3.24.42 \(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx\) [2342]

Optimal. Leaf size=402 \[ \frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c d^2-b d e+a e^2\right )^{9/2}} \]

[Out]

-1/5*e*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^5-7/40*e*(-b*e+2*c*d)*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+
c*d^2)^2/(e*x+d)^4-1/240*e*(108*c^2*d^2+35*b^2*e^2-4*c*e*(8*a*e+27*b*d))*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^
2)^3/(e*x+d)^3-1/256*(-4*a*c+b^2)*(-b*e+2*c*d)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(3*a*e+4*b*d))*arctanh(1/2*(b*d-2*a
*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(9/2)+1/128*(-b*e+2*c*d)
*(16*c^2*d^2+7*b^2*e^2-4*c*e*(3*a*e+4*b*d))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)
^4/(e*x+d)^2

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Rubi [A]
time = 0.34, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {758, 848, 820, 734, 738, 212} \begin {gather*} -\frac {e \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (8 a e+27 b d)+35 b^2 e^2+108 c^2 d^2\right )}{240 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) (-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 )^4}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \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 )^{9/2}}-\frac {7 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{40 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*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)^4*(d + e*x)^2) - (e*(a + b*x + c*x^2)^(3/2))/(5*(c*d^2 - b*d*e + a*e^2)*
(d + e*x)^5) - (7*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(40*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4) - (e*(10
8*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(27*b*d + 8*a*e))*(a + b*x + c*x^2)^(3/2))/(240*(c*d^2 - b*d*e + a*e^2)^3*(d +
e*x)^3) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*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)^(9/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 734

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] /; Free
Q[{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 738

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 758

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[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(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] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 820

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)/(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[S
implify[m + 2*p + 3], 0]

Rule 848

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])

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx &=-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {\int \frac {\left (\frac {1}{2} (-10 c d+7 b e)+2 c e x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx}{5 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}+\frac {\int \frac {\left (\frac {1}{4} \left (80 c^2 d^2+35 b^2 e^2-2 c e (47 b d+16 a e)\right )-\frac {7}{2} c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^4} \, dx}{20 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}+\frac {\left ((2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\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 )^3}\\ &=\frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (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 )^4 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}-\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\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 )^4}\\ &=\frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (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 )^4 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}+\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\right ) \text {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 )^4}\\ &=\frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (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 )^4 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \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 )^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 11.76, size = 367, normalized size = 0.91 \begin {gather*} -\frac {\frac {e (a+x (b+c x))^{3/2}}{(d+e x)^5}+\frac {7 e (2 c d-b e) (a+x (b+c x))^{3/2}}{8 \left (c d^2+e (-b d+a e)\right ) (d+e x)^4}-\frac {-\frac {e \left (108 c^2 d^2+35 b^2 e^2-4 c e (27 b d+8 a e)\right ) (a+x (b+c x))^{3/2}}{2 (d+e x)^3}+\frac {15}{4} (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {-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)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{24 \left (c d^2+e (-b d+a e)\right )^2}}{5 \left (c d^2+e (-b d+a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*((e*(a + x*(b + c*x))^(3/2))/(d + e*x)^5 + (7*e*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(8*(c*d^2 + e*(-(b
*d) + a*e))*(d + e*x)^4) - (-1/2*(e*(108*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(27*b*d + 8*a*e))*(a + x*(b + c*x))^(3/2
))/(d + e*x)^3 + (15*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*((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))))/4)/(24*(c*d^2 + e*(-(b*d) + a*e))^2))/(c*d^2 + e*(-(b*d) + a*e))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3911\) vs. \(2(376)=752\).
time = 0.93, size = 3912, normalized size = 9.73

method	result	size
default	\(\text {Expression too large to display}\)	\(3912\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

