3.3.93 \(\int \frac {\sqrt {b x+c x^2}}{(d+e x)^5} \, dx\) [293]

Optimal. Leaf size=258 \[ \frac {\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{64 d^3 (c d-b e)^3 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 d^2 (c d-b e)^2 (d+e x)^3}-\frac {b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{128 d^{7/2} (c d-b e)^{7/2}} \]

[Out]

-1/4*e*(c*x^2+b*x)^(3/2)/d/(-b*e+c*d)/(e*x+d)^4-5/24*e*(-b*e+2*c*d)*(c*x^2+b*x)^(3/2)/d^2/(-b*e+c*d)^2/(e*x+d)
^3-1/128*b^2*(5*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^(7/2)/(-b*e+c*d)^(7/2)+1/64*(5*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^3/(-b*e+c*d)^3/(e*x+d)^2

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Rubi [A]
time = 0.34, 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 = {758, 820, 734, 738, 212} \begin {gather*} -\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)} \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 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]

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 ) \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 )}{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*}

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Mathematica [A]
time = 10.48, size = 243, normalized size = 0.94 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (48 e x^{3/2} (b+c x)+\frac {40 e (2 c d-b e) x^{3/2} (b+c x) (d+e x)}{d (c d-b e)}+\frac {3 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (d+e x)^2 \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^{5/2} (c d-b e)^{5/2} \sqrt {b+c x}}\right )}{192 d (-c d+b e) \sqrt {x} (d+e x)^4} \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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2183\) vs. \(2(232)=464\).
time = 0.48, size = 2184, normalized size = 8.47

method	result	size
default	\(\text {Expression too large to display}\)	\(2184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(1/2)/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

1/e^5*(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)))))

