3.4.21 \(\int \frac {1}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx\) [321]

Optimal. Leaf size=180 \[ -\frac {e \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {b x+c x^2}}{4 d^2 (c d-b e)^2 (d+e x)}+\frac {\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 d^{5/2} (c d-b e)^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {758, 820, 738, 212} \begin {gather*} \frac {\left (3 b^2 e^2-8 b c d e+8 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 )}{8 d^{5/2} (c d-b e)^{5/2}}-\frac {3 e \sqrt {b x+c x^2} (2 c d-b e)}{4 d^2 (d+e x) (c d-b e)^2}-\frac {e \sqrt {b x+c x^2}}{2 d (d+e x)^2 (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*Sqrt[b*x + c*x^2]),x]

[Out]

-1/2*(e*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*(d + e*x)^2) - (3*e*(2*c*d - b*e)*Sqrt[b*x + c*x^2])/(4*d^2*(c*d - b
*e)^2*(d + e*x)) + ((8*c^2*d^2 - 8*b*c*d*e + 3*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])])/(8*d^(5/2)*(c*d - b*e)^(5/2))

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 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 {1}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx &=-\frac {e \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {\int \frac {\frac {1}{2} (-4 c d+3 b e)+c e x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx}{2 d (c d-b e)}\\ &=-\frac {e \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {b x+c x^2}}{4 d^2 (c d-b e)^2 (d+e x)}+\frac {\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 d^2 (c d-b e)^2}\\ &=-\frac {e \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {b x+c x^2}}{4 d^2 (c d-b e)^2 (d+e x)}-\frac {\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\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 )}{4 d^2 (c d-b e)^2}\\ &=-\frac {e \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {b x+c x^2}}{4 d^2 (c d-b e)^2 (d+e x)}+\frac {\left (8 c^2 d^2-8 b c d e+3 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 )}{8 d^{5/2} (c d-b e)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.83, size = 180, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x} \left (-\frac {\sqrt {d} e \sqrt {x} (b+c x) (2 c d (4 d+3 e x)-b e (5 d+3 e x))}{(c d-b e)^2 (d+e x)^2}-\frac {\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \sqrt {b+c x} \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}\right )}{4 d^{5/2} \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*Sqrt[b*x + c*x^2]),x]

[Out]

(Sqrt[x]*(-((Sqrt[d]*e*Sqrt[x]*(b + c*x)*(2*c*d*(4*d + 3*e*x) - b*e*(5*d + 3*e*x)))/((c*d - b*e)^2*(d + e*x)^2
)) - ((8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*Sqrt[b + c*x]*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x)
)/(Sqrt[d]*Sqrt[-(c*d) + b*e])])/(-(c*d) + b*e)^(5/2)))/(4*d^(5/2)*Sqrt[x*(b + c*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(158)=316\).
time = 0.45, size = 473, normalized size = 2.63

method	result	size
default	\(\frac {\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{2 d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 e \left (b e -2 c d \right ) \left (\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {e \left (b e -2 c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{4 d \left (b e -c d \right )}-\frac {c \,e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e^{3}}\)	\(473\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/e^3*(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)^(1/2)+3/4*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)^(1/2)-1/2*e*
(b*e-2*c*d)/d/(b*e-c*d)/(-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)))-1/2*c/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(1/(e*x+d)^3/(c*x^2+b*x)^(1/2),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. 361 vs. \(2 (168) = 336\).
time = 1.49, size = 734, normalized size = 4.08 \begin {gather*} \left [\frac {{\left (8 \, c^{2} d^{4} + 3 \, b^{2} x^{2} e^{4} - 2 \, {\left (4 \, b c d x^{2} - 3 \, b^{2} d x\right )} e^{3} + {\left (8 \, c^{2} d^{2} x^{2} - 16 \, b c d^{2} x + 3 \, b^{2} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{2} d^{3} x - b c d^{3}\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 (8 \, c^{2} d^{4} e + 3 \, b^{2} d x e^{4} - {\left (9 \, b c d^{2} x - 5 \, b^{2} d^{2}\right )} e^{3} + {\left (6 \, c^{2} d^{3} x - 13 \, b c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c^{3} d^{8} - b^{3} d^{3} x^{2} e^{5} + {\left (3 \, b^{2} c d^{4} x^{2} - 2 \, b^{3} d^{4} x\right )} e^{4} - {\left (3 \, b c^{2} d^{5} x^{2} - 6 \, b^{2} c d^{5} x + b^{3} d^{5}\right )} e^{3} + {\left (c^{3} d^{6} x^{2} - 6 \, b c^{2} d^{6} x + 3 \, b^{2} c d^{6}\right )} e^{2} + {\left (2 \, c^{3} d^{7} x - 3 \, b c^{2} d^{7}\right )} e\right )}}, \frac {{\left (8 \, c^{2} d^{4} + 3 \, b^{2} x^{2} e^{4} - 2 \, {\left (4 \, b c d x^{2} - 3 \, b^{2} d x\right )} e^{3} + {\left (8 \, c^{2} d^{2} x^{2} - 16 \, b c d^{2} x + 3 \, b^{2} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{2} d^{3} x - b c d^{3}\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 (8 \, c^{2} d^{4} e + 3 \, b^{2} d x e^{4} - {\left (9 \, b c d^{2} x - 5 \, b^{2} d^{2}\right )} e^{3} + {\left (6 \, c^{2} d^{3} x - 13 \, b c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c^{3} d^{8} - b^{3} d^{3} x^{2} e^{5} + {\left (3 \, b^{2} c d^{4} x^{2} - 2 \, b^{3} d^{4} x\right )} e^{4} - {\left (3 \, b c^{2} d^{5} x^{2} - 6 \, b^{2} c d^{5} x + b^{3} d^{5}\right )} e^{3} + {\left (c^{3} d^{6} x^{2} - 6 \, b c^{2} d^{6} x + 3 \, b^{2} c d^{6}\right )} e^{2} + {\left (2 \, c^{3} d^{7} x - 3 \, b c^{2} d^{7}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

