∫ (a+bx+cx2)3∕2 ______________________

3.21.21 $\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^6} \, dx$

Optimal. Leaf size=296 \[ -\frac {3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{16 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2} \]

________________________________________________________________________________________

Rubi [A] time = 0.23, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.182, Rules used = {730, 720, 724, 206} \begin {gather*} -\frac {3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{16 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(128*(c*d^2 - b*d*e + a

*e^2)^3*(d + e*x)^2) + ((2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(16*(c*d^2 - b*

d*e + a*e^2)^2*(d + e*x)^4) - (e*(a + b*x + c*x^2)^(5/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) + (3*(b^2 -

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

*x^2])])/(256*(c*d^2 - b*d*e + a*e^2)^(7/2))

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 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*

(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -

4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[

{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +

2*p + 2, 0] && GtQ[p, 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 730

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(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[(2*c*d - b*e)/(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, m, p}, x] && NeQ[b^2 - 4*a*c,

0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx &=-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {(2 c d-b e) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {\left (3 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx}{32 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{256 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{128 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A] time = 1.10, size = 275, normalized size = 0.93 \begin {gather*} -\frac {(2 c d-b e) \left (3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac {\sqrt {a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}\right )+\frac {2 (a+x (b+c x))^{3/2} (2 a e-b d+b e x-2 c d x)}{(d+e x)^4}\right )}{32 \left (e (a e-b d)+c d^2\right )^2}-\frac {e (a+x (b+c x))^{5/2}}{5 (d+e x)^5 \left (e (a e-b d)+c d^2\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/5*(e*(a + x*(b + c*x))^(5/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^5) - ((2*c*d - b*e)*((2*(-(b*d) + 2*a*e

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

*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2

*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)

))))/(32*(c*d^2 + e*(-(b*d) + a*e))^2)

________________________________________________________________________________________

IntegrateAlgebraic [F] time = 180.03, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

$Aborted

________________________________________________________________________________________

fricas [B] time = 75.94, size = 4830, normalized size = 16.32

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2560*(15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^5*e + (2*(b^4*c -

8*a*b^2*c^2 + 16*a^2*c^3)*d*e^5 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^5 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a

^2*c^3)*d^2*e^4 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*e^5)*x^4 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3*e

^3 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2*e^4)*x^3 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*e^2 - (b^5 -

8*a*b^3*c + 16*a^2*b*c^2)*d^3*e^3)*x^2 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^5*e - (b^5 - 8*a*b^3*c + 1

6*a^2*b*c^2)*d^4*e^2)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d

^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (

2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 4*(464*a^4*b

*d*e^6 - 128*a^5*e^7 - 10*(3*b^3*c^2 - 20*a*b*c^3)*d^7 + (45*b^4*c - 320*a*b^2*c^2 - 528*a^2*c^3)*d^6*e - (15*

b^5 - 100*a*b^3*c - 1424*a^2*b*c^2)*d^5*e^2 + (5*a*b^4 - 1064*a^2*b^2*c - 944*a^3*c^2)*d^4*e^3 + 2*(129*a^2*b^

3 + 724*a^3*b*c)*d^3*e^4 - 8*(73*a^3*b^2 + 68*a^4*c)*d^2*e^5 + (32*c^5*d^6*e - 96*b*c^4*d^5*e^2 + 4*(19*b^2*c^

3 + 44*a*c^4)*d^4*e^3 + 8*(b^3*c^2 - 44*a*b*c^3)*d^3*e^4 - (35*b^4*c - 256*a*b^2*c^2 - 16*a^2*c^3)*d^2*e^5 + (

15*b^5 - 80*a*b^3*c - 16*a^2*b*c^2)*d*e^6 - (15*a*b^4 - 100*a^2*b^2*c + 128*a^3*c^2)*e^7)*x^4 + 2*(80*c^5*d^7