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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)^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(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. 800 vs. \(2 (245) = 490\).
time = 2.01, size = 1612, normalized size = 6.25 \begin {gather*} \left [-\frac {3 \, {\left (16 \, b^{2} c^{2} d^{6} + 5 \, b^{4} x^{4} e^{6} - 4 \, {\left (4 \, b^{3} c d x^{4} - 5 \, b^{4} d x^{3}\right )} e^{5} + 2 \, {\left (8 \, b^{2} c^{2} d^{2} x^{4} - 32 \, b^{3} c d^{2} x^{3} + 15 \, b^{4} d^{2} x^{2}\right )} e^{4} + 4 \, {\left (16 \, b^{2} c^{2} d^{3} x^{3} - 24 \, b^{3} c d^{3} x^{2} + 5 \, b^{4} d^{3} x\right )} e^{3} + {\left (96 \, b^{2} c^{2} d^{4} x^{2} - 64 \, b^{3} c d^{4} x + 5 \, b^{4} d^{4}\right )} e^{2} + 16 \, {\left (4 \, b^{2} c^{2} d^{5} x - b^{3} c d^{5}\right )} e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {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}\right ) - 2 \, {\left (96 \, c^{4} d^{7} x + 48 \, b c^{3} d^{7} + 15 \, b^{4} d x^{3} e^{6} - {\left (53 \, b^{3} c d^{2} x^{3} - 55 \, b^{4} d^{2} x^{2}\right )} e^{5} + {\left (62 \, b^{2} c^{2} d^{3} x^{3} - 195 \, b^{3} c d^{3} x^{2} + 73 \, b^{4} d^{3} x\right )} e^{4} - {\left (40 \, b c^{3} d^{4} x^{3} - 244 \, b^{2} c^{2} d^{4} x^{2} + 271 \, b^{3} c d^{4} x + 15 \, b^{4} d^{4}\right )} e^{3} + {\left (16 \, c^{4} d^{5} x^{3} - 168 \, b c^{3} d^{5} x^{2} + 374 \, b^{2} c^{2} d^{5} x + 63 \, b^{3} c d^{5}\right )} e^{2} + 16 \, {\left (4 \, c^{4} d^{6} x^{2} - 17 \, b c^{3} d^{6} x - 6 \, b^{2} c^{2} d^{6}\right )} e\right )} \sqrt {c x^{2} + b x}}{384 \, {\left (c^{4} d^{12} + b^{4} d^{4} x^{4} e^{8} - 4 \, {\left (b^{3} c d^{5} x^{4} - b^{4} d^{5} x^{3}\right )} e^{7} + 2 \, {\left (3 \, b^{2} c^{2} d^{6} x^{4} - 8 \, b^{3} c d^{6} x^{3} + 3 \, b^{4} d^{6} x^{2}\right )} e^{6} - 4 \, {\left (b c^{3} d^{7} x^{4} - 6 \, b^{2} c^{2} d^{7} x^{3} + 6 \, b^{3} c d^{7} x^{2} - b^{4} d^{7} x\right )} e^{5} + {\left (c^{4} d^{8} x^{4} - 16 \, b c^{3} d^{8} x^{3} + 36 \, b^{2} c^{2} d^{8} x^{2} - 16 \, b^{3} c d^{8} x + b^{4} d^{8}\right )} e^{4} + 4 \, {\left (c^{4} d^{9} x^{3} - 6 \, b c^{3} d^{9} x^{2} + 6 \, b^{2} c^{2} d^{9} x - b^{3} c d^{9}\right )} e^{3} + 2 \, {\left (3 \, c^{4} d^{10} x^{2} - 8 \, b c^{3} d^{10} x + 3 \, b^{2} c^{2} d^{10}\right )} e^{2} + 4 \, {\left (c^{4} d^{11} x - b c^{3} d^{11}\right )} e\right )}}, -\frac {3 \, {\left (16 \, b^{2} c^{2} d^{6} + 5 \, b^{4} x^{4} e^{6} - 4 \, {\left (4 \, b^{3} c d x^{4} - 5 \, b^{4} d x^{3}\right )} e^{5} + 2 \, {\left (8 \, b^{2} c^{2} d^{2} x^{4} - 32 \, b^{3} c d^{2} x^{3} + 15 \, b^{4} d^{2} x^{2}\right )} e^{4} + 4 \, {\left (16 \, b^{2} c^{2} d^{3} x^{3} - 24 \, b^{3} c d^{3} x^{2} + 5 \, b^{4} d^{3} x\right )} e^{3} + {\left (96 \, b^{2} c^{2} d^{4} x^{2} - 64 \, b^{3} c d^{4} x + 5 \, b^{4} d^{4}\right )} e^{2} + 16 \, {\left (4 \, b^{2} c^{2} d^{5} x - b^{3} c d^{5}\right )} e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) - {\left (96 \, c^{4} d^{7} x + 48 \, b c^{3} d^{7} + 15 \, b^{4} d x^{3} e^{6} - {\left (53 \, b^{3} c d^{2} x^{3} - 55 \, b^{4} d^{2} x^{2}\right )} e^{5} + {\left (62 \, b^{2} c^{2} d^{3} x^{3} - 195 \, b^{3} c d^{3} x^{2} + 73 \, b^{4} d^{3} x\right )} e^{4} - {\left (40 \, b c^{3} d^{4} x^{3} - 244 \, b^{2} c^{2} d^{4} x^{2} + 271 \, b^{3} c d^{4} x + 15 \, b^{4} d^{4}\right )} e^{3} + {\left (16 \, c^{4} d^{5} x^{3} - 168 \, b c^{3} d^{5} x^{2} + 374 \, b^{2} c^{2} d^{5} x + 63 \, b^{3} c d^{5}\right )} e^{2} + 16 \, {\left (4 \, c^{4} d^{6} x^{2} - 17 \, b c^{3} d^{6} x - 6 \, b^{2} c^{2} d^{6}\right )} e\right )} \sqrt {c x^{2} + b x}}{192 \, {\left (c^{4} d^{12} + b^{4} d^{4} x^{4} e^{8} - 4 \, {\left (b^{3} c d^{5} x^{4} - b^{4} d^{5} x^{3}\right )} e^{7} + 2 \, {\left (3 \, b^{2} c^{2} d^{6} x^{4} - 8 \, b^{3} c d^{6} x^{3} + 3 \, b^{4} d^{6} x^{2}\right )} e^{6} - 4 \, {\left (b c^{3} d^{7} x^{4} - 6 \, b^{2} c^{2} d^{7} x^{3} + 6 \, b^{3} c d^{7} x^{2} - b^{4} d^{7} x\right )} e^{5} + {\left (c^{4} d^{8} x^{4} - 16 \, b c^{3} d^{8} x^{3} + 36 \, b^{2} c^{2} d^{8} x^{2} - 16 \, b^{3} c d^{8} x + b^{4} d^{8}\right )} e^{4} + 4 \, {\left (c^{4} d^{9} x^{3} - 6 \, b c^{3} d^{9} x^{2} + 6 \, b^{2} c^{2} d^{9} x - b^{3} c d^{9}\right )} e^{3} + 2 \, {\left (3 \, c^{4} d^{10} x^{2} - 8 \, b c^{3} d^{10} x + 3 \, b^{2} c^{2} d^{10}\right )} e^{2} + 4 \, {\left (c^{4} d^{11} x - b c^{3} d^{11}\right )} e\right )}}\right ] \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="fricas")

