3.3.94 \(\int \frac {\sqrt {b x+c x^2}}{(d+e x)^6} \, dx\) [294]

Optimal. Leaf size=337 \[ \frac {(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac {b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{256 d^{9/2} (c d-b e)^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.60, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {758, 848, 820, 734, 738, 212} \begin {gather*} -\frac {b^2 (2 c d-b e) \left (7 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 )}{256 d^{9/2} (c d-b e)^{9/2}}+\frac {\sqrt {b x+c x^2} (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac {e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )}{240 d^3 (d+e x)^3 (c d-b e)^3}-\frac {7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{40 d^2 (d+e x)^4 (c d-b e)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x + c*x^2]/(d + e*x)^6,x]

[Out]

((2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(128*d^4*(c*d
- b*e)^4*(d + e*x)^2) - (e*(b*x + c*x^2)^(3/2))/(5*d*(c*d - b*e)*(d + e*x)^5) - (7*e*(2*c*d - b*e)*(b*x + c*x^
2)^(3/2))/(40*d^2*(c*d - b*e)^2*(d + e*x)^4) - (e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2)*(b*x + c*x^2)^(3/2)
)/(240*d^3*(c*d - b*e)^3*(d + e*x)^3) - (b^2*(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*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])])/(256*d^(9/2)*(c*d - b*e)^(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 {b x+c x^2}}{(d+e x)^6} \, dx &=-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {\int \frac {\left (\frac {1}{2} (-10 c d+7 b e)+2 c e x\right ) \sqrt {b x+c x^2}}{(d+e x)^5} \, dx}{5 d (c d-b e)}\\ &=-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac {\int \frac {\left (\frac {1}{4} \left (80 c^2 d^2-94 b c d e+35 b^2 e^2\right )-\frac {7}{2} c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx}{20 d^2 (c d-b e)^2}\\ &=-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}+\frac {\left ((2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right )\right ) \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx}{32 d^3 (c d-b e)^3}\\ &=\frac {(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac {\left (b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{256 d^4 (c d-b e)^4}\\ &=\frac {(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}+\frac {\left (b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right )\right ) \text {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 )}{128 d^4 (c d-b e)^4}\\ &=\frac {(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}-\frac {e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac {b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 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 )}{256 d^{9/2} (c d-b e)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 10.78, size = 308, normalized size = 0.91 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (384 e x^{3/2} (b+c x)+\frac {336 e (2 c d-b e) x^{3/2} (b+c x) (d+e x)}{d (c d-b e)}+\frac {8 e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) x^{3/2} (b+c x) (d+e x)^2}{d^2 (c d-b e)^2}+\frac {15 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (d+e x)^3 \left (\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 \tanh ^{-1}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )}{d^{7/2} (c d-b e)^{7/2} \sqrt {b+c x}}\right )}{1920 d (-c d+b e) \sqrt {x} (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(384*e*x^(3/2)*(b + c*x) + (336*e*(2*c*d - b*e)*x^(3/2)*(b + c*x)*(d + e*x))/(d*(c*d - b*e)
) + (8*e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2)*x^(3/2)*(b + c*x)*(d + e*x)^2)/(d^2*(c*d - b*e)^2) + (15*(2*
c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(d + e*x)^3*(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^(7/
2)*(c*d - b*e)^(7/2)*Sqrt[b + c*x])))/(1920*d*(-(c*d) + b*e)*Sqrt[x]*(d + e*x)^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3381\) vs. \(2(307)=614\).
time = 0.53, size = 3382, normalized size = 10.04

method	result	size
default	\(\text {Expression too large to display}\)	\(3382\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^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-%e*b>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1209 vs. \(2 (325) = 650\).
time = 1.60, size = 2430, normalized size = 7.21 \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)^(1/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

