3.4.27 \(\int \frac {1}{(d+e x)^2 (b x+c x^2)^{3/2}} \, dx\) [327]

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

[Out]

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

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Rubi [A]
time = 0.16, 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 = {754, 820, 738, 212} \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 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 754

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 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)^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 ) \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 )}{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 = 0.87, size = 218, normalized size = 1.05 \begin {gather*} \frac {x \left (-\frac {\sqrt {d} (b+c x) \left (4 c^3 d^2 x (d+e x)+b^3 e^2 (2 d+3 e x)+2 b c^2 d \left (d^2-d e x-2 e^2 x^2\right )+b^2 c e \left (-4 d^2-2 d e x+3 e^2 x^2\right )\right )}{b^2 (c d-b e)^2 (d+e x)}-\frac {3 e^2 (2 c d-b e) \sqrt {x} (b+c x)^{3/2} \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 )}{d^{5/2} (x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(559\) vs. \(2(189)=378\).
time = 0.51, size = 560, normalized size = 2.71

method	result	size
risch	\(-\frac {2 \left (c x +b \right )}{b^{2} d^{2} \sqrt {x \left (c x +b \right )}}-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d^{2} \left (b e -c d \right )^{2} \left (x +\frac {d}{e}\right )}+\frac {3 b \,e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d^{2} \left (b e -c d \right )^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}-\frac {3 e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right ) c}{d \left (b e -c d \right )^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}-\frac {2 c^{2} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{b^{2} \left (b e -c d \right )^{2} \left (\frac {b}{c}+x \right )}\)	\(437\)
default	\(\frac {\frac {e^{2}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {3 e \left (b e -2 c d \right ) \left (-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e \left (b e -2 c d \right ) \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}+\frac {4 c \,e^{2} \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e^{2}}\)	\(560\)

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,method=_RETURNVERBOSE)

[Out]

