3.4.74 \(\int \frac {1}{\sqrt {d+e x} (b x+c x^2)^2} \, dx\) [374]

Optimal. Leaf size=154 \[ -\frac {\sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \left (b x+c x^2\right )}+\frac {(4 c d+b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{3/2}}-\frac {c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {754, 840, 1180, 214} \begin {gather*} -\frac {c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}+\frac {(b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{3/2}}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

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

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Mathematica [A]
time = 0.71, size = 149, normalized size = 0.97 \begin {gather*} \frac {-\frac {b \sqrt {d+e x} \left (-b c d+b^2 e-2 c^2 d x+b c e x\right )}{d (-c d+b e) x (b+c x)}-\frac {c^{3/2} (4 c d-5 b e) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{3/2}}+\frac {(4 c d+b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.52, size = 160, normalized size = 1.04

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {c^{2} \left (\frac {b e \sqrt {e x +d}}{2 \left (b e -c d \right ) \left (c \left (e x +d \right )+b e -c d \right )}+\frac {\left (5 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3}}+\frac {-\frac {b \sqrt {e x +d}}{2 d x}+\frac {\left (b e +4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 d^{\frac {3}{2}}}}{b^{3} e^{3}}\right )\)	\(160\)
default	\(2 e^{3} \left (\frac {c^{2} \left (\frac {b e \sqrt {e x +d}}{2 \left (b e -c d \right ) \left (c \left (e x +d \right )+b e -c d \right )}+\frac {\left (5 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3}}+\frac {-\frac {b \sqrt {e x +d}}{2 d x}+\frac {\left (b e +4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 d^{\frac {3}{2}}}}{b^{3} e^{3}}\right )\)	\(160\)
risch	\(-\frac {\sqrt {e x +d}}{d \,b^{2} x}+\frac {e \,c^{2} \sqrt {e x +d}}{b^{2} \left (b e -c d \right ) \left (c e x +b e \right )}+\frac {5 e \,c^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{2} \left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}}-\frac {4 d \,c^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} \left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}}+\frac {e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{d^{\frac {3}{2}} b^{2}}+\frac {4 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{\sqrt {d}\, b^{3}}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^3*(c^2/b^3/e^3*(1/2*b*e/(b*e-c*d)*(e*x+d)^(1/2)/(c*(e*x+d)+b*e-c*d)+1/2*(5*b*e-4*c*d)/(b*e-c*d)/((b*e-c*d)
*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)))+1/b^3/e^3*(-1/2*b/d*(e*x+d)^(1/2)/x+1/2*(b*e+4*c*d)/d^(
3/2)*arctanh((e*x+d)^(1/2)/d^(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)^(1/2)/(c*x^2+b*x)^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. 293 vs. \(2 (146) = 292\).
time = 3.13, size = 1212, normalized size = 7.87 \begin {gather*} \left [\frac {{\left (4 \, c^{3} d^{3} x^{2} + 4 \, b c^{2} d^{3} x - 5 \, {\left (b c^{2} d^{2} x^{2} + b^{2} c d^{2} x\right )} e\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {2 \, c d - 2 \, {\left (c d - b e\right )} \sqrt {x e + d} \sqrt {\frac {c}{c d - b e}} + {\left (c x - b\right )} e}{c x + b}\right ) + {\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - {\left (b^{2} c x^{2} + b^{3} x\right )} e^{2} - 3 