d\right )}{1920 a^5 x}+\frac{\sqrt{a+b x+c x^2} \left (120 a^2 (3 c e-4 a g)-350 a b^2 e-4 a b (161 c d-100 a f)+315 b^3 d\right )}{960 a^4 x^2}+\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right ) \left (48 a^2 b^2 (5 c e-2 a g)+48 a^2 b c (5 c d-4 a f)-32 a^3 c (3 c e-4 a g)-40 a b^3 (7 c d-2 a f)-70 a b^4 e+63 b^5 d\right )}{256 a^{11/2}}-\frac{\sqrt{a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{240 a^3 x^3}+\frac{\sqrt{a+b x+c x^2} (9 b d-10 a e)}{40 a^2 x^4}-\frac{d \sqrt{a+b x+c x^2}}{5 a x^5} \]

[Out]

-(d*Sqrt[a + b*x + c*x^2])/(5*a*x^5) + ((9*b*d - 10*a*e)*Sqrt[a + b*x + c*x^2])/(40*a^2*x^4) - ((63*b^2*d - 64
*a*c*d - 70*a*b*e + 80*a^2*f)*Sqrt[a + b*x + c*x^2])/(240*a^3*x^3) + ((315*b^3*d - 350*a*b^2*e - 4*a*b*(161*c*
d - 100*a*f) + 120*a^2*(3*c*e - 4*a*g))*Sqrt[a + b*x + c*x^2])/(960*a^4*x^2) - ((945*b^4*d - 1050*a*b^3*e - 60
*a*b^2*(49*c*d - 20*a*f) + 256*a^2*c*(4*c*d - 5*a*f) + 40*a^2*b*(55*c*e - 36*a*g))*Sqrt[a + b*x + c*x^2])/(192
0*a^5*x) + ((63*b^5*d - 70*a*b^4*e + 48*a^2*b*c*(5*c*d - 4*a*f) - 40*a*b^3*(7*c*d - 2*a*f) - 32*a^3*c*(3*c*e -
4*a*g) + 48*a^2*b^2*(5*c*e - 2*a*g))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(256*a^(11/2))

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Rubi [A]  time = 0.81717, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {1650, 834, 806, 724, 206} \[ -\frac{\sqrt{a+b x+c x^2} \left (40 a^2 b (55 c e-36 a g)+256 a^2 c (4 c d-5 a f)-60 a b^2 (49 c d-20 a f)-1050 a b^3 e+945 b^4 d\right )}{1920 a^5 x}+\frac{\sqrt{a+b x+c x^2} \left (120 a^2 (3 c e-4 a g)-350 a b^2 e-4 a b (161 c d-100 a f)+315 b^3 d\right )}{960 a^4 x^2}+\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right ) \left (48 a^2 b^2 (5 c e-2 a g)+48 a^2 b c (5 c d-4 a f)-32 a^3 c (3 c e-4 a g)-40 a b^3 (7 c d-2 a f)-70 a b^4 e+63 b^5 d\right )}{256 a^{11/2}}-\frac{\sqrt{a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{240 a^3 x^3}+\frac{\sqrt{a+b x+c x^2} (9 b d-10 a e)}{40 a^2 x^4}-\frac{d \sqrt{a+b x+c x^2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2 + g*x^3)/(x^6*Sqrt[a + b*x + c*x^2]),x]

[Out]

-(d*Sqrt[a + b*x + c*x^2])/(5*a*x^5) + ((9*b*d - 10*a*e)*Sqrt[a + b*x + c*x^2])/(40*a^2*x^4) - ((63*b^2*d - 64
*a*c*d - 70*a*b*e + 80*a^2*f)*Sqrt[a + b*x + c*x^2])/(240*a^3*x^3) + ((315*b^3*d - 350*a*b^2*e - 4*a*b*(161*c*
d - 100*a*f) + 120*a^2*(3*c*e - 4*a*g))*Sqrt[a + b*x + c*x^2])/(960*a^4*x^2) - ((945*b^4*d - 1050*a*b^3*e - 60
*a*b^2*(49*c*d - 20*a*f) + 256*a^2*c*(4*c*d - 5*a*f) + 40*a^2*b*(55*c*e - 36*a*g))*Sqrt[a + b*x + c*x^2])/(192
0*a^5*x) + ((63*b^5*d - 70*a*b^4*e + 48*a^2*b*c*(5*c*d - 4*a*f) - 40*a*b^3*(7*c*d - 2*a*f) - 32*a^3*c*(3*c*e -
4*a*g) + 48*a^2*b^2*(5*c*e - 2*a*g))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(256*a^(11/2))

