3.4.5 \(\int \frac {1}{(d+e x)^2 (b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=207 \[ -\frac {e \sqrt {b x+c x^2} \left (3 b^2 e^2-4 b c d e+4 c^2 d^2\right )}{b^2 d^2 (d+e x) (c d-b e)^2}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}+\frac {3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{5/2} (c d-b e)^{5/2}} \]

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Rubi [A]  time = 0.24, antiderivative size = 207, 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 = {740, 806, 724, 206} \begin {gather*} -\frac {e \sqrt {b x+c x^2} \left (3 b^2 e^2-4 b c d e+4 c^2 d^2\right )}{b^2 d^2 (d+e x) (c d-b e)^2}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}+\frac {3 e^2 (2 c d-b e) \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 )}{2 d^{5/2} (c d-b e)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

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

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Mathematica [A]  time = 0.18, size = 206, normalized size = 1.00 \begin {gather*} \frac {-3 b^2 e^2 \sqrt {x} \sqrt {b+c x} (d+e x) (b e-2 c d) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )-\sqrt {d} \sqrt {c d-b e} \left (b^3 e^2 (2 d+3 e x)+b^2 c e \left (-4 d^2-2 d e x+3 e^2 x^2\right )+2 b c^2 d \left (d^2-d e x-2 e^2 x^2\right )+4 c^3 d^2 x (d+e x)\right )}{b^2 d^{5/2} \sqrt {x (b+c x)} (d+e x) (c d-b e)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.03, size = 238, normalized size = 1.15 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-2 b^3 d e^2-3 b^3 e^3 x+4 b^2 c d^2 e+2 b^2 c d e^2 x-3 b^2 c e^3 x^2-2 b c^2 d^3+2 b c^2 d^2 e x+4 b c^2 d e^2 x^2-4 c^3 d^3 x-4 c^3 d^2 e x^2\right )}{b^2 d^2 x (b+c x) (d+e x) (b e-c d)^2}+\frac {3 \left (2 c d e^2-b e^3\right ) \tanh ^{-1}\left (\frac {-e \sqrt {b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.44, size = 914, normalized size = 4.42 \begin {gather*} \left [-\frac {3 \, {\left ({\left (2 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x^{3} + {\left (2 \, b^{2} c^{2} d^{2} e^{2} + b^{3} c d e^{3} - b^{4} e^{4}\right )} x^{2} + {\left (2 \, b^{3} c d^{2} e^{2} - b^{4} d e^{3}\right )} x\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + 2 \, {\left (2 \, b c^{3} d^{5} - 6 \, b^{2} c^{2} d^{4} e + 6 \, b^{3} c d^{3} e^{2} - 2 \, b^{4} d^{2} e^{3} + {\left (4 \, c^{4} d^{4} e - 8 \, b c^{3} d^{3} e^{2} + 7 \, b^{2} c^{2} d^{2} e^{3} - 3 \, b^{3} c d e^{4}\right )} x^{2} + {\left (4 \, c^{4} d^{5} - 6 \, b c^{3} d^{4} e + 5 \, b^{3} c d^{2} e^{3} - 3 \, b^{4} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{2 \, {\left ({\left (b^{2} c^{4} d^{6} e - 3 \, b^{3} c^{3} d^{5} e^{2} + 3 \, b^{4} c^{2} d^{4} e^{3} - b^{5} c d^{3} e^{4}\right )} x^{3} + {\left (b^{2} c^{4} d^{7} - 2 \, b^{3} c^{3} d^{6} e + 2 \, b^{5} c d^{4} e^{3} - b^{6} d^{3} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{7} - 3 \, b^{4} c^{2} d^{6} e + 3 \, b^{5} c d^{5} e^{2} - b^{6} d^{4} e^{3}\right )} x\right )}}, \frac {3 \, {\left ({\left (2 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x^{3} + {\left (2 \, b^{2} c^{2} d^{2} e^{2} + b^{3} c d e^{3} - b^{4} e^{4}\right )} x^{2} + {\left (2 \, b^{3} c d^{2} e^{2} - b^{4} d e^{3}\right )} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (2 \, b c^{3} d^{5} - 6 \, b^{2} c^{2} d^{4} e + 6 \, b^{3} c d^{3} e^{2} - 2 \, b^{4} d^{2} e^{3} + {\left (4 \, c^{4} d^{4} e - 8 \, b c^{3} d^{3} e^{2} + 7 \, b^{2} c^{2} d^{2} e^{3} - 3 \, b^{3} c d e^{4}\right )} x^{2} + {\left (4 \, c^{4} d^{5} - 6 \, b c^{3} d^{4} e + 5 \, b^{3} c d^{2} e^{3} - 3 \, b^{4} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{{\left (b^{2} c^{4} d^{6} e - 3 \, b^{3} c^{3} d^{5} e^{2} + 3 \, b^{4} c^{2} d^{4} e^{3} - b^{5} c d^{3} e^{4}\right )} x^{3} + {\left (b^{2} c^{4} d^{7} - 2 \, b^{3} c^{3} d^{6} e + 2 \, b^{5} c d^{4} e^{3} - b^{6} d^{3} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{7} - 3 \, b^{4} c^{2} d^{6} e + 3 \, b^{5} c d^{5} e^{2} - b^{6} d^{4} e^{3}\right )} x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.18, size = 776, normalized size = 3.75 \begin {gather*} \frac {1}{2} \, {\left (\frac {{\left (8 \, \sqrt {c d^{2} - b d e} c^{\frac {5}{2}} d^{2} e^{2} + 6 \, b^{2} c d e^{4} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} \right |}\right ) - 8 \, \sqrt {c d^{2} - b d e} b c^{\frac {3}{2}} d e^{3} - 3 \, b^{3} e^{5} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} \right |}\right ) + 6 \, \sqrt {c d^{2} - b d e} b^{2} \sqrt {c} e^{4}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{\sqrt {c d^{2} - b d e} b^{2} c^{2} d^{4} - 2 \, \sqrt {c d^{2} - b d e} b^{3} c d^{3} e + \sqrt {c d^{2} - b d e} b^{4} d^{2} e^{2}} + \frac {2 \, {\left (\frac {{\left (\frac {4 \, c^{3} d^{3} e^{8} - 6 \, b c^{2} d^{2} e^{9} + 8 \, b^{2} c d e^{10} - 3 \, b^{3} e^{11}}{b^{2} c^{2} d^{4} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 2 \, b^{3} c d^{3} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + b^{4} d^{2} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (b^{2} c d^{2} e^{11} - b^{3} d e^{12}\right )} e^{\left (-1\right )}}{{\left (b^{2} c^{2} d^{4} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 2 \, b^{3} c d^{3} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + b^{4} d^{2} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}}\right )} e^{\left (-1\right )}}{x e + d} - \frac {4 \, c^{3} d^{2} e^{7} - 4 \, b c^{2} d e^{8} + 3 \, b^{2} c e^{9}}{b^{2} c^{2} d^{4} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 2 \, b^{3} c d^{3} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + b^{4} d^{2} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )}}{\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 {3 \, {\left (2 \, c d e^{5} - b e^{6}\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^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} \sqrt {c d^{2} - b d e} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((8*sqrt(c*d^2 - b*d*e)*c^(5/2)*d^2*e^2 + 6*b^2*c*d*e^4*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c
))) - 8*sqrt(c*d^2 - b*d*e)*b*c^(3/2)*d*e^3 - 3*b^3*e^5*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c)))
+ 6*sqrt(c*d^2 - b*d*e)*b^2*sqrt(c)*e^4)*sgn(1/(x*e + d))/(sqrt(c*d^2 - b*d*e)*b^2*c^2*d^4 - 2*sqrt(c*d^2 - b*
d*e)*b^3*c*d^3*e + sqrt(c*d^2 - b*d*e)*b^4*d^2*e^2) + 2*(((4*c^3*d^3*e^8 - 6*b*c^2*d^2*e^9 + 8*b^2*c*d*e^10 -
3*b^3*e^11)/(b^2*c^2*d^4*e^5*sgn(1/(x*e + d)) - 2*b^3*c*d^3*e^6*sgn(1/(x*e + d)) + b^4*d^2*e^7*sgn(1/(x*e + d)
)) - (b^2*c*d^2*e^11 - b^3*d*e^12)*e^(-1)/((b^2*c^2*d^4*e^5*sgn(1/(x*e + d)) - 2*b^3*c*d^3*e^6*sgn(1/(x*e + d)
) + b^4*d^2*e^7*sgn(1/(x*e + d)))*(x*e + d)))*e^(-1)/(x*e + d) - (4*c^3*d^2*e^7 - 4*b*c^2*d*e^8 + 3*b^2*c*e^9)
/(b^2*c^2*d^4*e^5*sgn(1/(x*e + d)) - 2*b^3*c*d^3*e^6*sgn(1/(x*e + d)) + b^4*d^2*e^7*sgn(1/(x*e + d))))/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) - 3*(2*c*d*e^5 - b*e^6)*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^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3)*sqrt(c*d^2 -
b*d*e)*sgn(1/(x*e + d))))*e^(-2)

