3.3.71 \(\int \frac {\sqrt {b x+c x^2}}{(d+e x)^5} \, dx\)
Optimal. Leaf size=258 \[ -\frac {b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}+\frac {\sqrt {b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac {5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 d^2 (d+e x)^3 (c d-b e)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{4 d (d+e x)^4 (c d-b e)} \]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 258, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used =
{744, 806, 720, 724, 206} \begin {gather*} \frac {\sqrt {b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac {b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}-\frac {5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 d^2 (d+e x)^3 (c d-b e)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{4 d (d+e x)^4 (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[b*x + c*x^2]/(d + e*x)^5,x]
[Out]
((16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(64*d^3*(c*d - b*e)^3*(d + e
*x)^2) - (e*(b*x + c*x^2)^(3/2))/(4*d*(c*d - b*e)*(d + e*x)^4) - (5*e*(2*c*d - b*e)*(b*x + c*x^2)^(3/2))/(24*d
^2*(c*d - b*e)^2*(d + e*x)^3) - (b^2*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*
Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(128*d^(7/2)*(c*d - b*e)^(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 744
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 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]
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+c x^2}}{(d+e x)^5} \, dx &=-\frac {e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac {\int \frac {\left (\frac {1}{2} (-8 c d+5 b e)+c e x\right ) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx}{4 d (c d-b e)}\\ &=-\frac {e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 d^2 (c d-b e)^2 (d+e x)^3}+\frac {\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx}{16 d^2 (c d-b e)^2}\\ &=\frac {\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{64 d^3 (c d-b e)^3 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 d^2 (c d-b e)^2 (d+e x)^3}-\frac {\left (b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{128 d^3 (c d-b e)^3}\\ &=\frac {\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{64 d^3 (c d-b e)^3 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 d^2 (c d-b e)^2 (d+e x)^3}+\frac {\left (b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{64 d^3 (c d-b e)^3}\\ &=\frac {\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{64 d^3 (c d-b e)^3 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 d^2 (c d-b e)^2 (d+e x)^3}-\frac {b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.62, size = 243, normalized size = 0.94 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {3 (d+e