Rule 1650

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

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

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

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 x^3}{x^6 \sqrt{a+b x+c x^2}} \, dx &=-\frac{d \sqrt{a+b x+c x^2}}{5 a x^5}-\frac{\int \frac{\frac{1}{2} (9 b d-10 a e)+(4 c d-5 a f) x-5 a g x^2}{x^5 \sqrt{a+b x+c x^2}} \, dx}{5 a}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 b d-10 a e) \sqrt{a+b x+c x^2}}{40 a^2 x^4}+\frac{\int \frac{\frac{1}{4} \left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right )+\frac{1}{2} \left (27 b c d-30 a c e+40 a^2 g\right ) x}{x^4 \sqrt{a+b x+c x^2}} \, dx}{20 a^2}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 b d-10 a e) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt{a+b x+c x^2}}{240 a^3 x^3}-\frac{\int \frac{\frac{1}{8} \left (315 b^3 d-644 a b c d-350 a b^2 e+360 a^2 c e+400 a^2 b f-480 a^3 g\right )+\frac{1}{2} c \left (63 b^2 d-70 a b e-16 a (4 c d-5 a f)\right ) x}{x^3 \sqrt{a+b x+c x^2}} \, dx}{60 a^3}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 b d-10 a e) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt{a+b x+c x^2}}{240 a^3 x^3}+\frac{\left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) \sqrt{a+b x+c x^2}}{960 a^4 x^2}+\frac{\int \frac{\frac{1}{16} \left (945 b^4 d-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+256 a^2 c (4 c d-5 a f)+40 a^2 b (55 c e-36 a g)\right )+\frac{1}{8} c \left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{120 a^4}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 b d-10 a e) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt{a+b x+c x^2}}{240 a^3 x^3}+\frac{\left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) \sqrt{a+b x+c x^2}}{960 a^4 x^2}-\frac{\left (945 b^4 d-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+256 a^2 c (4 c d-5 a f)+40 a^2 b (55 c e-36 a g)\right ) \sqrt{a+b x+c x^2}}{1920 a^5 x}-\frac{\left (63 b^5 d-70 a b^4 e+48 a^2 b c (5 c d-4 a f)-40 a b^3 (7 c d-2 a f)-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{256 a^5}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 b d-10 a e) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt{a+b x+c x^2}}{240 a^3 x^3}+\frac{\left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) \sqrt{a+b x+c x^2}}{960 a^4 x^2}-\frac{\left (945 b^4 d-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+256 a^2 c (4 c d-5 a f)+40 a^2 b (55 c e-36 a g)\right ) \sqrt{a+b x+c x^2}}{1920 a^5 x}+\frac{\left (63 b^5 d-70 a b^4 e+48 a^2 b c (5 c d-4 a f)-40 a b^3 (7 c d-2 a f)-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x}{\sqrt{a+b x+c x^2}}\right )}{128 a^5}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 b d-10 a e) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt{a+b x+c x^2}}{240 a^3 x^3}+\frac{\left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) \sqrt{a+b x+c x^2}}{960 a^4 x^2}-\frac{\left (945 b^4 d-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+256 a^2 c (4 c d-5 a f)+40 a^2 b (55 c e-36 a g)\right ) \sqrt{a+b x+c x^2}}{1920 a^5 x}+\frac{\left (63 b^5 d-70 a b^4 e+48 a^2 b c (5 c d-4 a f)-40 a b^3 (7 c d-2 a f)-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.78017, size = 299, normalized size = 0.81 \[ \frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right ) \left (-48 a^2 b^2 (2 a g-5 c e)-48 a^2 b c (4 a f-5 c d)+32 a^3 c (4 a g-3 c e)+40 a b^3 (2 a f-7 c d)-70 a b^4 e+63 b^5 d\right )}{256 a^{11/2}}-\frac{\sqrt{a+x (b+c x)} \left (4 a^2 x^2 \left (b^2 (126 d+25 x (7 e+12 f x))+2 b c x (161 d+275 e x)+256 c^2 d x^2\right )-16 a^3 x (b (27 d+5 x (7 e+2 x (5 f+9 g x)))+c x (32 d+5 x (9 e+16 f x)))+32 a^4 \left (12 d+5 x \left (3 e+4 f x+6 g x^2\right )\right )-210 a b^2 x^3 (3 b d+5 b e x+14 c d x)+945 b^4 d x^4\right )}{1920 a^5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2 + g*x^3)/(x^6*Sqrt[a + b*x + c*x^2]),x]