- 248*b*c^4*d^6*e + 2*(107*b^2*c^3 + 220*a*c^4)*d^5*e^2 + (b^3*c^2 - 924*a*b*c^3)*d^4*e^3 - 2*(41*b^4*c - 334*

a*b^2*c^2 - 80*a^2*c^3)*d^3*e^4 + (35*b^5 - 174*a*b^3*c - 224*a^2*b*c^2)*d^2*e^5 - 2*(20*a*b^4 - 127*a^2*b^2*c

+ 100*a^3*c^2)*d*e^6 + (5*a^2*b^3 - 28*a^3*b*c)*e^7)*x^3 + 2*(120*b*c^4*d^7 - 2*(227*b^2*c^3 - 116*a*c^4)*d^6

*e + 9*(63*b^3*c^2 - 4*a*b*c^3)*d^5*e^2 - 3*(99*b^4*c + 152*a*b^2*c^2 + 128*a^2*c^3)*d^4*e^3 + 2*(32*b^5 + 151

*a*b^3*c + 504*a^2*b*c^2)*d^3*e^4 - 3*(29*a*b^4 + 122*a^2*b^2*c + 248*a^3*c^2)*d^2*e^5 + 3*(9*a^2*b^3 + 148*a^

3*b*c)*d*e^6 - 4*(a^3*b^2 + 32*a^4*c)*e^7)*x^2 - 2*(88*a^4*b*e^7 - 10*(b^2*c^3 + 20*a*c^4)*d^7 + (85*b^3*c^2 +

468*a*b*c^3)*d^6*e - 2*(55*b^4*c + 382*a*b^2*c^2 - 80*a^2*c^3)*d^5*e^2 + (35*b^5 + 794*a*b^3*c - 16*a^2*b*c^2

)*d^4*e^3 - 2*(134*a*b^4 + 409*a^2*b^2*c - 220*a^3*c^2)*d^3*e^4 + 3*(163*a^2*b^3 + 28*a^3*b*c)*d^2*e^5 - 8*(43

*a^3*b^2 - 10*a^4*c)*d*e^6)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^13 - 4*b*c^3*d^12*e - 4*a^3*b*d^6*e^7 + a^4*d^5*e

^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^11*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^10*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^9*e^

4 - 4*(a*b^3 + 3*a^2*b*c)*d^8*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^7*e^6 + (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 - 4*a^3*b

*d*e^12 + a^4*e^13 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^7 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^8 + (b^4 + 12*a*b^2*c + 6*a

^2*c^2)*d^4*e^9 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^10 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^11)*x^5 + 5*(c^4*d^9*e^4 - 4*

b*c^3*d^8*e^5 - 4*a^3*b*d^2*e^11 + a^4*d*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^6 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^

7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^9 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^10)

*x^4 + 10*(c^4*d^10*e^3 - 4*b*c^3*d^9*e^4 - 4*a^3*b*d^3*e^10 + a^4*d^2*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^5

- 4*(b^3*c + 3*a*b*c^2)*d^7*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^8 + 2*(

3*a^2*b^2 + 2*a^3*c)*d^4*e^9)*x^3 + 10*(c^4*d^11*e^2 - 4*b*c^3*d^10*e^3 - 4*a^3*b*d^4*e^9 + a^4*d^3*e^10 + 2*(

3*b^2*c^2 + 2*a*c^3)*d^9*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^6 - 4*(a*b

^3 + 3*a^2*b*c)*d^6*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^8)*x^2 + 5*(c^4*d^12*e - 4*b*c^3*d^11*e^2 - 4*a^3*b*d^

5*e^8 + a^4*d^4*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^10*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^9*e^4 + (b^4 + 12*a*b^2*c + 6

*a^2*c^2)*d^8*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^7*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^7)*x), 1/1280*(15*(2*(b^4*c

- 8*a*b^2*c^2 + 16*a^2*c^3)*d^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^5*e + (2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^

3)*d*e^5 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^5 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2*e^4 - (b^5