1/e^6*(-1/5/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^5*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3
/2)-7/10*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(-1/4/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^4*(c*(x+d/e)^2+1/e*(b*e-2*c*d
)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-5/8*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(-1/3/(a*e^2-b*d*e+c*d^2)*e^2/(
x+d/e)^3*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/2*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*
d^2)*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/
2)-1/4*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d
/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)
+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b
*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1
/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(
b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))+2*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/4*(2*c*(x+d/e)+1/e
*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^
2)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d
/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))))+1/2*c/(a*e^2-b*d*e+c*d^2)*e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e
^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*
c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*l
n((2*(a*e^2-b*d*e+c*d^2)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2
*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))))-1/4*c/(a*e^2-b*d*e+c*d^2)*e^2*(-1/2/(a*e^2-b*d*e+c*d
^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/4*e*(b*e-2*c*d)/(a*e^2
-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2
)^(3/2)+1/2*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(
1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*
d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^
2)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b
*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))+2*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^
2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)
/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2
)^(1/2))))+1/2*c/(a*e^2-b*d*e+c*d^2)*e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+
1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^
2+1/e*(b*e-2*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+
c*d^2)/e^2)^(1/2))/(x+d/e)))))-2/5*c/(a*e^2-b*d*e+c*d^2)*e^2*(-1/3/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e
)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/2*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(-1/2/(a*e^2-
b*d*e+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/4*e*(b*e-2*c*
d)/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c
*d^2)/e^2)^(3/2)+1/2*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b
*d*e+c*d^2)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)
+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))+2*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c
*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-
2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2))))+1/2*c/(a*e^2-b*d*e+c*d^2)*e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^
2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^
2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+
c*d^2)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*((a*e^2-b*...

________________________________________________________________________________________

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)^(1/2)/(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(c*d^2-%e*b*d+%e^2*a>0)', see `
assume?` for