[-1/3840*(15*(32*b^2*c^3*d^8 - 7*b^5*x^5*e^8 + 5*(6*b^4*c*d*x^5 - 7*b^5*d*x^4)*e^7 - 2*(24*b^3*c^2*d^2*x^5 - 7
5*b^4*c*d^2*x^4 + 35*b^5*d^2*x^3)*e^6 + 2*(16*b^2*c^3*d^3*x^5 - 120*b^3*c^2*d^3*x^4 + 150*b^4*c*d^3*x^3 - 35*b
^5*d^3*x^2)*e^5 + 5*(32*b^2*c^3*d^4*x^4 - 96*b^3*c^2*d^4*x^3 + 60*b^4*c*d^4*x^2 - 7*b^5*d^4*x)*e^4 + (320*b^2*
c^3*d^5*x^3 - 480*b^3*c^2*d^5*x^2 + 150*b^4*c*d^5*x - 7*b^5*d^5)*e^3 + 10*(32*b^2*c^3*d^6*x^2 - 24*b^3*c^2*d^6
*x + 3*b^4*c*d^6)*e^2 + 16*(10*b^2*c^3*d^7*x - 3*b^3*c^2*d^7)*e)*sqrt(c*d^2 - b*d*e)*log((2*c*d*x - b*x*e + b*
d + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(x*e + d)) - 2*(960*c^5*d^9*x + 480*b*c^4*d^9 - 105*b^5*d*x^4*e^8
+ 5*(97*b^4*c*d^2*x^4 - 98*b^5*d^2*x^3)*e^7 - 4*(214*b^3*c^2*d^3*x^4 - 567*b^4*c*d^3*x^3 + 224*b^5*d^3*x^2)*e
^6 + 2*(334*b^2*c^3*d^4*x^4 - 2007*b^3*c^2*d^4*x^3 + 2079*b^4*c*d^4*x^2 - 395*b^5*d^4*x)*e^5 - (288*b*c^4*d^5*
x^4 - 3244*b^2*c^3*d^5*x^3 + 7458*b^3*c^2*d^5*x^2 - 3740*b^4*c*d^5*x - 105*b^5*d^5)*e^4 + (96*c^5*d^6*x^4 - 14
88*b*c^4*d^6*x^3 + 6356*b^2*c^3*d^6*x^2 - 6970*b^3*c^2*d^6*x - 555*b^4*c*d^6)*e^3 + 30*(16*c^5*d^7*x^3 - 104*b
*c^4*d^7*x^2 + 214*b^2*c^3*d^7*x + 39*b^3*c^2*d^7)*e^2 + 240*(4*c^5*d^8*x^2 - 14*b*c^4*d^8*x - 5*b^2*c^3*d^8)*
e)*sqrt(c*x^2 + b*x))/(c^5*d^15 - b^5*d^5*x^5*e^10 + 5*(b^4*c*d^6*x^5 - b^5*d^6*x^4)*e^9 - 5*(2*b^3*c^2*d^7*x^
5 - 5*b^4*c*d^7*x^4 + 2*b^5*d^7*x^3)*e^8 + 10*(b^2*c^3*d^8*x^5 - 5*b^3*c^2*d^8*x^4 + 5*b^4*c*d^8*x^3 - b^5*d^8
*x^2)*e^7 - 5*(b*c^4*d^9*x^5 - 10*b^2*c^3*d^9*x^4 + 20*b^3*c^2*d^9*x^3 - 10*b^4*c*d^9*x^2 + b^5*d^9*x)*e^6 + (
c^5*d^10*x^5 - 25*b*c^4*d^10*x^4 + 100*b^2*c^3*d^10*x^3 - 100*b^3*c^2*d^10*x^2 + 25*b^4*c*d^10*x - b^5*d^10)*e
^5 + 5*(c^5*d^11*x^4 - 10*b*c^4*d^11*x^3 + 20*b^2*c^3*d^11*x^2 - 10*b^3*c^2*d^11*x + b^4*c*d^11)*e^4 + 10*(c^5
*d^12*x^3 - 5*b*c^4*d^12*x^2 + 5*b^2*c^3*d^12*x - b^3*c^2*d^12)*e^3 + 5*(2*c^5*d^13*x^2 - 5*b*c^4*d^13*x + 2*b