- 8*a*b^3*c + 16*a^2*b*c^2)*d*e^5)*x^4 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3*e^3 - (b^5 - 8*a*b^3*c +

16*a^2*b*c^2)*d^2*e^4)*x^3 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*e^2 - (b^5 - 8*a*b^3*c + 16*a^2*b*c

^2)*d^3*e^3)*x^2 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^5*e - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^4*e^2)*x

)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (

2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^

2)*x)) + 2*(464*a^4*b*d*e^6 - 128*a^5*e^7 - 10*(3*b^3*c^2 - 20*a*b*c^3)*d^7 + (45*b^4*c - 320*a*b^2*c^2 - 528*

a^2*c^3)*d^6*e - (15*b^5 - 100*a*b^3*c - 1424*a^2*b*c^2)*d^5*e^2 + (5*a*b^4 - 1064*a^2*b^2*c - 944*a^3*c^2)*d^

4*e^3 + 2*(129*a^2*b^3 + 724*a^3*b*c)*d^3*e^4 - 8*(73*a^3*b^2 + 68*a^4*c)*d^2*e^5 + (32*c^5*d^6*e - 96*b*c^4*d

^5*e^2 + 4*(19*b^2*c^3 + 44*a*c^4)*d^4*e^3 + 8*(b^3*c^2 - 44*a*b*c^3)*d^3*e^4 - (35*b^4*c - 256*a*b^2*c^2 - 16

*a^2*c^3)*d^2*e^5 + (15*b^5 - 80*a*b^3*c - 16*a^2*b*c^2)*d*e^6 - (15*a*b^4 - 100*a^2*b^2*c + 128*a^3*c^2)*e^7)

*x^4 + 2*(80*c^5*d^7 - 248*b*c^4*d^6*e + 2*(107*b^2*c^3 + 220*a*c^4)*d^5*e^2 + (b^3*c^2 - 924*a*b*c^3)*d^4*e^3

- 2*(41*b^4*c - 334*a*b^2*c^2 - 80*a^2*c^3)*d^3*e^4 + (35*b^5 - 174*a*b^3*c - 224*a^2*b*c^2)*d^2*e^5 - 2*(20*

a*b^4 - 127*a^2*b^2*c + 100*a^3*c^2)*d*e^6 + (5*a^2*b^3 - 28*a^3*b*c)*e^7)*x^3 + 2*(120*b*c^4*d^7 - 2*(227*b^2

*c^3 - 116*a*c^4)*d^6*e + 9*(63*b^3*c^2 - 4*a*b*c^3)*d^5*e^2 - 3*(99*b^4*c + 152*a*b^2*c^2 + 128*a^2*c^3)*d^4*

e^3 + 2*(32*b^5 + 151*a*b^3*c + 504*a^2*b*c^2)*d^3*e^4 - 3*(29*a*b^4 + 122*a^2*b^2*c + 248*a^3*c^2)*d^2*e^5 +

3*(9*a^2*b^3 + 148*a^3*b*c)*d*e^6 - 4*(a^3*b^2 + 32*a^4*c)*e^7)*x^2 - 2*(88*a^4*b*e^7 - 10*(b^2*c^3 + 20*a*c^4

)*d^7 + (85*b^3*c^2 + 468*a*b*c^3)*d^6*e - 2*(55*b^4*c + 382*a*b^2*c^2 - 80*a^2*c^3)*d^5*e^2 + (35*b^5 + 794*a

*b^3*c - 16*a^2*b*c^2)*d^4*e^3 - 2*(134*a*b^4 + 409*a^2*b^2*c - 220*a^3*c^2)*d^3*e^4 + 3*(163*a^2*b^3 + 28*a^3