[Out]

[-1/384*(3*(16*b^2*c^2*d^6 + 5*b^4*x^4*e^6 - 4*(4*b^3*c*d*x^4 - 5*b^4*d*x^3)*e^5 + 2*(8*b^2*c^2*d^2*x^4 - 32*b
^3*c*d^2*x^3 + 15*b^4*d^2*x^2)*e^4 + 4*(16*b^2*c^2*d^3*x^3 - 24*b^3*c*d^3*x^2 + 5*b^4*d^3*x)*e^3 + (96*b^2*c^2
*d^4*x^2 - 64*b^3*c*d^4*x + 5*b^4*d^4)*e^2 + 16*(4*b^2*c^2*d^5*x - b^3*c*d^5)*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*(96*c^4*d^7*x + 48*b*c^3*d^7 + 15*
b^4*d*x^3*e^6 - (53*b^3*c*d^2*x^3 - 55*b^4*d^2*x^2)*e^5 + (62*b^2*c^2*d^3*x^3 - 195*b^3*c*d^3*x^2 + 73*b^4*d^3
*x)*e^4 - (40*b*c^3*d^4*x^3 - 244*b^2*c^2*d^4*x^2 + 271*b^3*c*d^4*x + 15*b^4*d^4)*e^3 + (16*c^4*d^5*x^3 - 168*
b*c^3*d^5*x^2 + 374*b^2*c^2*d^5*x + 63*b^3*c*d^5)*e^2 + 16*(4*c^4*d^6*x^2 - 17*b*c^3*d^6*x - 6*b^2*c^2*d^6)*e)
*sqrt(c*x^2 + b*x))/(c^4*d^12 + b^4*d^4*x^4*e^8 - 4*(b^3*c*d^5*x^4 - b^4*d^5*x^3)*e^7 + 2*(3*b^2*c^2*d^6*x^4 -
8*b^3*c*d^6*x^3 + 3*b^4*d^6*x^2)*e^6 - 4*(b*c^3*d^7*x^4 - 6*b^2*c^2*d^7*x^3 + 6*b^3*c*d^7*x^2 - b^4*d^7*x)*e^
5 + (c^4*d^8*x^4 - 16*b*c^3*d^8*x^3 + 36*b^2*c^2*d^8*x^2 - 16*b^3*c*d^8*x + b^4*d^8)*e^4 + 4*(c^4*d^9*x^3 - 6*
b*c^3*d^9*x^2 + 6*b^2*c^2*d^9*x - b^3*c*d^9)*e^3 + 2*(3*c^4*d^10*x^2 - 8*b*c^3*d^10*x + 3*b^2*c^2*d^10)*e^2 +
4*(c^4*d^11*x - b*c^3*d^11)*e), -1/192*(3*(16*b^2*c^2*d^6 + 5*b^4*x^4*e^6 - 4*(4*b^3*c*d*x^4 - 5*b^4*d*x^3)*e^
5 + 2*(8*b^2*c^2*d^2*x^4 - 32*b^3*c*d^2*x^3 + 15*b^4*d^2*x^2)*e^4 + 4*(16*b^2*c^2*d^3*x^3 - 24*b^3*c*d^3*x^2 +
5*b^4*d^3*x)*e^3 + (96*b^2*c^2*d^4*x^2 - 64*b^3*c*d^4*x + 5*b^4*d^4)*e^2 + 16*(4*b^2*c^2*d^5*x - b^3*c*d^5)*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)) - (96*c^4*d^7*x + 48*b*
c^3*d^7 + 15*b^4*d*x^3*e^6 - (53*b^3*c*d^2*x^3 - 55*b^4*d^2*x^2)*e^5 + (62*b^2*c^2*d^3*x^3 - 195*b^3*c*d^3*x^2
+ 73*b^4*d^3*x)*e^4 - (40*b*c^3*d^4*x^3 - 244*b^2*c^2*d^4*x^2 + 271*b^3*c*d^4*x + 15*b^4*d^4)*e^3 + (16*c^4*d
^5*x^3 - 168*b*c^3*d^5*x^2 + 374*b^2*c^2*d^5*x + 63*b^3*c*d^5)*e^2 + 16*(4*c^4*d^6*x^2 - 17*b*c^3*d^6*x - 6*b^
2*c^2*d^6)*e)*sqrt(c*x^2 + b*x))/(c^4*d^12 + b^4*d^4*x^4*e^8 - 4*(b^3*c*d^5*x^4 - b^4*d^5*x^3)*e^7 + 2*(3*b^2*
c^2*d^6*x^4 - 8*b^3*c*d^6*x^3 + 3*b^4*d^6*x^2)*e^6 - 4*(b*c^3*d^7*x^4 - 6*b^2*c^2*d^7*x^3 + 6*b^3*c*d^7*x^2 -
b^4*d^7*x)*e^5 + (c^4*d^8*x^4 - 16*b*c^3*d^8*x^3 + 36*b^2*c^2*d^8*x^2 - 16*b^3*c*d^8*x + b^4*d^8)*e^4 + 4*(c^4
*d^9*x^3 - 6*b*c^3*d^9*x^2 + 6*b^2*c^2*d^9*x - b^3*c*d^9)*e^3 + 2*(3*c^4*d^10*x^2 - 8*b*c^3*d^10*x + 3*b^2*c^2
*d^10)*e^2 + 4*(c^4*d^11*x - b*c^3*d^11)*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 )^{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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1144 vs. \(2 (245) = 490\).