x)^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \left (b^2 (d+e x)^2 \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )+\sqrt {d} \sqrt {x} \sqrt {b+c x} \sqrt {c d-b e} (-b d+b e x-2 c d x)\right )}{d^{5/2} \sqrt {b+c x} (c d-b e)^{5/2}}+\frac {40 e x^{3/2} (b+c x) (d+e x) (2 c d-b e)}{d (c d-b e)}+48 e x^{3/2} (b+c x)\right )}{192 d \sqrt {x} (d+e x)^4 (b e-c d)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^5,x]
[Out]
(Sqrt[x*(b + c*x)]*(48*e*x^(3/2)*(b + c*x) + (40*e*(2*c*d - b*e)*x^(3/2)*(b + c*x)*(d + e*x))/(d*(c*d - b*e))
+ (3*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(d + e*x)^2*(Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[x]*Sqrt[b + c*x]*(-(b*d)
- 2*c*d*x + b*e*x) + b^2*(d + e*x)^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))/(d^(5/2)*(c*
d - b*e)^(5/2)*Sqrt[b + c*x])))/(192*d*(-(c*d) + b*e)*Sqrt[x]*(d + e*x)^4)
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 11.38, size = 2552, normalized size = 9.89 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[b*x + c*x^2]/(d + e*x)^5,x]
[Out]
(Sqrt[b*x + c*x^2]*(15*b^7*d^3*e^3 - 73*b^7*d^2*e^4*x - 55*b^7*d*e^5*x^2 - 15*b^7*e^6*x^3) + c*Sqrt[b*x + c*x^
2]*(72*b^6*d^4*e^2 + 1158*b^6*d^3*e^3*x - 1476*b^6*d^2*e^4*x^2 - 1242*b^6*d*e^5*x^3 - 360*b^6*e^6*x^4) + Sqrt[
c]*(-15*b^7*d^4*e^2 - 180*b^7*d^3*e^3*x + 494*b^7*d^2*e^4*x^2 + 380*b^7*d*e^5*x^3 + 105*b^7*e^6*x^4) + c^2*Sqr
t[b*x + c*x^2]*(-256*b^5*d^5*e - 1728*b^5*d^4*e^2*x + 11608*b^5*d^3*e^3*x^2 - 1240*b^5*d^2*e^4*x^3 - 3088*b^5*
d*e^5*x^4 - 1200*b^5*e^6*x^5) + c^(3/2)*(38*b^6*d^5*e + 56*b^6*d^4*e^2*x - 4596*b^6*d^3*e^3*x^2 + 2576*b^6*d^2
*e^4*x^3 + 2654*b^6*d*e^5*x^4 + 840*b^6*e^6*x^5) + c^6*Sqrt[b*x + c*x^2]*(3072*b*d^6*x^3 + 43008*b*d^5*e*x^4 +
12288*b*d^4*e^2*x^5 + 3072*b*d^3*e^3*x^6) + c^5*Sqrt[b*x + c*x^2]*(4608*b^2*d^6*x^2 + 41216*b^2*d^5*e*x^3 - 3
2768*b^2*d^4*e^2*x^4 - 9472*b^2*d^3*e^3*x^5 - 3072*b^2*d^2*e^4*x^6) + c^4*Sqrt[b*x + c*x^2]*(1920*b^3*d^6*x +
11136*b^3*d^5*e*x^2 - 54144*b^3*d^4*e^2*x^3 + 8064*b^3*d^3*e^3*x^4 + 6912*b^3*d^2*e^4*x^5 + 3072*b^3*d*e^5*x^6
) + c^3*Sqrt[b*x + c*x^2]*(192*b^4*d^6 - 640*b^4*d^5*e*x - 21376*b^4*d^4*e^2*x^2 + 26256*b^4*d^3*e^3*x^3 + 825
6*b^4*d^2*e^4*x^4 + 480*b^4*d*e^5*x^5 - 960*b^4*e^6*x^6) + c^(5/2)*(-24*b^5*d^6 + 736*b^5*d^5*e*x + 7952*b^5*d
^4*e^2*x^2 - 23072*b^5*d^3*e^3*x^3 - 1832*b^5*d^2*e^4*x^4 + 3232*b^5*d*e^5*x^5 + 1680*b^5*e^6*x^6) + c^7*Sqrt[
b*x + c*x^2]*(12288*d^5*e*x^5 + 8192*d^4*e^2*x^6 + 2048*d^3*e^3*x^7) + c^(13/2)*(-3072*b*d^6*x^4 - 49152*b*d^5
*e*x^5 - 16384*b*d^4*e^2*x^6 - 4096*b*d^3*e^3*x^7) + c^(11/2)*(-6144*b^2*d^6*x^3 - 61184*b^2*d^5*e*x^4 + 27648
*b^2*d^4*e^2*x^5 + 8192*b^2*d^3*e^3*x^6 + 3072*b^2*d^2*e^4*x^7) + c^(9/2)*(-3840*b^3*d^6*x^2 - 27136*b^3*d^5*e