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {d} \log \left (\frac {x e + 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) - 2 \, {\left (2 \, b c^{2} d^{2} x + b^{2} c d^{2} - {\left (b^{2} c d x + b^{3} d\right )} e\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c^{2} d^{3} x^{2} + b^{4} c d^{3} x - {\left (b^{4} c d^{2} x^{2} + b^{5} d^{2} x\right )} e\right )}}, -\frac {2 \, {\left (4 \, c^{3} d^{3} x^{2} + 4 \, b c^{2} d^{3} x - 5 \, {\left (b c^{2} d^{2} x^{2} + b^{2} c d^{2} x\right )} e\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {x e + d} \sqrt {-\frac {c}{c d - b e}}}{c x e + c d}\right ) - {\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - {\left (b^{2} c x^{2} + b^{3} x\right )} e^{2} - 3 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {d} \log \left (\frac {x e + 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (2 \, b c^{2} d^{2} x + b^{2} c d^{2} - {\left (b^{2} c d x + b^{3} d\right )} e\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c^{2} d^{3} x^{2} + b^{4} c d^{3} x - {\left (b^{4} c d^{2} x^{2} + b^{5} d^{2} x\right )} e\right )}}, -\frac {2 \, {\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - {\left (b^{2} c x^{2} + b^{3} x\right )} e^{2} - 3 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) - {\left (4 \, c^{3} d^{3} x^{2} + 4 \, b c^{2} d^{3} x - 5 \, {\left (b c^{2} d^{2} x^{2} + b^{2} c d^{2} x\right )} e\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {2 \, c d - 2 \, {\left (c d - b e\right )} \sqrt {x e + d} \sqrt {\frac {c}{c d - b e}} + {\left (c x - b\right )} e}{c x + b}\right ) + 2 \, {\left (2 \, b c^{2} d^{2} x + b^{2} c d^{2} - {\left (b^{2} c d x + b^{3} d\right )} e\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c^{2} d^{3} x^{2} + b^{4} c d^{3} x - {\left (b^{4} c d^{2} x^{2} + b^{5} d^{2} x\right )} e\right )}}, -\frac {{\left (4 \, c^{3} d^{3} x^{2} + 4 \, b c^{2} d^{3} x - 5 \, {\left (b c^{2} d^{2} x^{2} + b^{2} c d^{2} x\right )} e\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {x e + d} \sqrt {-\frac {c}{c d - b e}}}{c x e + c d}\right ) + {\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - {\left (b^{2} c x^{2} + b^{3} x\right )} e^{2} - 3 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (2 \, b c^{2} d^{2} x + b^{2} c d^{2} - {\left (b^{2} c d x + b^{3} d\right )} e\right )} \sqrt {x e + d}}{b^{3} c^{2} d^{3} x^{2} + b^{4} c d^{3} x - {\left (b^{4} c d^{2} x^{2} + b^{5} d^{2} x\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((4*c^3*d^3*x^2 + 4*b*c^2*d^3*x - 5*(b*c^2*d^2*x^2 + b^2*c*d^2*x)*e)*sqrt(c/(c*d - b*e))*log((2*c*d - 2*(
c*d - b*e)*sqrt(x*e + d)*sqrt(c/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)) + (4*c^3*d^2*x^2 + 4*b*c^2*d^2*x - (b^2
*c*x^2 + b^3*x)*e^2 - 3*(b*c^2*d*x^2 + b^2*c*d*x)*e)*sqrt(d)*log((x*e + 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) - 2*
(2*b*c^2*d^2*x + b^2*c*d^2 - (b^2*c*d*x + b^3*d)*e)*sqrt(x*e + d))/(b^3*c^2*d^3*x^2 + b^4*c*d^3*x - (b^4*c*d^2
*x^2 + b^5*d^2*x)*e), -1/2*(2*(4*c^3*d^3*x^2 + 4*b*c^2*d^3*x - 5*(b*c^2*d^2*x^2 + b^2*c*d^2*x)*e)*sqrt(-c/(c*d
- b*e))*arctan(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) - (4*c^3*d^2*x^2 + 4*b*c^2*d^2*
x - (b^2*c*x^2 + b^3*x)*e^2 - 3*(b*c^2*d*x^2 + b^2*c*d*x)*e)*sqrt(d)*log((x*e + 2*sqrt(x*e + d)*sqrt(d) + 2*d)
/x) + 2*(2*b*c^2*d^2*x + b^2*c*d^2 - (b^2*c*d*x + b^3*d)*e)*sqrt(x*e + d))/(b^3*c^2*d^3*x^2 + b^4*c*d^3*x - (b