^2*c^3*d^13)*e^2 + 5*(c^5*d^14*x - b*c^4*d^14)*e), -1/1920*(15*(32*b^2*c^3*d^8 - 7*b^5*x^5*e^8 + 5*(6*b^4*c*d*
x^5 - 7*b^5*d*x^4)*e^7 - 2*(24*b^3*c^2*d^2*x^5 - 75*b^4*c*d^2*x^4 + 35*b^5*d^2*x^3)*e^6 + 2*(16*b^2*c^3*d^3*x^
5 - 120*b^3*c^2*d^3*x^4 + 150*b^4*c*d^3*x^3 - 35*b^5*d^3*x^2)*e^5 + 5*(32*b^2*c^3*d^4*x^4 - 96*b^3*c^2*d^4*x^3
+ 60*b^4*c*d^4*x^2 - 7*b^5*d^4*x)*e^4 + (320*b^2*c^3*d^5*x^3 - 480*b^3*c^2*d^5*x^2 + 150*b^4*c*d^5*x - 7*b^5*
d^5)*e^3 + 10*(32*b^2*c^3*d^6*x^2 - 24*b^3*c^2*d^6*x + 3*b^4*c*d^6)*e^2 + 16*(10*b^2*c^3*d^7*x - 3*b^3*c^2*d^7
)*e)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x - b*x*e)) - (960*c^5*d^9*x + 4
80*b*c^4*d^9 - 105*b^5*d*x^4*e^8 + 5*(97*b^4*c*d^2*x^4 - 98*b^5*d^2*x^3)*e^7 - 4*(214*b^3*c^2*d^3*x^4 - 567*b^
4*c*d^3*x^3 + 224*b^5*d^3*x^2)*e^6 + 2*(334*b^2*c^3*d^4*x^4 - 2007*b^3*c^2*d^4*x^3 + 2079*b^4*c*d^4*x^2 - 395*
b^5*d^4*x)*e^5 - (288*b*c^4*d^5*x^4 - 3244*b^2*c^3*d^5*x^3 + 7458*b^3*c^2*d^5*x^2 - 3740*b^4*c*d^5*x - 105*b^5
*d^5)*e^4 + (96*c^5*d^6*x^4 - 1488*b*c^4*d^6*x^3 + 6356*b^2*c^3*d^6*x^2 - 6970*b^3*c^2*d^6*x - 555*b^4*c*d^6)*
e^3 + 30*(16*c^5*d^7*x^3 - 104*b*c^4*d^7*x^2 + 214*b^2*c^3*d^7*x + 39*b^3*c^2*d^7)*e^2 + 240*(4*c^5*d^8*x^2 -
14*b*c^4*d^8*x - 5*b^2*c^3*d^8)*e)*sqrt(c*x^2 + b*x))/(c^5*d^15 - b^5*d^5*x^5*e^10 + 5*(b^4*c*d^6*x^5 - b^5*d^
6*x^4)*e^9 - 5*(2*b^3*c^2*d^7*x^5 - 5*b^4*c*d^7*x^4 + 2*b^5*d^7*x^3)*e^8 + 10*(b^2*c^3*d^8*x^5 - 5*b^3*c^2*d^8
*x^4 + 5*b^4*c*d^8*x^3 - b^5*d^8*x^2)*e^7 - 5*(b*c^4*d^9*x^5 - 10*b^2*c^3*d^9*x^4 + 20*b^3*c^2*d^9*x^3 - 10*b^
4*c*d^9*x^2 + b^5*d^9*x)*e^6 + (c^5*d^10*x^5 - 25*b*c^4*d^10*x^4 + 100*b^2*c^3*d^10*x^3 - 100*b^3*c^2*d^10*x^2
+ 25*b^4*c*d^10*x - b^5*d^10)*e^5 + 5*(c^5*d^11*x^4 - 10*b*c^4*d^11*x^3 + 20*b^2*c^3*d^11*x^2 - 10*b^3*c^2*d^
11*x + b^4*c*d^11)*e^4 + 10*(c^5*d^12*x^3 - 5*b*c^4*d^12*x^2 + 5*b^2*c^3*d^12*x - b^3*c^2*d^12)*e^3 + 5*(2*c^5
*d^13*x^2 - 5*b*c^4*d^13*x + 2*b^2*c^3*d^13)*e^2 + 5*(c^5*d^14*x - b*c^4*d^14)*e)]