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

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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(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. 453 vs. \(2 (199) = 398\).
time = 1.58, size = 919, normalized size = 4.44 \begin {gather*} \left [\frac {3 \, \sqrt {c d^{2} - b d e} {\left ({\left (b^{3} c x^{3} + b^{4} x^{2}\right )} e^{4} - {\left (2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2} - b^{4} d x\right )} e^{3} - 2 \, {\left (b^{2} c^{2} d^{2} x^{2} + b^{3} c d^{2} x\right )} e^{2}\right )} \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 (4 \, c^{4} d^{5} x + 2 \, b c^{3} d^{5} - 3 \, {\left (b^{3} c d x^{2} + b^{4} d x\right )} e^{4} + {\left (7 \, b^{2} c^{2} d^{2} x^{2} + 5 \, b^{3} c d^{2} x - 2 \, b^{4} d^{2}\right )} e^{3} - 2 \, {\left (4 \, b c^{3} d^{3} x^{2} - 3 \, b^{3} c d^{3}\right )} e^{2} + 2 \, {\left (2 \, c^{4} d^{4} x^{2} - 3 \, b c^{3} d^{4} x - 3 \, b^{2} c^{2} d^{4}\right )} e\right )} \sqrt {c x^{2} + b x}}{2 \, {\left (b^{2} c^{4} d^{7} x^{2} + b^{3} c^{3} d^{7} x - {\left (b^{5} c d^{3} x^{3} + b^{6} d^{3} x^{2}\right )} e^{4} + {\left (3 \, b^{4} c^{2} d^{4} x^{3} + 2 \, b^{5} c d^{4} x^{2} - b^{6} d^{4} x\right )} e^{3} - 3 \, {\left (b^{3} c^{3} d^{5} x^{3} - b^{5} c d^{5} x\right )} e^{2} + {\left (b^{2} c^{4} d^{6} x^{3} - 2 \, b^{3} c^{3} d^{6} x^{2} - 3 \, b^{4} c^{2} d^{6} x\right )} e\right )}}, -\frac {3 \, \sqrt {-c d^{2} + b d e} {\left ({\left (b^{3} c x^{3} + b^{4} x^{2}\right )} e^{4} - {\left (2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2} - b^{4} d x\right )} e^{3} - 2 \, {\left (b^{2} c^{2} d^{2} x^{2} + b^{3} c d^{2} x\right )} e^{2}\right )} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) + {\left (4 \, c^{4} d^{5} x + 2 \, b c^{3} d^{5} - 3 \, {\left (b^{3} c d x^{2} + b^{4} d x\right )} e^{4} + {\left (7 \, b^{2} c^{2} d^{2} x^{2} + 5 \, b^{3} c d^{2} x - 2 \, b^{4} d^{2}\right )} e^{3} - 2 \, {\left (4 \, b c^{3} d^{3} x^{2} - 3 \, b^{3} c d^{3}\right )} e^{2} + 2 \, {\left (2 \, c^{4} d^{4} x^{2} - 3 \, b c^{3} d^{4} x - 3 \, b^{2} c^{2} d^{4}\right )} e\right )} \sqrt {c x^{2} + b x}}{b^{2} c^{4} d^{7} x^{2} + b^{3} c^{3} d^{7} x - {\left (b^{5} c d^{3} x^{3} + b^{6} d^{3} x^{2}\right )} e^{4} + {\left (3 \, b^{4} c^{2} d^{4} x^{3} + 2 \, b^{5} c d^{4} x^{2} - b^{6} d^{4} x\right )} e^{3} - 3 \, {\left (b^{3} c^{3} d^{5} x^{3} - b^{5} c d^{5} x\right )} e^{2} + {\left (b^{2} c^{4} d^{6} x^{3} - 2 \, b^{3} c^{3} d^{6} x^{2} - 3 \, b^{4} c^{2} d^{6} x\right )} e}\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*sqrt(c*d^2 - b*d*e)*((b^3*c*x^3 + b^4*x^2)*e^4 - (2*b^2*c^2*d*x^3 + b^3*c*d*x^2 - b^4*d*x)*e^3 - 2*(b^
2*c^2*d^2*x^2 + b^3*c*d^2*x)*e^2)*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*(4*c^4*d^5*x + 2*b*c^3*d^5 - 3*(b^3*c*d*x^2 + b^4*d*x)*e^4 + (7*b^2*c^2*d^2*x^2 + 5*b^3*c*d^2*x - 2*b
^4*d^2)*e^3 - 2*(4*b*c^3*d^3*x^2 - 3*b^3*c*d^3)*e^2 + 2*(2*c^4*d^4*x^2 - 3*b*c^3*d^4*x - 3*b^2*c^2*d^4)*e)*sqr
t(c*x^2 + b*x))/(b^2*c^4*d^7*x^2 + b^3*c^3*d^7*x - (b^5*c*d^3*x^3 + b^6*d^3*x^2)*e^4 + (3*b^4*c^2*d^4*x^3 + 2*
b^5*c*d^4*x^2 - b^6*d^4*x)*e^3 - 3*(b^3*c^3*d^5*x^3 - b^5*c*d^5*x)*e^2 + (b^2*c^4*d^6*x^3 - 2*b^3*c^3*d^6*x^2
- 3*b^4*c^2*d^6*x)*e), -(3*sqrt(-c*d^2 + b*d*e)*((b^3*c*x^3 + b^4*x^2)*e^4 - (2*b^2*c^2*d*x^3 + b^3*c*d*x^2 -
b^4*d*x)*e^3 - 2*(b^2*c^2*d^2*x^2 + b^3*c*d^2*x)*e^2)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x -
b*x*e)) + (4*c^4*d^5*x + 2*b*c^3*d^5 - 3*(b^3*c*d*x^2 + b^4*d*x)*e^4 + (7*b^2*c^2*d^2*x^2 + 5*b^3*c*d^2*x - 2*
b^4*d^2)*e^3 - 2*(4*b*c^3*d^3*x^2 - 3*b^3*c*d^3)*e^2 + 2*(2*c^4*d^4*x^2 - 3*b*c^3*d^4*x - 3*b^2*c^2*d^4)*e)*sq
rt(c*x^2 + b*x))/(b^2*c^4*d^7*x^2 + b^3*c^3*d^7*x - (b^5*c*d^3*x^3 + b^6*d^3*x^2)*e^4 + (3*b^4*c^2*d^4*x^3 + 2
*b^5*c*d^4*x^2 - b^6*d^4*x)*e^3 - 3*(b^3*c^3*d^5*x^3 - b^5*c*d^5*x)*e^2 + (b^2*c^4*d^6*x^3 - 2*b^3*c^3*d^6*x^2
- 3*b^4*c^2*d^6*x)*e)]

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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (199) = 398\).
time = 1.82, 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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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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