*b*c)*d^2*e^5 - 8*(43*a^3*b^2 - 10*a^4*c)*d*e^6)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^13 - 4*b*c^3*d^12*e - 4*a^3*

b*d^6*e^7 + a^4*d^5*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^11*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^10*e^3 + (b^4 + 12*a*b^2*

c + 6*a^2*c^2)*d^9*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^8*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^7*e^6 + (c^4*d^8*e^5 - 4*b*

c^3*d^7*e^6 - 4*a^3*b*d*e^12 + a^4*e^13 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^7 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^8 + (b

^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^9 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^10 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^11)*x^5

+ 5*(c^4*d^9*e^4 - 4*b*c^3*d^8*e^5 - 4*a^3*b*d^2*e^11 + a^4*d*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^6 - 4*(b^3*

c + 3*a*b*c^2)*d^6*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^9 + 2*(3*a^2*b^2

+ 2*a^3*c)*d^3*e^10)*x^4 + 10*(c^4*d^10*e^3 - 4*b*c^3*d^9*e^4 - 4*a^3*b*d^3*e^10 + a^4*d^2*e^11 + 2*(3*b^2*c^

2 + 2*a*c^3)*d^8*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^7 - 4*(a*b^3 + 3*a

^2*b*c)*d^5*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^9)*x^3 + 10*(c^4*d^11*e^2 - 4*b*c^3*d^10*e^3 - 4*a^3*b*d^4*e^9

+ a^4*d^3*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*

c^2)*d^7*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^8)*x^2 + 5*(c^4*d^12*e - 4*b*c^3*

d^11*e^2 - 4*a^3*b*d^5*e^8 + a^4*d^4*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^10*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^9*e^4 +

(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^8*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^7*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^7)*x)]

________________________________________________________________________________________

giac [B] time = 3.69, size = 8184, normalized size = 27.65

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/128*(2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2*b*c^2*e)*arctan(-((sqrt(c)*x -

sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e

^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3

*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/640*(2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*c^(13/2)*d^8*e + 1024*

(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^7*d^9 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*c^6*d^7*e^2 + 3072*

(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^6*d^8*e + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(13/2)*d^9

+ 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*c^(11/2)*d^6*e^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b*c

^(11/2)*d^7*e^2 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(11/2)*d^8*e - 2560*(sqrt(c)*x - sqrt(c*x^2

+ b*x + a))^4*a*c^(13/2)*d^8*e + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^6*d^9 - 3840*(sqrt(c)*x - s

qrt(c*x^2 + b*x + a))^7*b*c^5*d^6*e^3 - 7936*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^5*d^7*e^2 - 512*(sqrt

(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^6*d^7*e^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c^5*d^8*e - 51

20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^6*d^8*e + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(11/

2)*d^9 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b*c^(9/2)*d^5*e^4 - 6400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a

))^6*b^2*c^(9/2)*d^6*e^3 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*c^(11/2)*d^6*e^3 - 6400*(sqrt(c)*x - s

qrt(c*x^2 + b*x + a))^4*b^3*c^(9/2)*d^7*e^2 + 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(11/2)*d^7*e^2

- 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*c^(9/2)*d^8*e - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*

b^2*c^(11/2)*d^8*e + 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c^5*d^9 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x

+ a))^7*b^2*c^4*d^5*e^4 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*c^5*d^5*e^4 - 1280*(sqrt(c)*x - sqrt(c*

x^2 + b*x + a))^5*b^3*c^4*d^6*e^3 + 24832*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c^5*d^6*e^3 - 1600*(sqrt(c

)*x - sqrt(c*x^2 + b*x + a))^3*b^4*c^4*d^7*e^2 + 17920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c^5*d^7*e^2

+ 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^6*d^7*e^2 - 640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*c^

4*d^8*e - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c^5*d^8*e + 32*b^5*c^(9/2)*d^9 + 3840*(sqrt(c)*x - sq

rt(c*x^2 + b*x + a))^8*b^2*c^(7/2)*d^4*e^5 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^(9/2)*d^4*e^5 + 12