^4*c*d^2*x^2 + b^5*d^2*x)*e), -1/2*(2*(4*c^3*d^2*x^2 + 4*b*c^2*d^2*x - (b^2*c*x^2 + b^3*x)*e^2 - 3*(b*c^2*d*x^
2 + b^2*c*d*x)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) - (4*c^3*d^3*x^2 + 4*b*c^2*d^3*x - 5*(b*c^2*d^2*x^
2 + b^2*c*d^2*x)*e)*sqrt(c/(c*d - b*e))*log((2*c*d - 2*(c*d - b*e)*sqrt(x*e + d)*sqrt(c/(c*d - b*e)) + (c*x -
b)*e)/(c*x + b)) + 2*(2*b*c^2*d^2*x + b^2*c*d^2 - (b^2*c*d*x + b^3*d)*e)*sqrt(x*e + d))/(b^3*c^2*d^3*x^2 + b^4
*c*d^3*x - (b^4*c*d^2*x^2 + b^5*d^2*x)*e), -((4*c^3*d^3*x^2 + 4*b*c^2*d^3*x - 5*(b*c^2*d^2*x^2 + b^2*c*d^2*x)*
e)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) + (4*c^3*d^2*x^2
+ 4*b*c^2*d^2*x - (b^2*c*x^2 + b^3*x)*e^2 - 3*(b*c^2*d*x^2 + b^2*c*d*x)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt
(-d)/d) + (2*b*c^2*d^2*x + b^2*c*d^2 - (b^2*c*d*x + b^3*d)*e)*sqrt(x*e + d))/(b^3*c^2*d^3*x^2 + b^4*c*d^3*x -
(b^4*c*d^2*x^2 + b^5*d^2*x)*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.35, size = 253, normalized size = 1.64 \begin {gather*} \frac {{\left (4 \, c^{3} d - 5 \, b c^{2} e\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c d - b^{4} e\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d e - 2 \, \sqrt {x e + d} c^{2} d^{2} e - {\left (x e + d\right )}^{\frac {3}{2}} b c e^{2} + 2 \, \sqrt {x e + d} b c d e^{2} - \sqrt {x e + d} b^{2} e^{3}}{{\left (b^{2} c d^{2} - b^{3} d e\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}} - \frac {{\left (4 \, c d + b e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.22, size = 2500, normalized size = 16.23 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d + e*x)^(1/2)*(b^2*e^3 + 2*c^2*d^2*e - 2*b*c*d*e^2))/(b^2*(c*d^2 - b*d*e)) + (c*(b*e^2 - 2*c*d*e)*(d + e*x
)^(3/2))/(b^2*(c*d^2 - b*d*e)))/((b*e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + c*d^2 - b*d*e) + (atan((((-c^3*(b*e
- c*d)^3)^(1/2)*((2*(d + e*x)^(1/2)*(b^4*c^3*e^6 + 32*c^7*d^4*e^2 - 64*b*c^6*d^3*e^3 + 6*b^3*c^4*d*e^5 + 26*b
^2*c^5*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) - ((-c^3*(b*e - c*d)^3)^(1/2)*(5*b*e - 4*c*d)*((4
*b^9*c^2*d*e^6 + 8*b^6*c^5*d^4*e^3 - 16*b^7*c^4*d^3*e^4 + 4*b^8*c^3*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^
7*c*d^3*e) + ((-c^3*(b*e - c*d)^3)^(1/2)*(5*b*e - 4*c*d)*(d + e*x)^(1/2)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e
^3 + 16*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5))/((b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)*(b^6*e^3 - b^3*c^3*
d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2))))/(2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*(5*b*
e - 4*c*d)*1i)/(2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)) + ((-c^3*(b*e - c*d)^3)^(1/2)*((2
*(d + e*x)^(1/2)*(b^4*c^3*e^6 + 32*c^7*d^4*e^2 - 64*b*c^6*d^3*e^3 + 6*b^3*c^4*d*e^5 + 26*b^2*c^5*d^2*e^4))/(b^
4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) + ((-c^3*(b*e - c*d)^3)^(1/2)*(5*b*e - 4*c*d)*((4*b^9*c^2*d*e^6 + 8*b
^6*c^5*d^4*e^3 - 16*b^7*c^4*d^3*e^4 + 4*b^8*c^3*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) - ((-c^3*
(b*e - c*d)^3)^(1/2)*(5*b*e - 4*c*d)*(d + e*x)^(1/2)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d^3*
e^4 - 4*b^9*c^2*d^2*e^5))/((b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*
e - 3*b^5*c*d*e^2))))/(2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*(5*b*e - 4*c*d)*1i)/(2*(b