time = 2.76, size = 1144, normalized size = 4.43 \begin {gather*} \frac {1}{384} \, {\left (2 \, \sqrt {c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}} {\left (\frac {2 \, {\left (\frac {4 \, {\left (\frac {2 \, c^{3} d^{5} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 5 \, b c^{2} d^{4} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 4 \, b^{2} c d^{3} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b^{3} d^{2} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{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}} - \frac {6 \, {\left (c^{3} d^{6} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 3 \, b c^{2} d^{5} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, b^{2} c d^{4} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b^{3} d^{3} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} e^{\left (-1\right )}}{{\left (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}\right )} {\left (x e + d\right )}}\right )} e^{\left (-1\right )}}{x e + d} + \frac {8 \, c^{3} d^{4} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 16 \, b c^{2} d^{3} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 13 \, b^{2} c d^{2} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 5 \, b^{3} d e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{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}}\right )} e^{\left (-1\right )}}{x e + d} + \frac {16 \, c^{3} d^{3} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 24 \, b c^{2} d^{2} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 38 \, b^{2} c d e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 15 \, b^{3} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{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}}\right )} - \frac {{\left (48 \, b^{2} c^{2} d^{2} e^{2} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} \right |}\right ) + 32 \, \sqrt {c d^{2} - b d e} c^{\frac {7}{2}} d^{3} - 48 \, \sqrt {c d^{2} - b d e} b c^{\frac {5}{2}} d^{2} e - 48 \, b^{3} c d e^{3} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} \right |}\right ) + 76 \, \sqrt {c d^{2} - b d e} b^{2} c^{\frac {3}{2}} d e^{2} + 15 \, b^{4} e^{4} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} \right |}\right ) - 30 \, \sqrt {c d^{2} - b d e} b^{3} \sqrt {c} e^{3}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{\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}} + \frac {3 \, {\left (16 \, b^{2} c^{2} d^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 16 \, b^{3} c d e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 5 \, b^{4} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} {\left (\sqrt {c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}} + \frac {\sqrt {c d^{2} e^{2} - b d e^{3}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{{\left (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}\right )} \sqrt {c d^{2} - b d e}}\right )} e^{2} \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="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

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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 )}^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)

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