*x^3 + 71552*b^3*d^4*e^2*x^4 - 3072*b^3*d^3*e^3*x^5 - 5376*b^3*d^2*e^4*x^6 - 3072*b^3*d*e^5*x^7) + c^(7/2)*(-7
68*b^4*d^6*x - 1984*b^4*d^5*e*x^2 + 43904*b^4*d^4*e^2*x^3 - 31584*b^4*d^3*e^3*x^4 - 12096*b^4*d^2*e^4*x^5 - 20
16*b^4*d*e^5*x^6 + 960*b^4*e^6*x^7) + c^(15/2)*(-12288*d^5*e*x^6 - 8192*d^4*e^2*x^7 - 2048*d^3*e^3*x^8))/(-192
*b^7*d^3*e^4*(d + e*x)^4 + 24576*c^7*d^6*e*x^4*(d + e*x)^4 + 192*c*d^3*e*(d + e*x)^4*(3*b^6*d*e^2 - 32*b^6*e^3
*x) + 1536*b^6*Sqrt[c]*d^3*e^4*(d + e*x)^4*Sqrt[b*x + c*x^2] - 24576*c^(13/2)*d^6*e*x^3*(d + e*x)^4*Sqrt[b*x +
c*x^2] + 192*c^(3/2)*d^3*e*(d + e*x)^4*(-24*b^5*d*e^2 + 80*b^5*e^3*x)*Sqrt[b*x + c*x^2] + 192*c^(5/2)*d^3*e*(
d + e*x)^4*Sqrt[b*x + c*x^2]*(24*b^4*d^2*e - 240*b^4*d*e^2*x + 192*b^4*e^3*x^2) + 192*c^2*d^3*e*(d + e*x)^4*(-
3*b^5*d^2*e + 96*b^5*d*e^2*x - 160*b^5*e^3*x^2) + 192*c^(11/2)*d^3*e*(d + e*x)^4*Sqrt[b*x + c*x^2]*(-192*b*d^3
*x^2 + 384*b*d^2*e*x^3) + 192*c^(9/2)*d^3*e*(d + e*x)^4*Sqrt[b*x + c*x^2]*(-80*b^2*d^3*x + 576*b^2*d^2*e*x^2 -
384*b^2*d*e^2*x^3) + 192*c^(7/2)*d^3*e*(d + e*x)^4*Sqrt[b*x + c*x^2]*(-8*b^3*d^3 + 240*b^3*d^2*e*x - 576*b^3*
d*e^2*x^2 + 128*b^3*e^3*x^3) + 192*c^3*d^3*e*(d + e*x)^4*(b^4*d^3 - 96*b^4*d^2*e*x + 480*b^4*d*e^2*x^2 - 256*b
^4*e^3*x^3) + 192*c^6*d^3*e*(d + e*x)^4*(256*b*d^3*x^3 - 384*b*d^2*e*x^4) + 192*c^5*d^3*e*(d + e*x)^4*(160*b^2
*d^3*x^2 - 768*b^2*d^2*e*x^3 + 384*b^2*d*e^2*x^4) + 192*c^4*d^3*e*(d + e*x)^4*(32*b^3*d^3*x - 480*b^3*d^2*e*x^
2 + 768*b^3*d*e^2*x^3 - 128*b^3*e^3*x^4)) - (b^2*c^5*d^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d])/Sqrt[c*d - b*e] + (Sqrt
[c]*e*x)/(Sqrt[d]*Sqrt[c*d - b*e]) - (e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/(4*(c*d - b*e)^(13/2))
+ (b^3*c^4*Sqrt[d]*e*ArcTanh[(Sqrt[c]*Sqrt[d])/Sqrt[c*d - b*e] + (Sqrt[c]*e*x)/(Sqrt[d]*Sqrt[c*d - b*e]) - (e*
Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/(c*d - b*e)^(13/2) - (101*b^4*c^3*e^2*ArcTanh[(Sqrt[c]*Sqrt[d])
/Sqrt[c*d - b*e] + (Sqrt[c]*e*x)/(Sqrt[d]*Sqrt[c*d - b*e]) - (e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])
/(64*Sqrt[d]*(c*d - b*e)^(13/2)) + (79*b^5*c^2*e^3*ArcTanh[(Sqrt[c]*Sqrt[d])/Sqrt[c*d - b*e] + (Sqrt[c]*e*x)/(
Sqrt[d]*Sqrt[c*d - b*e]) - (e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/(64*d^(3/2)*(c*d - b*e)^(13/2)) -
(31*b^6*c*e^4*ArcTanh[(Sqrt[c]*Sqrt[d])/Sqrt[c*d - b*e] + (Sqrt[c]*e*x)/(Sqrt[d]*Sqrt[c*d - b*e]) - (e*Sqrt[b
*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/(64*d^(5/2)*(c*d - b*e)^(13/2)) + (5*b^7*e^5*ArcTanh[(Sqrt[c]*Sqrt[d]
)/Sqrt[c*d - b*e] + (Sqrt[c]*e*x)/(Sqrt[d]*Sqrt[c*d - b*e]) - (e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])]
)/(64*d^(7/2)*(c*d - b*e)^(13/2))
________________________________________________________________________________________
fricas [B] time = 0.45, size = 1611, normalized size = 6.24
