3.4.52 \(\int \frac {1}{\sqrt {d+e x} (b x+c x^2)^2} \, dx\)

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

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Rubi [A]  time = 0.26, 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 = {740, 826, 1166, 208} \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 208

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

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 826

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 1166

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 \operatorname {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 ) \operatorname {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)) \operatorname {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 = 0.46, size = 148, normalized size = 0.96 \begin {gather*} \frac {\frac {b \sqrt {d+e x} \left (b^2 e+b c (e x-d)-2 c^2 d x\right )}{x (b+c x) (c d-b e)}+\frac {c^{3/2} d (5 b e-4 c d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{(c d-b e)^{3/2}}+\frac {(b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}}{b^3 d} \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^2*e - 2*c^2*d*x + b*c*(-d + e*x)))/((c*d - b*e)*x*(b + c*x)) + ((4*c*d + b*e)*ArcTanh[Sqr
t[d + e*x]/Sqrt[d]])/Sqrt[d] + (c^(3/2)*d*(-4*c*d + 5*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(
c*d - b*e)^(3/2))/(b^3*d)

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IntegrateAlgebraic [A]  time = 0.77, size = 195, normalized size = 1.27 \begin {gather*} \frac {\left (4 c^{5/2} d-5 b c^{3/2} e\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b^3 (b e-c d)^{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} \left (b^2 e^2+b c e (d+e x)-2 b c d e+2 c^2 d^2-2 c^2 d (d+e x)\right )}{b^2 d x (b e-c d) (b e+c (d+e x)-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.57, size = 1164, normalized size = 7.56

result too large to display

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

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giac [A]  time = 0.18, 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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maple [A]  time = 0.06, size = 202, normalized size = 1.31 \begin {gather*} \frac {5 c^{2} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{2}}-\frac {4 c^{3} d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {\sqrt {e x +d}\, c^{2} e}{\left (b e -c d \right ) \left (c e x +b e \right ) b^{2}}+\frac {e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2} d^{\frac {3}{2}}}+\frac {4 c \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3} \sqrt {d}}-\frac {\sqrt {e x +d}}{b^{2} d x} \end {gather*}

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)

[Out]

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

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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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 1.22, size = 3784, normalized size = 24.57

result too large to display

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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