80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^3*c^(7/2)*d^5*e^4 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b*

c^(9/2)*d^5*e^4 + 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*c^(7/2)*d^6*e^3 + 19200*(sqrt(c)*x - sqrt(c*x

^2 + b*x + a))^4*a*b^2*c^(9/2)*d^6*e^3 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(11/2)*d^6*e^3 + 160

*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^5*c^(7/2)*d^7*e^2 + 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3

*c^(9/2)*d^7*e^2 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^(11/2)*d^7*e^2 - 64*b^6*c^(7/2)*d^8*e -

160*a*b^4*c^(9/2)*d^8*e + 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^3*c^3*d^4*e^5 - 3840*(sqrt(c)*x - sqrt(

c*x^2 + b*x + a))^7*a*b*c^4*d^4*e^5 + 4280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^4*c^3*d^5*e^4 - 18880*(sqrt

(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c^4*d^5*e^4 - 25216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^5*d^5*e

^4 + 2080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c^3*d^6*e^3 - 24320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*

a^2*b*c^5*d^6*e^3 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^6*c^3*d^7*e^2 + 4000*(sqrt(c)*x - sqrt(c*x^2 + b

*x + a))*a*b^4*c^4*d^7*e^2 + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c^5*d^7*e^2 - 1280*(sqrt(c)*x -

sqrt(c*x^2 + b*x + a))^8*b^3*c^(5/2)*d^3*e^6 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b*c^(7/2)*d^3*e^6

+ 7420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^4*c^(5/2)*d^4*e^5 - 3040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*

a*b^2*c^(7/2)*d^4*e^5 - 16960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^(9/2)*d^4*e^5 + 2860*(sqrt(c)*x - sq

rt(c*x^2 + b*x + a))^4*b^5*c^(5/2)*d^5*e^4 - 21600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^3*c^(7/2)*d^5*e^4

- 40000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b*c^(9/2)*d^5*e^4 + 860*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))

^2*b^6*c^(5/2)*d^6*e^3 - 5200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^(7/2)*d^6*e^3 - 24000*(sqrt(c)*x -

sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^(9/2)*d^6*e^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(11/2)*d^

6*e^3 + 12*b^7*c^(5/2)*d^7*e^2 + 464*a*b^5*c^(7/2)*d^7*e^2 + 320*a^2*b^3*c^(9/2)*d^7*e^2 - 4780*(sqrt(c)*x - s

qrt(c*x^2 + b*x + a))^7*b^4*c^2*d^3*e^6 - 7840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^2*c^3*d^3*e^6 - 7360*

(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*c^4*d^3*e^6 + 1448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^5*c^2*d^4

*e^5 + 8640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^3*c^3*d^4*e^5 + 12160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)

)^5*a^2*b*c^4*d^4*e^5 + 540*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^6*c^2*d^5*e^4 - 7520*(sqrt(c)*x - sqrt(c*x

^2 + b*x + a))^3*a*b^4*c^3*d^5*e^4 - 13120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*c^4*d^5*e^4 + 12800*(

sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*c^5*d^5*e^4 + 200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^7*c^2*d^6*e^3

- 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c^3*d^6*e^3 - 9600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b

^3*c^4*d^6*e^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c^5*d^6*e^3 - 270*(sqrt(c)*x - sqrt(c*x^2 + b*

x + a))^8*b^4*c^(3/2)*d^2*e^7 + 6000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^2*c^(5/2)*d^2*e^7 - 480*(sqrt(c

)*x - sqrt(c*x^2 + b*x + a))^8*a^2*c^(7/2)*d^2*e^7 - 5330*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^5*c^(3/2)*d^

3*e^6 - 9840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^3*c^(5/2)*d^3*e^6 + 8160*(sqrt(c)*x - sqrt(c*x^2 + b*x

+ a))^6*a^2*b*c^(7/2)*d^3*e^6 - 1390*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^6*c^(3/2)*d^4*e^5 + 9620*(sqrt(c)