[Out]

-(Sqrt[a + x*(b + c*x)]*(945*b^4*d*x^4 - 210*a*b^2*x^3*(3*b*d + 14*c*d*x + 5*b*e*x) + 32*a^4*(12*d + 5*x*(3*e
+ 4*f*x + 6*g*x^2)) + 4*a^2*x^2*(256*c^2*d*x^2 + 2*b*c*x*(161*d + 275*e*x) + b^2*(126*d + 25*x*(7*e + 12*f*x))
) - 16*a^3*x*(c*x*(32*d + 5*x*(9*e + 16*f*x)) + b*(27*d + 5*x*(7*e + 2*x*(5*f + 9*g*x))))))/(1920*a^5*x^5) + (
(63*b^5*d - 70*a*b^4*e + 40*a*b^3*(-7*c*d + 2*a*f) - 48*a^2*b*c*(-5*c*d + 4*a*f) - 48*a^2*b^2*(-5*c*e + 2*a*g)
+ 32*a^3*c*(-3*c*e + 4*a*g))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(256*a^(11/2))

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Maple [B]  time = 0.062, size = 859, 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((g*x^3+f*x^2+e*x+d)/x^6/(c*x^2+b*x+a)^(1/2),x)

[Out]

7/24*e*b/a^2/x^3*(c*x^2+b*x+a)^(1/2)+3/4*g*b/a^2/x*(c*x^2+b*x+a)^(1/2)+5/12*f*b/a^2/x^2*(c*x^2+b*x+a)^(1/2)-5/
8*f*b^2/a^3/x*(c*x^2+b*x+a)^(1/2)-3/4*f*b/a^(5/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+2/3*f*c/a^2/
x*(c*x^2+b*x+a)^(1/2)+21/64*d*b^3/a^4/x^2*(c*x^2+b*x+a)^(1/2)+9/40*d*b/a^2/x^4*(c*x^2+b*x+a)^(1/2)-21/80*d*b^2
/a^3/x^3*(c*x^2+b*x+a)^(1/2)+15/16*e*b^2/a^(7/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-35/96*e*b^2/a
^3/x^2*(c*x^2+b*x+a)^(1/2)+63/256*d*b^5/a^(11/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/4*e/a/x^4*(c*
x^2+b*x+a)^(1/2)-35/128*e*b^4/a^(9/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-3/8*e*c^2/a^(5/2)*ln((2*a+
b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/2*g/a/x^2*(c*x^2+b*x+a)^(1/2)-3/8*g*b^2/a^(5/2)*ln((2*a+b*x+2*a^(1/2)*
(c*x^2+b*x+a)^(1/2))/x)+1/2*g*c/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/3*f/a/x^3*(c*x^2+b*x+a
)^(1/2)+5/16*f*b^3/a^(7/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+3/8*e*c/a^2/x^2*(c*x^2+b*x+a)^(1/2)+3
5/64*e*b^3/a^4/x*(c*x^2+b*x+a)^(1/2)-63/128*d*b^4/a^5/x*(c*x^2+b*x+a)^(1/2)-35/32*d*b^3/a^(9/2)*c*ln((2*a+b*x+
2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+15/16*d*b/a^(7/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+4/15*d*c
/a^2/x^3*(c*x^2+b*x+a)^(1/2)-8/15*d*c^2/a^3/x*(c*x^2+b*x+a)^(1/2)-1/5*d*(c*x^2+b*x+a)^(1/2)/a/x^5-55/48*e*b/a^
3*c/x*(c*x^2+b*x+a)^(1/2)+49/32*d*b^2/a^4*c/x*(c*x^2+b*x+a)^(1/2)-161/240*d*b/a^3*c/x^2*(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((g*x^3+f*x^2+e*x+d)/x^6/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 115.592, size = 1708, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f*x^2+e*x+d)/x^6/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x + f x^{2} + g x^{3}}{x^{6} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**3+f*x**2+e*x+d)/x**6/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((d + e*x + f*x**2 + g*x**3)/(x**6*sqrt(a + b*x + c*x**2)), x)