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x)^(1/2)/(e*x+d)^5,x, algorithm="fricas")
[Out]
[-1/384*(3*(16*b^2*c^2*d^6 - 16*b^3*c*d^5*e + 5*b^4*d^4*e^2 + (16*b^2*c^2*d^2*e^4 - 16*b^3*c*d*e^5 + 5*b^4*e^6
)*x^4 + 4*(16*b^2*c^2*d^3*e^3 - 16*b^3*c*d^2*e^4 + 5*b^4*d*e^5)*x^3 + 6*(16*b^2*c^2*d^4*e^2 - 16*b^3*c*d^3*e^3
+ 5*b^4*d^2*e^4)*x^2 + 4*(16*b^2*c^2*d^5*e - 16*b^3*c*d^4*e^2 + 5*b^4*d^3*e^3)*x)*sqrt(c*d^2 - b*d*e)*log((b*
d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(48*b*c^3*d^7 - 96*b^2*c^2*d^6*e
+ 63*b^3*c*d^5*e^2 - 15*b^4*d^4*e^3 + (16*c^4*d^5*e^2 - 40*b*c^3*d^4*e^3 + 62*b^2*c^2*d^3*e^4 - 53*b^3*c*d^2*
e^5 + 15*b^4*d*e^6)*x^3 + (64*c^4*d^6*e - 168*b*c^3*d^5*e^2 + 244*b^2*c^2*d^4*e^3 - 195*b^3*c*d^3*e^4 + 55*b^4
*d^2*e^5)*x^2 + (96*c^4*d^7 - 272*b*c^3*d^6*e + 374*b^2*c^2*d^5*e^2 - 271*b^3*c*d^4*e^3 + 73*b^4*d^3*e^4)*x)*s
qrt(c*x^2 + b*x))/(c^4*d^12 - 4*b*c^3*d^11*e + 6*b^2*c^2*d^10*e^2 - 4*b^3*c*d^9*e^3 + b^4*d^8*e^4 + (c^4*d^8*e
^4 - 4*b*c^3*d^7*e^5 + 6*b^2*c^2*d^6*e^6 - 4*b^3*c*d^5*e^7 + b^4*d^4*e^8)*x^4 + 4*(c^4*d^9*e^3 - 4*b*c^3*d^8*e
^4 + 6*b^2*c^2*d^7*e^5 - 4*b^3*c*d^6*e^6 + b^4*d^5*e^7)*x^3 + 6*(c^4*d^10*e^2 - 4*b*c^3*d^9*e^3 + 6*b^2*c^2*d^
8*e^4 - 4*b^3*c*d^7*e^5 + b^4*d^6*e^6)*x^2 + 4*(c^4*d^11*e - 4*b*c^3*d^10*e^2 + 6*b^2*c^2*d^9*e^3 - 4*b^3*c*d^
8*e^4 + b^4*d^7*e^5)*x), -1/192*(3*(16*b^2*c^2*d^6 - 16*b^3*c*d^5*e + 5*b^4*d^4*e^2 + (16*b^2*c^2*d^2*e^4 - 16
*b^3*c*d*e^5 + 5*b^4*e^6)*x^4 + 4*(16*b^2*c^2*d^3*e^3 - 16*b^3*c*d^2*e^4 + 5*b^4*d*e^5)*x^3 + 6*(16*b^2*c^2*d^
4*e^2 - 16*b^3*c*d^3*e^3 + 5*b^4*d^2*e^4)*x^2 + 4*(16*b^2*c^2*d^5*e - 16*b^3*c*d^4*e^2 + 5*b^4*d^3*e^3)*x)*sqr
t(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (48*b*c^3*d^7 - 96*b^2*c^2
*d^6*e + 63*b^3*c*d^5*e^2 - 15*b^4*d^4*e^3 + (16*c^4*d^5*e^2 - 40*b*c^3*d^4*e^3 + 62*b^2*c^2*d^3*e^4 - 53*b^3*
c*d^2*e^5 + 15*b^4*d*e^6)*x^3 + (64*c^4*d^6*e - 168*b*c^3*d^5*e^2 + 244*b^2*c^2*d^4*e^3 - 195*b^3*c*d^3*e^4 +
55*b^4*d^2*e^5)*x^2 + (96*c^4*d^7 - 272*b*c^3*d^6*e + 374*b^2*c^2*d^5*e^2 - 271*b^3*c*d^4*e^3 + 73*b^4*d^3*e^4
)*x)*sqrt(c*x^2 + b*x))/(c^4*d^12 - 4*b*c^3*d^11*e + 6*b^2*c^2*d^10*e^2 - 4*b^3*c*d^9*e^3 + b^4*d^8*e^4 + (c^4
*d^8*e^4 - 4*b*c^3*d^7*e^5 + 6*b^2*c^2*d^6*e^6 - 4*b^3*c*d^5*e^7 + b^4*d^4*e^8)*x^4 + 4*(c^4*d^9*e^3 - 4*b*c^3
*d^8*e^4 + 6*b^2*c^2*d^7*e^5 - 4*b^3*c*d^6*e^6 + b^4*d^5*e^7)*x^3 + 6*(c^4*d^10*e^2 - 4*b*c^3*d^9*e^3 + 6*b^2*
c^2*d^8*e^4 - 4*b^3*c*d^7*e^5 + b^4*d^6*e^6)*x^2 + 4*(c^4*d^11*e - 4*b*c^3*d^10*e^2 + 6*b^2*c^2*d^9*e^3 - 4*b^
3*c*d^8*e^4 + b^4*d^7*e^5)*x)]
________________________________________________________________________________________
giac [B] time = 1.09, size = 1144, normalized size = 4.43
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x)^(1/2)/(e*x+d)^5,x, algorithm="giac")
[Out]