*x - sqrt(c*x^2 + b*x + a))^4*a*b^4*c^(5/2)*d^4*e^5 + 37120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^2*c^(7

/2)*d^4*e^5 + 37440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*c^(9/2)*d^4*e^5 - 230*(sqrt(c)*x - sqrt(c*x^2 +

b*x + a))^2*b^7*c^(3/2)*d^5*e^4 - 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^5*c^(5/2)*d^5*e^4 + 5920*(sqrt

(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^3*c^(7/2)*d^5*e^4 + 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b*c

^(9/2)*d^5*e^4 + 20*b^8*c^(3/2)*d^6*e^3 - 204*a*b^6*c^(5/2)*d^6*e^3 - 1360*a^2*b^4*c^(7/2)*d^6*e^3 - 320*a^3*b

^2*c^(9/2)*d^6*e^3 - 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^4*c*d*e^8 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x

+ a))^9*a*b^2*c^2*d*e^8 - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*c^3*d*e^8 + 330*(sqrt(c)*x - sqrt(c*x^

2 + b*x + a))^7*b^5*c*d^2*e^7 + 8880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^3*c^2*d^2*e^7 + 9120*(sqrt(c)*x

- sqrt(c*x^2 + b*x + a))^7*a^2*b*c^3*d^2*e^7 - 2626*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^6*c*d^3*e^6 - 626

0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^4*c^2*d^3*e^6 - 21760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^

2*c^3*d^3*e^6 + 29120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*c^4*d^3*e^6 - 930*(sqrt(c)*x - sqrt(c*x^2 + b*

x + a))^3*b^7*c*d^4*e^5 + 4760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^5*c^2*d^4*e^5 + 14240*(sqrt(c)*x - sq

rt(c*x^2 + b*x + a))^3*a^2*b^3*c^3*d^4*e^5 + 42880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b*c^4*d^4*e^5 - 1

20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^8*c*d^5*e^4 + 260*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^6*c^2*d^5*e

^4 + 3880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^4*c^3*d^5*e^4 + 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*

a^3*b^2*c^4*d^5*e^4 + 640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*c^5*d^5*e^4 + 135*(sqrt(c)*x - sqrt(c*x^2 +

b*x + a))^8*b^5*sqrt(c)*d*e^8 - 1080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^3*c^(3/2)*d*e^8 - 1680*(sqrt(c)

*x - sqrt(c*x^2 + b*x + a))^8*a^2*b*c^(5/2)*d*e^8 + 490*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^6*sqrt(c)*d^2*

e^7 + 12420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^4*c^(3/2)*d^2*e^7 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x +

a))^6*a^2*b^2*c^(5/2)*d^2*e^7 + 18240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*c^(7/2)*d^2*e^7 - 640*(sqrt(c

)*x - sqrt(c*x^2 + b*x + a))^4*b^7*sqrt(c)*d^3*e^6 + 1390*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^5*c^(3/2)*

d^3*e^6 - 41840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^3*c^(5/2)*d^3*e^6 - 2080*(sqrt(c)*x - sqrt(c*x^2 +

b*x + a))^4*a^3*b*c^(7/2)*d^3*e^6 - 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^8*sqrt(c)*d^4*e^5 + 1720*(sqr

t(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^6*c^(3/2)*d^4*e^5 - 2380*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^4*c

^(5/2)*d^4*e^5 + 10720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^2*c^(7/2)*d^4*e^5 - 7360*(sqrt(c)*x - sqrt(

c*x^2 + b*x + a))^2*a^4*c^(9/2)*d^4*e^5 - 15*b^9*sqrt(c)*d^5*e^4 + 30*a*b^7*c^(3/2)*d^5*e^4 + 532*a^2*b^5*c^(5

/2)*d^5*e^4 + 2240*a^3*b^3*c^(7/2)*d^5*e^4 + 320*a^4*b*c^(9/2)*d^5*e^4 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)