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3138 vs. \(2 (393) = 786\).
time = 214.16, size = 6319, normalized size = 15.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/7680*(15*(32*(b^2*c^3 - 4*a*c^4)*d^8 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*x^5*e^8 + (6*(5*b^4*c - 24*a*b^
2*c^2 + 16*a^2*c^3)*d*x^5 - 5*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d*x^4)*e^7 - 2*(24*(b^3*c^2 - 4*a*b*c^3)*d^2
*x^5 - 15*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d^2*x^4 + 5*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d^2*x^3)*e^6 +
2*(16*(b^2*c^3 - 4*a*c^4)*d^3*x^5 - 120*(b^3*c^2 - 4*a*b*c^3)*d^3*x^4 + 30*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c
^3)*d^3*x^3 - 5*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d^3*x^2)*e^5 + 5*(32*(b^2*c^3 - 4*a*c^4)*d^4*x^4 - 96*(b^3
*c^2 - 4*a*b*c^3)*d^4*x^3 + 12*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d^4*x^2 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*
c^2)*d^4*x)*e^4 + (320*(b^2*c^3 - 4*a*c^4)*d^5*x^3 - 480*(b^3*c^2 - 4*a*b*c^3)*d^5*x^2 + 30*(5*b^4*c - 24*a*b^
2*c^2 + 16*a^2*c^3)*d^5*x - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d^5)*e^3 + 2*(160*(b^2*c^3 - 4*a*c^4)*d^6*x^2
- 120*(b^3*c^2 - 4*a*b*c^3)*d^6*x + 3*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d^6)*e^2 + 16*(10*(b^2*c^3 - 4*a*c
^4)*d^7*x - 3*(b^3*c^2 - 4*a*b*c^3)*d^7)*e)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b
^2 + 4*a*c)*d^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b
*x + (b^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e +
d^2)) - 4*(960*c^5*d^9*x + 480*b*c^4*d^9 - (48*a^4*b*x + 384*a^5 - (105*a*b^4 - 460*a^2*b^2*c + 256*a^3*c^2)*x
^4 + 2*(35*a^2*b^3 - 116*a^3*b*c)*x^3 - 8*(7*a^3*b^2 - 16*a^4*c)*x^2)*e^9 + (1872*a^4*b*d - (105*b^5 - 80*a*b^
3*c - 1072*a^2*b*c^2)*d*x^4 + 4*(140*a*b^4 - 529*a^2*b^2*c + 140*a^3*c^2)*d*x^3 - 6*(63*a^2*b^3 - 164*a^3*b*c)
*d*x^2 + 16*(19*a^3*b^2 - 10*a^4*c)*d*x)*e^8 + ((485*b^4*c - 1312*a*b^2*c^2 - 1072*a^2*c^3)*d^2*x^4 - 2*(245*b
^5 - 18*a*b^3*c - 2768*a^2*b*c^2)*d^2*x^3 + 6*(203*a*b^4 - 586*a^2*b^2*c - 24*a^3*c^2)*d^2*x^2 - 6*(139*a^2*b^
3 - 116*a^3*b*c)*d^2*x - 8*(449*a^3*b^2 + 244*a^4*c)*d^2)*e^7 - 2*(4*(107*b^3*c^2 - 308*a*b*c^3)*d^3*x^4 - 2*(
567*b^4*c - 1378*a*b^2*c^2 - 1280*a^2*c^3)*d^3*x^3 + 2*(224*b^5 + 217*a*b^3*c - 2592*a^2*b*c^2)*d^3*x^2 - 2*(3
42*a*b^4 - 343*a^2*b^2*c - 140*a^3*c^2)*d^3*x - (1657*a^2*b^3 + 3812*a^3*b*c)*d^3)*e^6 + (4*(167*b^2*c^3 - 308
*a*c^4)*d^4*x^4 - 6*(669*b^3*c^2 - 1756*a*b*c^3)*d^4*x^3 + 6*(693*b^4*c - 1336*a*b^2*c^2 - 1504*a^2*c^3)*d^4*x
^2 - 2*(395*b^5 + 1242*a*b^3*c - 3488*a^2*b*c^2)*d^4*x - (1315*a*b^4 + 11032*a^2*b^2*c + 3952*a^3*c^2)*d^4)*e^
5 - (288*b*c^4*d^5*x^4 - 4*(811*b^2*c^3 - 1300*a*c^4)*d^5*x^3 + 6*(1243*b^3*c^2 - 2724*a*b*c^3)*d^5*x^2 - 4*(9
35*b^4*c - 754*a*b^2*c^2 - 1600*a^2*c^3)*d^5*x - (105*b^5 + 6020*a*b^3*c + 12592*a^2*b*c^2)*d^5)*e^4 + (96*c^5
*d^6*x^4 - 1488*b*c^4*d^6*x^3 + 4*(1589*b^2*c^3 - 2012*a*c^4)*d^6*x^2 - 2*(3485*b^3*c^2 - 4796*a*b*c^3)*d^6*x
- (555*b^4*c + 10720*a*b^2*c^2 + 5264*a^2*c^3)*d^6)*e^3 + 30*(16*c^5*d^7*x^3 - 104*b*c^4*d^7*x^2 + 2*(107*b^2*
c^3 - 84*a*c^4)*d^7*x + 3*(13*b^3*c^2 + 100*a*b*c^3)*d^7)*e^2 + 240*(4*c^5*d^8*x^2 - 14*b*c^4*d^8*x - (5*b^2*c
^3 + 12*a*c^4)*d^8)*e)*sqrt(c*x^2 + b*x + a))/(c^5*d^15 + a^5*x^5*e^15 - 5*(a^4*b*d*x^5 - a^5*d*x^4)*e^14 - 5*
(5*a^4*b*d^2*x^4 - 2*a^5*d^2*x^3 - (2*a^3*b^2 + a^4*c)*d^2*x^5)*e^13 - 5*(10*a^4*b*d^3*x^3 - 2*a^5*d^3*x^2 + 2
*(a^2*b^3 + 2*a^3*b*c)*d^3*x^5 - 5*(2*a^3*b^2 + a^4*c)*d^3*x^4)*e^12 - 5*(10*a^4*b*d^4*x^2 - a^5*d^4*x - (a*b^
4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^4*x^5 + 10*(a^2*b^3 + 2*a^3*b*c)*d^4*x^4 - 10*(2*a^3*b^2 + a^4*c)*d^4*x^3)*e^11
- (25*a^4*b*d^5*x + (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^5*x^5 - a^5*d^5 - 25*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2