1/384*(2*sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2)*(2*(4*((2*c^3*d^5*e
^6*sgn(1/(x*e + d)) - 5*b*c^2*d^4*e^7*sgn(1/(x*e + d)) + 4*b^2*c*d^3*e^8*sgn(1/(x*e + d)) - b^3*d^2*e^9*sgn(1/
(x*e + d)))/(c^3*d^6*e^8 - 3*b*c^2*d^5*e^9 + 3*b^2*c*d^4*e^10 - b^3*d^3*e^11) - 6*(c^3*d^6*e^7*sgn(1/(x*e + d)
) - 3*b*c^2*d^5*e^8*sgn(1/(x*e + d)) + 3*b^2*c*d^4*e^9*sgn(1/(x*e + d)) - b^3*d^3*e^10*sgn(1/(x*e + d)))*e^(-1
)/((c^3*d^6*e^8 - 3*b*c^2*d^5*e^9 + 3*b^2*c*d^4*e^10 - b^3*d^3*e^11)*(x*e + d)))*e^(-1)/(x*e + d) + (8*c^3*d^4
*e^5*sgn(1/(x*e + d)) - 16*b*c^2*d^3*e^6*sgn(1/(x*e + d)) + 13*b^2*c*d^2*e^7*sgn(1/(x*e + d)) - 5*b^3*d*e^8*sg
n(1/(x*e + d)))/(c^3*d^6*e^8 - 3*b*c^2*d^5*e^9 + 3*b^2*c*d^4*e^10 - b^3*d^3*e^11))*e^(-1)/(x*e + d) + (16*c^3*
d^3*e^4*sgn(1/(x*e + d)) - 24*b*c^2*d^2*e^5*sgn(1/(x*e + d)) + 38*b^2*c*d*e^6*sgn(1/(x*e + d)) - 15*b^3*e^7*sg
n(1/(x*e + d)))/(c^3*d^6*e^8 - 3*b*c^2*d^5*e^9 + 3*b^2*c*d^4*e^10 - b^3*d^3*e^11)) - (48*b^2*c^2*d^2*e^2*log(a
bs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) + 32*sqrt(c*d^2 - b*d*e)*c^(7/2)*d^3 - 48*sqrt(c*d^2 - b*d*e)
*b*c^(5/2)*d^2*e - 48*b^3*c*d*e^3*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) + 76*sqrt(c*d^2 - b*d*
e)*b^2*c^(3/2)*d*e^2 + 15*b^4*e^4*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) - 30*sqrt(c*d^2 - b*d*
e)*b^3*sqrt(c)*e^3)*sgn(1/(x*e + d))/(sqrt(c*d^2 - b*d*e)*c^3*d^6*e^4 - 3*sqrt(c*d^2 - b*d*e)*b*c^2*d^5*e^5 +
3*sqrt(c*d^2 - b*d*e)*b^2*c*d^4*e^6 - sqrt(c*d^2 - b*d*e)*b^3*d^3*e^7) + 3*(16*b^2*c^2*d^2*sgn(1/(x*e + d)) -
16*b^3*c*d*e*sgn(1/(x*e + d)) + 5*b^4*e^2*sgn(1/(x*e + d)))*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*(sqrt(
c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2) + sqrt(c*d^2*e^2 - b*d*e^3)*e^(-1
)/(x*e + d))))/((c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 + 3*b^2*c*d^4*e^4 - b^3*d^3*e^5)*sqrt(c*d^2 - b*d*e)))*e^2
________________________________________________________________________________________
maple [B] time = 0.05, size = 4819, normalized size = 18.68 \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)^(1/2)/(e*x+d)^5,x)
[Out]
5/8/(b*e-c*d)^4/d*c^4*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x+3/4/e^2/(b*e-c*d)^3*c^(7/2)*
ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))-5/8/e/(b*e
-c*d)^4*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c^4+17/32/(b*e-c*d)^3/d^2*c^(3/2)*ln(((x+d/e
)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*b^2+13/16/(b*e-c*d)^
3/d^2*c^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b-5/8/(b*e-c*d)^4/d/(x+d/e)*((x+d/e)^2*c-(
b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c^3+5/64*e^3/(b*e-c*d)^4/d^4/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+