)^9*b^5*e^9 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c*e^9 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^

9*a^2*b*c^2*e^9 + 70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^6*d*e^8 - 420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))

^7*a*b^4*c*d*e^8 - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^2*c^2*d*e^8 + 4800*(sqrt(c)*x - sqrt(c*x^

2 + b*x + a))^7*a^3*c^3*d*e^8 + 128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^7*d^2*e^7 + 9026*(sqrt(c)*x - sqrt

(c*x^2 + b*x + a))^5*a*b^5*c*d^2*e^7 + 1520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^3*c^2*d^2*e^7 + 11040*

(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b*c^3*d^2*e^7 - 70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^8*d^3*e^6

+ 2280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^6*c*d^3*e^6 - 20420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^

2*b^4*c^2*d^3*e^6 - 20320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^2*c^3*d^3*e^6 - 24640*(sqrt(c)*x - sqrt(

c*x^2 + b*x + a))^3*a^4*c^4*d^3*e^6 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^9*d^4*e^5 + 450*(sqrt(c)*x - sq

rt(c*x^2 + b*x + a))*a*b^7*c*d^4*e^5 - 2080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^5*c^2*d^4*e^5 - 1120*(sq

rt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^3*c^3*d^4*e^5 - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b*c^4*d^4*

e^5 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^3*c^(5/2)*e^9 - 490*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a

*b^5*sqrt(c)*d*e^8 - 11440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^3*c^(3/2)*d*e^8 - 1440*(sqrt(c)*x - sqr

t(c*x^2 + b*x + a))^6*a^3*b*c^(5/2)*d*e^8 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^6*sqrt(c)*d^2*e^7 +

5120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^4*c^(3/2)*d^2*e^7 + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)

)^4*a^3*b^2*c^(5/2)*d^2*e^7 - 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*c^(7/2)*d^2*e^7 + 630*(sqrt(c)*x

- sqrt(c*x^2 + b*x + a))^2*a*b^7*sqrt(c)*d^3*e^6 - 4430*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^5*c^(3/2)

*d^3*e^6 - 2480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^3*c^(5/2)*d^3*e^6 - 22240*(sqrt(c)*x - sqrt(c*x^2

+ b*x + a))^2*a^4*b*c^(7/2)*d^3*e^6 + 60*a*b^8*sqrt(c)*d^4*e^5 - 330*a^2*b^6*c^(3/2)*d^4*e^5 - 260*a^3*b^4*c^(

5/2)*d^4*e^5 - 2560*a^4*b^2*c^(7/2)*d^4*e^5 - 64*a^5*c^(9/2)*d^4*e^5 - 70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^

7*a*b^5*e^9 + 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^3*c*e^9 + 2720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a

))^7*a^3*b*c^2*e^9 - 256*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^6*d*e^8 - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*

x + a))^5*a^2*b^4*c*d*e^8 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^4*c^3*d*e^8 + 210*(sqrt(c)*x - sqrt(c

*x^2 + b*x + a))^3*a*b^7*d^2*e^7 + 230*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^5*c*d^2*e^7 + 26480*(sqrt(c

)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^3*c^2*d^2*e^7 + 6240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b*c^3*d^2*

e^7 + 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^8*d^3*e^6 - 750*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^6*c

*d^3*e^6 + 2820*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^4*c^2*d^3*e^6 - 5760*(sqrt(c)*x - sqrt(c*x^2 + b*x +

a))*a^4*b^2*c^3*d^3*e^6 + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*c^4*d^3*e^6 + 5120*(sqrt(c)*x - sqrt(c

*x^2 + b*x + a))^6*a^3*b^2*c^(3/2)*e^9 - 3200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^5*sqrt(c)*d*e^8 - 76

80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^3*c^(3/2)*d*e^8 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^