)*d^5*x^4 + 100*(a^2*b^3 + 2*a^3*b*c)*d^5*x^3 - 50*(2*a^3*b^2 + a^4*c)*d^5*x^2)*e^10 + 5*((b^4*c + 6*a*b^2*c^2
+ 2*a^2*c^3)*d^6*x^5 - a^4*b*d^6 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^6*x^4 + 10*(a*b^4 + 6*a^2*b^2*c + 2*a^
3*c^2)*d^6*x^3 - 20*(a^2*b^3 + 2*a^3*b*c)*d^6*x^2 + 5*(2*a^3*b^2 + a^4*c)*d^6*x)*e^9 - 5*(2*(b^3*c^2 + 2*a*b*c
^3)*d^7*x^5 - 5*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^7*x^4 + 2*(b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^7*x^3 - 10*(
a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^7*x^2 + 10*(a^2*b^3 + 2*a^3*b*c)*d^7*x - (2*a^3*b^2 + a^4*c)*d^7)*e^8 + 5*(
(2*b^2*c^3 + a*c^4)*d^8*x^5 - 10*(b^3*c^2 + 2*a*b*c^3)*d^8*x^4 + 10*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^8*x^3
- 2*(b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^8*x^2 + 5*(a*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^8*x - 2*(a^2*b^3 + 2*a^3
*b*c)*d^8)*e^7 - 5*(b*c^4*d^9*x^5 - 5*(2*b^2*c^3 + a*c^4)*d^9*x^4 + 20*(b^3*c^2 + 2*a*b*c^3)*d^9*x^3 - 10*(b^4
*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^9*x^2 + (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^9*x - (a*b^4 + 6*a^2*b^2*c + 2*a^3
*c^2)*d^9)*e^6 + (c^5*d^10*x^5 - 25*b*c^4*d^10*x^4 + 50*(2*b^2*c^3 + a*c^4)*d^10*x^3 - 100*(b^3*c^2 + 2*a*b*c^
3)*d^10*x^2 + 25*(b^4*c + 6*a*b^2*c^2 + 2*a^2*c^3)*d^10*x - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^10)*e^5 + 5*(c
^5*d^11*x^4 - 10*b*c^4*d^11*x^3 + 10*(2*b^2*c^3 + a*c^4)*d^11*x^2 - 10*(b^3*c^2 + 2*a*b*c^3)*d^11*x + (b^4*c +
6*a*b^2*c^2 + 2*a^2*c^3)*d^11)*e^4 + 5*(2*c^5*d^12*x^3 - 10*b*c^4*d^12*x^2 + 5*(2*b^2*c^3 + a*c^4)*d^12*x - 2
*(b^3*c^2 + 2*a*b*c^3)*d^12)*e^3 + 5*(2*c^5*d^1...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b x + c x^{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)**(1/2)/(e*x+d)**6,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 8452 vs. \(2 (393) = 786\).
time = 1.28, size = 8452, normalized size = 21.02 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(32*b^2*c^3*d^3 - 128*a*c^4*d^3 - 48*b^3*c^2*d^2*e + 192*a*b*c^3*d^2*e + 30*b^4*c*d*e^2 - 144*a*b^2*c^2
*d*e^2 + 96*a^2*c^3*d*e^2 - 7*b^5*e^3 + 40*a*b^3*c*e^3 - 48*a^2*b*c^2*e^3)*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^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3
*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b
^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*sqrt(-c*d^2 +
b*d*e - a*e^2)) + 1/1920*(7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*c^(13/2)*d^8*e + 3072*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^5*c^7*d^9 + 9216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^6*d^8*e + 7680*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^4*b*c^(13/2)*d^9 - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b*c^(11/2)*d^7*e^2 - 3840*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(11/2)*d^8*e - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(13/2)*
d^8*e + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^6*d^9 - 50048*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b
^2*c^5*d^7*e^2 - 57856*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^6*d^7*e^2 - 11520*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^3*b^3*c^5*d^8*e - 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^6*d^8*e + 3840*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^2*b^3*c^(11/2)*d^9 + 70720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^2*c^(9/2)*d^6*e^3 - 67840
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*c^(11/2)*d^6*e^3 - 17600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*c^
(9/2)*d^7*e^2 - 113920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(11/2)*d^7*e^2 - 7200*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2*b^4*c^(9/2)*d^8*e - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(11/2)*d^8*e + 960*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c^5*d^9 + 15040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^2*c^4*d^5*e^4 -
