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

Optimal. Leaf size=125 \[ \frac {(4 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 \sqrt {d}}-\frac {\sqrt {c} (4 c d-3 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 \sqrt {c d-b e}}-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )} \]

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Rubi [A]  time = 0.23, antiderivative size = 125, 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 = {736, 826, 1166, 208} \begin {gather*} -\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \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 \sqrt {d}}-\frac {\sqrt {c} (4 c d-3 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 \sqrt {c d-b e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(b*x + c*x^2)^2,x]

[Out]

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

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 736

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*(b + 2*
c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d
, e}, 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] && GtQ[m
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && 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 {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx &=-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}+\frac {\int \frac {-2 c d+\frac {b e}{2}-c e x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {c d e+e \left (-2 c d+\frac {b e}{2}\right )-c 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}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}+\frac {(c (4 c d-3 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}-\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}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \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 \sqrt {d}}-\frac {\sqrt {c} (4 c d-3 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 \sqrt {c d-b e}}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 169, normalized size = 1.35 \begin {gather*} \frac {\sqrt {d} \left (b (b+2 c x) \sqrt {d+e x} (c d-b e)+\sqrt {c} x (b+c x) (4 c d-3 b e) \sqrt {c d-b e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )-x (b+c x) \left (b^2 e^2-5 b c d e+4 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 \sqrt {d} x (b+c x) (b e-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.70, size = 157, normalized size = 1.26 \begin {gather*} \frac {\left (3 b \sqrt {c} e-4 c^{3/2} d\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b^3 \sqrt {b e-c d}}+\frac {(4 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 \sqrt {d}}-\frac {\sqrt {d+e x} (b e+2 c (d+e x)-2 c d)}{b^2 x (b e+c (d+e x)-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 798, normalized size = 6.38 \begin {gather*} \left [-\frac {{\left ({\left (4 \, c^{2} d^{2} - 3 \, b c d e\right )} x^{2} + {\left (4 \, b c d^{2} - 3 \, b^{2} d e\right )} x\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, {\left (c d - b e\right )} \sqrt {e x + d} \sqrt {\frac {c}{c d - b e}}}{c x + b}\right ) + {\left ({\left (4 \, c^{2} d - b c e\right )} x^{2} + {\left (4 \, b c d - b^{2} e\right )} x\right )} \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (2 \, b c d x + b^{2} d\right )} \sqrt {e x + d}}{2 \, {\left (b^{3} c d x^{2} + b^{4} d x\right )}}, -\frac {2 \, {\left ({\left (4 \, c^{2} d^{2} - 3 \, b c d e\right )} x^{2} + {\left (4 \, b c d^{2} - 3 \, b^{2} d e\right )} x\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {e x + d} \sqrt {-\frac {c}{c d - b e}}}{c e x + c d}\right ) + {\left ({\left (4 \, c^{2} d - b c e\right )} x^{2} + {\left (4 \, b c d - b^{2} e\right )} x\right )} \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (2 \, b c d x + b^{2} d\right )} \sqrt {e x + d}}{2 \, {\left (b^{3} c d x^{2} + b^{4} d x\right )}}, -\frac {2 \, {\left ({\left (4 \, c^{2} d - b c e\right )} x^{2} + {\left (4 \, b c d - b^{2} e\right )} x\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left ({\left (4 \, c^{2} d^{2} - 3 \, b c d e\right )} x^{2} + {\left (4 \, b c d^{2} - 3 \, b^{2} d e\right )} x\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, {\left (c d - b e\right )} \sqrt {e x + d} \sqrt {\frac {c}{c d - b e}}}{c x + b}\right ) + 2 \, {\left (2 \, b c d x + b^{2} d\right )} \sqrt {e x + d}}{2 \, {\left (b^{3} c d x^{2} + b^{4} d x\right )}}, -\frac {{\left ({\left (4 \, c^{2} d^{2} - 3 \, b c d e\right )} x^{2} + {\left (4 \, b c d^{2} - 3 \, b^{2} d e\right )} x\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {e x + d} \sqrt {-\frac {c}{c d - b e}}}{c e x + c d}\right ) + {\left ({\left (4 \, c^{2} d - b c e\right )} x^{2} + {\left (4 \, b c d - b^{2} e\right )} x\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (2 \, b c d x + b^{2} d\right )} \sqrt {e x + d}}{b^{3} c d x^{2} + b^{4} d x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 181, normalized size = 1.45 \begin {gather*} \frac {{\left (4 \, c^{2} d - 3 \, b c e\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3}} - \frac {{\left (4 \, c d - b e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c e - 2 \, \sqrt {x e + d} c d e + \sqrt {x e + d} b e^{2}}{{\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 )} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 167, normalized size = 1.34 \begin {gather*} -\frac {3 c e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {4 c^{2} d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{3}}-\frac {\sqrt {e x +d}\, c e}{\left (c e x +b e \right ) b^{2}}-\frac {e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2} \sqrt {d}}+\frac {4 c \sqrt {d}\, \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3}}-\frac {\sqrt {e x +d}}{b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e/b^2*c*(e*x+d)^(1/2)/(c*e*x+b*e)-3*e/b^2*c/((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^2/((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*(e*x+d)^(1/2)/x-e/b^2/d^(1/2)
*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((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 = 0.53, size = 1174, normalized size = 9.39