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maple [B]  time = 0.07, size = 893, normalized size = 4.31 \begin {gather*} \frac {3 b \,e^{2} \ln \left (\frac {-\frac {2 \left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{x +\frac {d}{e}}\right )}{2 \left (b e -c d \right )^{2} \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, d^{2}}+\frac {12 c^{2} e x}{\left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b d}-\frac {12 c^{3} x}{\left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b^{2}}-\frac {3 c e \ln \left (\frac {-\frac {2 \left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{x +\frac {d}{e}}\right )}{\left (b e -c d \right )^{2} \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, d}-\frac {3 c \,e^{2} x}{\left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, d^{2}}-\frac {3 b \,e^{2}}{\left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, d^{2}}-\frac {6 c^{2}}{\left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b}+\frac {9 c e}{\left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, d}-\frac {8 c^{2} x}{\left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b^{2} d}-\frac {4 c}{\left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b d}+\frac {1}{\left (b e -c d \right ) \left (x +\frac {d}{e}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(b*e-c*d)/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)^(1/2)-3*e^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)*b+9*e/(b*e-c*d)^2/d/((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*e^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)*x*c+12*e/(
b*e-c*d)^2/d/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^2-12/(b*e-c*d)^2/b^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*c^3-6/(b*e-c*d)^2/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(
x+d/e)/e)^(1/2)*c^2+3/2*e^2/(b*e-c*d)^2/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-3*e/(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))*c-8*c^2/(b*e-c*d)/d/b^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-4*c/(b*e-c*d)/d/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1
/2)

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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)^2/(c*x^2+b*x)^(3/2),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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive, negative or zero?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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