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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 )^{6}}\, 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)**6,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2101 vs. \(2 (325) = 650\).
time = 2.46, size = 2101, normalized size = 6.23 \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)^(1/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

1/128*(32*b^2*c^3*d^3 - 48*b^3*c^2*d^2*e + 30*b^4*c*d*e^2 - 7*b^5*e^3)*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))
*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^
4*e^4)*sqrt(-c*d^2 + b*d*e)) + 1/1920*(7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*c^(13/2)*d^8*e + 3072*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^5*c^7*d^9 + 9216*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b*c^6*d^8*e + 7680*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^4*b*c^(13/2)*d^9 - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b*c^(11/2)*d^7*e^2 - 3840*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^4*b^2*c^(11/2)*d^8*e + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*c^6*d^9 - 50048*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^5*b^2*c^5*d^7*e^2 - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^3*c^5*d^8*e + 3840*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^2*b^3*c^(11/2)*d^9 + 70720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^2*c^(9/2)*d^6*e^3 -
17600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^3*c^(9/2)*d^7*e^2 - 7200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^4*c^(9
/2)*d^8*e + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^4*c^5*d^9 + 15040*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*b^2*c^4*
d^5*e^4 + 129280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^3*c^4*d^6*e^3 + 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b
^4*c^4*d^7*e^2 - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^5*c^4*d^8*e + 96*b^5*c^(9/2)*d^9 + 4320*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^8*b^2*c^(7/2)*d^4*e^5 - 52000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^3*c^(7/2)*d^5*e^4 + 81920*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^4*c^(7/2)*d^6*e^3 + 13760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^5*c^(7/2)*d^
7*e^2 - 192*b^6*c^(7/2)*d^8*e + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*b^2*c^3*d^3*e^6 - 20320*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^7*b^3*c^3*d^4*e^5 - 120680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^4*c^3*d^5*e^4 + 14080*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^3*b^5*c^3*d^6*e^3 + 4280*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^6*c^3*d^7*e^2 - 6480*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^8*b^3*c^(5/2)*d^3*e^6 + 7260*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^4*c^(5/2)*d^4*e^5 -
85780*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^5*c^(5/2)*d^5*e^4 - 6340*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^6*c^(5/
2)*d^6*e^3 + 476*b^7*c^(5/2)*d^7*e^2 - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*b^3*c^2*d^2*e^7 + 10740*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^7*b^4*c^2*d^3*e^6 + 47944*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^5*c^2*d^4*e^5 - 25220*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*b^6*c^2*d^5*e^4 - 3080*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^7*c^2*d^6*e^3 + 4050*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^8*b^4*c^(3/2)*d^2*e^7 + 9310*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^5*c^(3/2)*d^3*
e^6 + 35330*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^6*c^(3/2)*d^4*e^5 - 1750*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^7
*c^(3/2)*d^5*e^4 - 380*b^8*c^(3/2)*d^6*e^3 + 450*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*b^4*c*d*e^8 - 1190*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^7*b^5*c*d^2*e^7 - 4658*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^6*c*d^3*e^6 + 10510*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^3*b^7*c*d^4*e^5 + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^8*c*d^5*e^4 - 945*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^8*b^5*sqrt(c)*d*e^8 - 3430*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^6*sqrt(c)*d^2*e^7 - 4480*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^7*sqrt(c)*d^3*e^6 + 1470*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^8*sqrt(c)*d^4*
e^5 + 105*b^9*sqrt(c)*d^5*e^4 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*b^5*e^9 - 490*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^7*b^6*d*e^8 - 896*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^7*d^2*e^7 - 790*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*
b^8*d^3*e^6 + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^9*d^4*e^5)/((c^4*d^8*e^2 - 4*b*c^3*d^7*e^3 + 6*b^2*c^2*d^6
*e^4 - 4*b^3*c*d^5*e^5 + b^4*d^4*e^6)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))
*sqrt(c)*d + b*d)^5)

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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}}{{\left (d+e\,x\right )}^6} \,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)^6,x)

[Out]

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

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