^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))/((2*(5*b^3*c^4*e^6 + 32*c^7*d^3*e^3 - 48*b*c^6*d^2*e
^4 + 6*b^2*c^5*d*e^5))/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) + ((-c^3*(b*e - c*d)^3)^(1/2)*((2*(d + e*x)
^(1/2)*(b^4*c^3*e^6 + 32*c^7*d^4*e^2 - 64*b*c^6*d^3*e^3 + 6*b^3*c^4*d*e^5 + 26*b^2*c^5*d^2*e^4))/(b^4*c^2*d^4
+ b^6*d^2*e^2 - 2*b^5*c*d^3*e) - ((-c^3*(b*e - c*d)^3)^(1/2)*(5*b*e - 4*c*d)*((4*b^9*c^2*d*e^6 + 8*b^6*c^5*d^4
*e^3 - 16*b^7*c^4*d^3*e^4 + 4*b^8*c^3*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) + ((-c^3*(b*e - c*d
)^3)^(1/2)*(5*b*e - 4*c*d)*(d + e*x)^(1/2)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d^3*e^4 - 4*b^
9*c^2*d^2*e^5))/((b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*
c*d*e^2))))/(2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*(5*b*e - 4*c*d))/(2*(b^6*e^3 - b^3*
c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)) - ((-c^3*(b*e - c*d)^3)^(1/2)*((2*(d + e*x)^(1/2)*(b^4*c^3*e^6 + 3
2*c^7*d^4*e^2 - 64*b*c^6*d^3*e^3 + 6*b^3*c^4*d*e^5 + 26*b^2*c^5*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c
*d^3*e) + ((-c^3*(b*e - c*d)^3)^(1/2)*(5*b*e - 4*c*d)*((4*b^9*c^2*d*e^6 + 8*b^6*c^5*d^4*e^3 - 16*b^7*c^4*d^3*e
^4 + 4*b^8*c^3*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) - ((-c^3*(b*e - c*d)^3)^(1/2)*(5*b*e - 4*c
*d)*(d + e*x)^(1/2)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5))/((b^4*c
^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2))))/(2*(b^6*e^3
- b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*(5*b*e - 4*c*d))/(2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2
*e - 3*b^5*c*d*e^2))))*(-c^3*(b*e - c*d)^3)^(1/2)*(5*b*e - 4*c*d)*1i)/(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e
- 3*b^5*c*d*e^2) + (atan(((((2*(d + e*x)^(1/2)*(b^4*c^3*e^6 + 32*c^7*d^4*e^2 - 64*b*c^6*d^3*e^3 + 6*b^3*c^4*d
*e^5 + 26*b^2*c^5*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) - (((4*b^9*c^2*d*e^6 + 8*b^6*c^5*d^4*e
^3 - 16*b^7*c^4*d^3*e^4 + 4*b^8*c^3*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) + ((b*e + 4*c*d)*(d +
e*x)^(1/2)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5))/(b^3*(d^3)^(1/2
)*(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)))*(b*e + 4*c*d))/(2*b^3*(d^3)^(1/2)))*(b*e + 4*c*d)*1i)/(2*b^3*(
d^3)^(1/2)) + (((2*(d + e*x)^(1/2)*(b^4*c^3*e^6 + 32*c^7*d^4*e^2 - 64*b*c^6*d^3*e^3 + 6*b^3*c^4*d*e^5 + 26*b^2
*c^5*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) + (((4*b^9*c^2*d*e^6 + 8*b^6*c^5*d^4*e^3 - 16*b^7*c
^4*d^3*e^4 + 4*b^8*c^3*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) - ((b*e + 4*c*d)*(d + e*x)^(1/2)*(
8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5))/(b^3*(d^3)^(1/2)*(b^4*c^2*d^
4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)))*(b*e + 4*c*d))/(2*b^3*(d^3)^(1/2)))*(b*e + 4*c*d)*1i)/(2*b^3*(d^3)^(1/2)))/
((2*(5*b^3*c^4*e^6 + 32*c^7*d^3*e^3 - 48*b*c^6*d^2*e^4 + 6*b^2*c^5*d*e^5))/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*
c*d^3*e) + (((2*(d + e*x)^(1/2)*(b^4*c^3*e^6 + 32*c^7*d^4*e^2 - 64*b*c^6*d^3*e^3 + 6*b^3*c^4*d*e^5 + 26*b^2*c^
5*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) - (((4*b^9*c^2*d*e^6 + 8*b^6*c^5*d^4*e^3 - 16*b^7*c^4*
d^3*e^4 + 4*b^8*c^3*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) + ((b*e + 4*c*d)*(d + e*x)^(1/2)*(8*b
^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^...

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