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atanh((2*c^2*e^6*(d + e*x)^(1/2))/(d^(3/2)*((8*c^3*e^5)/b - (2*c^2*e^6)/d)) - (8*c^3*e^5*(d + e*x)^(1/2))/(d^
(1/2)*(8*c^3*e^5 - (2*b*c^2*e^6)/d)))*(b*e - 4*c*d))/(b^3*d^(1/2)) - ((2*c*e*(d + e*x)^(3/2))/b^2 + (e*(b*e -
2*c*d)*(d + e*x)^(1/2))/b^2)/((b*e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + c*d^2 - b*d*e) + (atan(((((4*(d + e*x)
^(1/2)*(5*b^2*c^3*e^4 + 16*c^5*d^2*e^2 - 16*b*c^4*d*e^3))/b^4 - ((-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*((2*(2
*b^7*c^2*e^4 - 4*b^6*c^3*d*e^3))/b^6 - (2*(2*b^7*c^2*e^3 - 4*b^6*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*
c*d)*(d + e*x)^(1/2))/(b^4*(b^4*e - b^3*c*d))))/(2*(b^4*e - b^3*c*d)))*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*
1i)/(2*(b^4*e - b^3*c*d)) + (((4*(d + e*x)^(1/2)*(5*b^2*c^3*e^4 + 16*c^5*d^2*e^2 - 16*b*c^4*d*e^3))/b^4 + ((-c
*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*((2*(2*b^7*c^2*e^4 - 4*b^6*c^3*d*e^3))/b^6 + (2*(2*b^7*c^2*e^3 - 4*b^6*c^3
*d*e^2)*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*(d + e*x)^(1/2))/(b^4*(b^4*e - b^3*c*d))))/(2*(b^4*e - b^3*c*d)
))*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*1i)/(2*(b^4*e - b^3*c*d)))/((4*(3*b^2*c^3*e^5 + 16*c^5*d^2*e^3 - 16*
b*c^4*d*e^4))/b^6 - (((4*(d + e*x)^(1/2)*(5*b^2*c^3*e^4 + 16*c^5*d^2*e^2 - 16*b*c^4*d*e^3))/b^4 - ((-c*(b*e -
c*d))^(1/2)*(3*b*e - 4*c*d)*((2*(2*b^7*c^2*e^4 - 4*b^6*c^3*d*e^3))/b^6 - (2*(2*b^7*c^2*e^3 - 4*b^6*c^3*d*e^2)*
(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*(d + e*x)^(1/2))/(b^4*(b^4*e - b^3*c*d))))/(2*(b^4*e - b^3*c*d)))*(-c*(
b*e - c*d))^(1/2)*(3*b*e - 4*c*d))/(2*(b^4*e - b^3*c*d)) + (((4*(d + e*x)^(1/2)*(5*b^2*c^3*e^4 + 16*c^5*d^2*e^
2 - 16*b*c^4*d*e^3))/b^4 + ((-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*((2*(2*b^7*c^2*e^4 - 4*b^6*c^3*d*e^3))/b^6
+ (2*(2*b^7*c^2*e^3 - 4*b^6*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*(d + e*x)^(1/2))/(b^4*(b^4*e - b
^3*c*d))))/(2*(b^4*e - b^3*c*d)))*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d))/(2*(b^4*e - b^3*c*d))))*(-c*(b*e - c
*d))^(1/2)*(3*b*e - 4*c*d)*1i)/(b^4*e - b^3*c*d)