[1/8*((8*c^2*d^4 + 3*b^2*x^2*e^4 - 2*(4*b*c*d*x^2 - 3*b^2*d*x)*e^3 + (8*c^2*d^2*x^2 - 16*b*c*d^2*x + 3*b^2*d^2
)*e^2 + 8*(2*c^2*d^3*x - b*c*d^3)*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)*sq
rt(c*x^2 + b*x))/(x*e + d)) - 2*(8*c^2*d^4*e + 3*b^2*d*x*e^4 - (9*b*c*d^2*x - 5*b^2*d^2)*e^3 + (6*c^2*d^3*x -
13*b*c*d^3)*e^2)*sqrt(c*x^2 + b*x))/(c^3*d^8 - b^3*d^3*x^2*e^5 + (3*b^2*c*d^4*x^2 - 2*b^3*d^4*x)*e^4 - (3*b*c^
2*d^5*x^2 - 6*b^2*c*d^5*x + b^3*d^5)*e^3 + (c^3*d^6*x^2 - 6*b*c^2*d^6*x + 3*b^2*c*d^6)*e^2 + (2*c^3*d^7*x - 3*
b*c^2*d^7)*e), 1/4*((8*c^2*d^4 + 3*b^2*x^2*e^4 - 2*(4*b*c*d*x^2 - 3*b^2*d*x)*e^3 + (8*c^2*d^2*x^2 - 16*b*c*d^2
*x + 3*b^2*d^2)*e^2 + 8*(2*c^2*d^3*x - b*c*d^3)*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)) - (8*c^2*d^4*e + 3*b^2*d*x*e^4 - (9*b*c*d^2*x - 5*b^2*d^2)*e^3 + (6*c^2*d^3*x - 13*b
*c*d^3)*e^2)*sqrt(c*x^2 + b*x))/(c^3*d^8 - b^3*d^3*x^2*e^5 + (3*b^2*c*d^4*x^2 - 2*b^3*d^4*x)*e^4 - (3*b*c^2*d^
5*x^2 - 6*b^2*c*d^5*x + b^3*d^5)*e^3 + (c^3*d^6*x^2 - 6*b*c^2*d^6*x + 3*b^2*c*d^6)*e^2 + (2*c^3*d^7*x - 3*b*c^
2*d^7)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(1/(sqrt(x*(b + c*x))*(d + e*x)**3), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (168) = 336\).
time = 1.42, size = 487, normalized size = 2.71 \begin {gather*} -\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e}} - \frac {8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{2} d^{2} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {3}{2}} d^{2} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b c^{2} d^{3} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c d e^{2} - 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e + 6 \, b^{2} c^{\frac {3}{2}} d^{3} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} - 3 \, b^{3} \sqrt {c} d^{2} e + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2}}{4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

-1/4*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 +
b*d*e))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) - 1/4*(8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
3*c^2*d^2*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*c^(5/2)*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b*c^(3/2
)*d^2*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b*c^2*d^3 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c*d*e^2 - 20*(s
qrt(c)*x - sqrt(c*x^2 + b*x))*b^2*c*d^2*e + 6*b^2*c^(3/2)*d^3 + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*sqrt(c
)*d*e^2 - 3*b^3*sqrt(c)*d^2*e + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*e^3 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)*b^3*d*e^2)/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*((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)^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^(1/2)*(d + e*x)^3),x)

[Out]

int(1/((b*x + c*x^2)^(1/2)*(d + e*x)^3), x)

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