(b*e-2*c*d)*(x+d/e)/e)^(3/2)*b^3+5/24/e/(b*e-c*d)^2/d^2/(x+d/e)^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+
d/e)/e)^(3/2)*b-5/12/e^2/(b*e-c*d)^2/d/(x+d/e)^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c+3
5/16/e/(b*e-c*d)^4/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^
(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2*c^3-3/2/e^2/(b*e-c*d)^3*c^3/(-(b
*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e
-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b+5/8/e^3/(b*e-c*d)^4*d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(
b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/
e)/e)^(1/2))/(x+d/e))*c^5+3/4/e^3/(b*e-c*d)^3*d*c^4/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d
)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))-35/
64*e/(b*e-c*d)^4/d^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)
/e)^(1/2))*c^(3/2)*b^3-7/32*e/(b*e-c*d)^3/d^3*c*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^2-
5/64*e/(b*e-c*d)^3/d^3*c^(1/2)*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*
d)*(x+d/e)/e)^(1/2))*b^3+5/64*e^2/(b*e-c*d)^4/d^3*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c
*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(1/2)*b^4-5/128*e^2/(b*e-c*d)^4/d^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*
(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d
/e)/e)^(1/2))/(x+d/e))*b^5-1/16/e*c^(3/2)/(b*e-c*d)^2/d^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*
c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*b+1/8/e*c/(b*e-c*d)^2/d^2/(x+d/e)^2*((x+d/e)^2*c-(b*e-c*d)*d/e
^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)+35/64*e^2/(b*e-c*d)^4/d^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^
(1/2)*b^3*c-45/32*e/(b*e-c*d)^4/d^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^2*c^2+5/32*e/(
b*e-c*d)^3/d^3/(x+d/e)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*b^2-9/8/e/(b*e-c*d)^3/d*c^(
5/2)*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*b+5/8
/e/(b*e-c*d)^3/d/(x+d/e)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c^2+25/16/(b*e-c*d)^4/d*(
(x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*c^3+1/8*c^3/(b*e-c*d)^3/d^2*((x+d/e)^2*c-(b*e-c*d)*
d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x-1/8*c^2/(b*e-c*d)^3/d^2/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*
(x+d/e)/e)^(3/2)+45/32/(b*e-c*d)^4/d*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*
e-2*c*d)*(x+d/e)/e)^(1/2))*c^(5/2)*b^2-25/16/e/(b*e-c*d)^4*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2
*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(7/2)*b-5/64*e^3/(b*e-c*d)^4/d^4*c*((x+d/e)^2*c-(b*e-c*d)*d
/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*b^3-1/16*e*c^2/(b*e-c*d)^3/d^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x