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sympy [B]  time = 55.42, size = 790, normalized size = 6.32 \begin {gather*} \frac {2 c^{2} d e \sqrt {d + e x}}{2 b^{4} e^{2} - 2 b^{3} c d e + 2 b^{3} c e^{2} x - 2 b^{2} c^{2} d e x} - \frac {2 c e^{2} \sqrt {d + e x}}{2 b^{3} e^{2} - 2 b^{2} c d e + 2 b^{2} c e^{2} x - 2 b c^{2} d e x} + \frac {c e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {c e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {c^{2} d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} + \frac {c^{2} d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} - \frac {d e \sqrt {\frac {1}{d^{3}}} \log {\left (- d^{2} \sqrt {\frac {1}{d^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} + \frac {d e \sqrt {\frac {1}{d^{3}}} \log {\left (d^{2} \sqrt {\frac {1}{d^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} - \frac {2 e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e}{c} - d}} \right )}}{b^{2} \sqrt {\frac {b e}{c} - d}} + \frac {2 e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b^{2} \sqrt {- d}} - \frac {\sqrt {d + e x}}{b^{2} x} + \frac {4 c d \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e}{c} - d}} \right )}}{b^{3} \sqrt {\frac {b e}{c} - d}} - \frac {4 c d \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b^{3} \sqrt {- d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c**2*d*e*sqrt(d + e*x)/(2*b**4*e**2 - 2*b**3*c*d*e + 2*b**3*c*e**2*x - 2*b**2*c**2*d*e*x) - 2*c*e**2*sqrt(d
+ e*x)/(2*b**3*e**2 - 2*b**2*c*d*e + 2*b**2*c*e**2*x - 2*b*c**2*d*e*x) + c*e**2*sqrt(-1/(c*(b*e - c*d)**3))*lo
g(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) - c**2*d**2*sqrt(-1/(c*(b*e -
c*d)**3)) + sqrt(d + e*x))/(2*b) - c*e**2*sqrt(-1/(c*(b*e - c*d)**3))*log(b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3
)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b) - c*
*2*d*e*sqrt(-1/(c*(b*e - c*d)**3))*log(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(b*e - c*
d)**3)) - c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b**2) + c**2*d*e*sqrt(-1/(c*(b*e - c*d)**3
))*log(b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c*(b
*e - c*d)**3)) + sqrt(d + e*x))/(2*b**2) - d*e*sqrt(d**(-3))*log(-d**2*sqrt(d**(-3)) + sqrt(d + e*x))/(2*b**2)
+ d*e*sqrt(d**(-3))*log(d**2*sqrt(d**(-3)) + sqrt(d + e*x))/(2*b**2) - 2*e*atan(sqrt(d + e*x)/sqrt(b*e/c - d)
)/(b**2*sqrt(b*e/c - d)) + 2*e*atan(sqrt(d + e*x)/sqrt(-d))/(b**2*sqrt(-d)) - sqrt(d + e*x)/(b**2*x) + 4*c*d*a
tan(sqrt(d + e*x)/sqrt(b*e/c - d))/(b**3*sqrt(b*e/c - d)) - 4*c*d*atan(sqrt(d + e*x)/sqrt(-d))/(b**3*sqrt(-d))

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