60160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*c^5*d^5*e^4 + 129280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*c
^4*d^6*e^3 - 1024*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c^5*d^6*e^3 + 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*b^4*c^4*d^7*e^2 - 90880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c^5*d^7*e^2 + 15360*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^3*a^2*c^6*d^7*e^2 - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*c^4*d^8*e - 3840*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c^5*d^8*e + 96*b^5*c^(9/2)*d^9 + 4320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
8*b^2*c^(7/2)*d^4*e^5 - 17280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^(9/2)*d^4*e^5 - 52000*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^6*b^3*c^(7/2)*d^5*e^4 - 7040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b*c^(9/2)*d^5*e^4 + 8
1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*c^(7/2)*d^6*e^3 + 114240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*
a*b^2*c^(9/2)*d^6*e^3 + 167680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(11/2)*d^6*e^3 + 13760*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*b^5*c^(7/2)*d^7*e^2 - 37760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3*c^(9/2)*d^7*
e^2 + 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^(11/2)*d^7*e^2 - 192*b^6*c^(7/2)*d^8*e - 480*a*b^4*c
^(9/2)*d^8*e + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^2*c^3*d^3*e^6 - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^9*a*c^4*d^3*e^6 - 20320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^3*c^3*d^4*e^5 + 81280*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^7*a*b*c^4*d^4*e^5 - 120680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^4*c^3*d^5*e^4 - 122240*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c^4*d^5*e^4 + 226432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^5*d
^5*e^4 + 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c^3*d^6*e^3 + 88320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^3*a*b^3*c^4*d^6*e^3 + 281600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c^5*d^6*e^3 + 4280*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*b^6*c^3*d^7*e^2 - 8480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*c^4*d^7*e^2 + 11520*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c^5*d^7*e^2 - 6480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^3*c^(5/2)*d
^3*e^6 + 25920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b*c^(7/2)*d^3*e^6 + 7260*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^6*b^4*c^(5/2)*d^4*e^5 + 59680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^(7/2)*d^4*e^5 + 182720*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^(9/2)*d^4*e^5 - 85780*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^5*c^(5/2)
*d^5*e^4 - 237120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^3*c^(7/2)*d^5*e^4 + 63040*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^4*a^2*b*c^(9/2)*d^5*e^4 - 6340*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^6*c^(5/2)*d^6*e^3 + 22000*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^(7/2)*d^6*e^3 + 176640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*
b^2*c^(9/2)*d^6*e^3 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(11/2)*d^6*e^3 + 476*b^7*c^(5/2)*d^7*e^
2 - 848*a*b^5*c^(7/2)*d^7*e^2 + 1920*a^2*b^3*c^(9/2)*d^7*e^2 - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c
^2*d^2*e^7 + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b*c^3*d^2*e^7 + 10740*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^7*b^4*c^2*d^3*e^6 - 56480*(sqrt(c)*x - sq...

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

[Out]

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^6, x)

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