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Giac [B]  time = 1.26123, size = 2939, normalized size = 7.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f*x^2+e*x+d)/x^6/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/128*(63*b^5*d - 280*a*b^3*c*d + 240*a^2*b*c^2*d + 80*a^2*b^3*f - 192*a^3*b*c*f - 96*a^3*b^2*g + 128*a^4*c*g
- 70*a*b^4*e + 240*a^2*b^2*c*e - 96*a^3*c^2*e)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a
)*a^5) + 1/1920*(945*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^5*d - 4200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*
a*b^3*c*d + 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b*c^2*d + 1200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
9*a^2*b^3*f - 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*b*c*f - 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
9*a^3*b^2*g + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^4*c*g - 1050*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*
a*b^4*e + 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b^2*c*e - 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*
a^3*c^2*e - 4410*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^5*d + 19600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a
^2*b^3*c*d - 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b*c^2*d - 5600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^7*a^3*b^3*f + 13440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^4*b*c*f + 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^7*a^4*b^2*g - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^5*c*g + 4900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^7*a^2*b^4*e - 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b^2*c*e + 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^7*a^4*c^2*e + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^5*c^(3/2)*f + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^6*a^5*b*sqrt(c)*g + 8064*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^5*d - 35840*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*a^3*b^3*c*d + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^4*b*c^2*d + 10240*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^5*a^4*b^3*f - 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^5*b*c*f - 11520*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*a^5*b^2*g - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^4*e + 30720*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*a^4*b^2*c*e + 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^5*c^(5/2)*d + 3840*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^4*a^5*b^2*sqrt(c)*f - 17920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^6*c^(3/2)*f
- 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^6*b*sqrt(c)*g + 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a
^5*b*c^(3/2)*e - 7110*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^5*d + 31600*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^3*a^4*b^3*c*d + 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*b*c^2*d - 8480*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^3*a^5*b^3*f + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^6*b*c*f + 8640*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^3*a^6*b^2*g + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^7*c*g + 7900*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*a^4*b^4*e - 13920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*b^2*c*e - 6720*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^3*a^6*c^2*e + 38400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^2*c^(3/2)*d - 10240*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^2*a^6*c^(5/2)*d - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^6*b^2*sqrt(c)*f + 12800*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^7*c^(3/2)*f + 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^7*b*sqrt(c)*g
+ 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^3*sqrt(c)*e - 25600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
a^6*b*c^(3/2)*e + 2895*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^5*d + 4200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*a^5*b^3*c*d - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b*c^2*d + 2640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*a^6*b^3*f + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^7*b*c*f - 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^
7*b^2*g - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^8*c*g - 2790*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*b^4*
e - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b^2*c*e + 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^7*c^2*e
+ 3840*a^5*b^4*sqrt(c)*d - 7680*a^6*b^2*c^(3/2)*d + 2048*a^7*c^(5/2)*d + 3840*a^7*b^2*sqrt(c)*f - 2560*a^8*c^(
3/2)*f - 3840*a^8*b*sqrt(c)*g - 3840*a^6*b^3*sqrt(c)*e + 5120*a^7*b*c^(3/2)*e)/(((sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2 - a)^5*a^5)