+d/e)/e)^(1/2)*x*b-1/8/e^2*c^2/(b*e-c*d)^2/d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/
e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b+45/128*e
/(b*e-c*d)^4/d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1
/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^4*c-15/16*e/(b*e-c*d)^4/d^2*c^3*((x+
d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*b+1/16*e*c/(b*e-c*d)^3/d^3/(x+d/e)*((x+d/e)^2*c-(b*e-c
*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*b-15/8/e^2/(b*e-c*d)^4*d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^
2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/
(x+d/e))*c^4*b+15/16*e/(b*e-c*d)^4/d^2/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c^2*b
-15/32*e^2/(b*e-c*d)^4/d^3/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*b^2*c+15/16/e/(b*
e-c*d)^3/d*c^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2
)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2+15/32*e^2/(b*e-c*d)^4/d^3*c^2*((x+d/
e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*b^2-1/8/e*c^2/(b*e-c*d)^2/d^2*((x+d/e)^2*c-(b*e-c*d)*d/e
^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)+1/8/e^3*c^3/(b*e-c*d)^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2
*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))
+1/8/e^2*c^(5/2)/(b*e-c*d)^2/d*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*
d)*(x+d/e)/e)^(1/2))-3/4/e/(b*e-c*d)^3/d*c^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)-5/64*e^
3/(b*e-c*d)^4/d^4*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^4+5/8/e^2/(b*e-c*d)^4*d*ln(((x+d
/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(9/2)+1/4/e^3/(b
*e-c*d)/d/(x+d/e)^4*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)-5/4/(b*e-c*d)^4/d/(-(b*e-c*d)*d/
e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^
2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^3*c^2-5/8/(b*e-c*d)^3/d^2/(x+d/e)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b
*e-2*c*d)*(x+d/e)/e)^(3/2)*b*c-3/16/(b*e-c*d)^3/d^2*c/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c
*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b
^3
________________________________________________________________________________________
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)^(1/2)/(e*x+d)^5,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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d zero or nonzero?
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x + c*x^2)^(1/2)/(d + e*x)^5,x)
[Out]
int((b*x + c*x^2)^(1/2)/(d + e*x)^5, x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**2+b*x)**(1/2)/(e*x+d)**5,x)
[Out]
Integral(sqrt(x*(b + c*x))/(d + e*x)**5, x)
________________________________________________________________________________________