4*b*c^(5/2)*d*e^8 - 630*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^6*sqrt(c)*d^2*e^7 + 7180*(sqrt(c)*x - sqrt

(c*x^2 + b*x + a))^2*a^3*b^4*c^(3/2)*d^2*e^7 + 7360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^2*c^(5/2)*d^2*

e^7 + 11200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*c^(7/2)*d^2*e^7 - 90*a^2*b^7*sqrt(c)*d^3*e^6 + 610*a^3*b

^5*c^(3/2)*d^3*e^6 - 560*a^4*b^3*c^(5/2)*d^3*e^6 + 1568*a^5*b*c^(7/2)*d^3*e^6 + 128*(sqrt(c)*x - sqrt(c*x^2 +

b*x + a))^5*a^2*b^5*e^9 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^3*c*e^9 + 3840*(sqrt(c)*x - sqrt(c*

x^2 + b*x + a))^5*a^4*b*c^2*e^9 - 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^6*d*e^8 - 3580*(sqrt(c)*x -

sqrt(c*x^2 + b*x + a))^3*a^3*b^4*c*d*e^8 - 13760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b^2*c^2*d*e^8 + 544

0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*c^3*d*e^8 - 90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^7*d^2*e^7

+ 750*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^5*c*d^2*e^7 - 400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^3

*c^2*d^2*e^7 + 6880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*b*c^3*d^2*e^7 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x

+ a))^4*a^3*b^4*sqrt(c)*e^9 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*b^2*c^(3/2)*e^9 + 2560*(sqrt(c)*

x - sqrt(c*x^2 + b*x + a))^4*a^5*c^(5/2)*e^9 + 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^5*sqrt(c)*d*e^8

- 6800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^3*c^(3/2)*d*e^8 - 3040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))

^2*a^5*b*c^(5/2)*d*e^8 + 60*a^3*b^6*sqrt(c)*d^2*e^7 - 450*a^4*b^4*c^(3/2)*d^2*e^7 + 976*a^5*b^2*c^(5/2)*d^2*e^

7 - 288*a^6*c^(7/2)*d^2*e^7 + 70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^5*e^9 + 2000*(sqrt(c)*x - sqrt(c*

x^2 + b*x + a))^3*a^4*b^3*c*e^9 + 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*b*c^2*e^9 + 60*(sqrt(c)*x - s

qrt(c*x^2 + b*x + a))*a^3*b^6*d*e^8 - 450*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^4*c*d*e^8 - 1840*(sqrt(c)*

x - sqrt(c*x^2 + b*x + a))*a^5*b^2*c^2*d*e^8 - 2080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*c^3*d*e^8 + 2560*(

sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^2*c^(3/2)*e^9 - 15*a^4*b^5*sqrt(c)*d*e^8 + 120*a^5*b^3*c^(3/2)*d*e^

8 - 752*a^6*b*c^(5/2)*d*e^8 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^5*e^9 + 120*(sqrt(c)*x - sqrt(c*x^2

+ b*x + a))*a^5*b^3*c*e^9 + 1040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b*c^2*e^9 + 256*a^7*c^(5/2)*e^9)/((c

^3*d^6*e^4 - 3*b*c^2*d^5*e^5 + 3*b^2*c*d^4*e^6 + 3*a*c^2*d^4*e^6 - b^3*d^3*e^7 - 6*a*b*c*d^3*e^7 + 3*a*b^2*d^2

*e^8 + 3*a^2*c*d^2*e^8 - 3*a^2*b*d*e^9 + a^3*e^10)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - s

qrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^5)

________________________________________________________________________________________

maple [B] time = 0.08, size = 20477, normalized size = 69.18 \begin {gather*} \text {output too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?`

for more details)Is a*e^2-b*d*e +c*d^2 zero or nonzero?

________________________________________________________________________________________

mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^6} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^6,x)

[Out]

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^6, x)

________________________________________________________________________________________

sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]

3.21.21 \(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^6} \, dx\)