3.17.20 \(\int \frac {(b+2 c x) \sqrt {d+e x}}{(a+b x+c x^2)^2} \, dx\) [1620]

Optimal. Leaf size=223 \[ -\frac {\sqrt {d+e x}}{a+b x+c x^2}-\frac {\sqrt {2} \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.24, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {782, 721, 1107, 214} \begin {gather*} -\frac {\sqrt {2} \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x}}{a+b x+c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 721

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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}, 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]

Rule 782

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] - Dist[e*g*(m/(2*c*(p + 1))), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]

Rule 1107

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 1.20, size = 247, normalized size = 1.11 \begin {gather*} -\frac {\sqrt {d+e x}}{a+x (b+c x)}-\frac {i \sqrt {2} \sqrt {c} e \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {i \sqrt {2} \sqrt {c} e \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.92, size = 252, normalized size = 1.13

method	result	size
derivativedivides	\(2 e^{2} \left (-\frac {\sqrt {e x +d}}{2 \left (\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )}+2 c \left (-\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\right )\)	\(252\)
default	\(2 e^{2} \left (-\frac {\sqrt {e x +d}}{2 \left (\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )}+2 c \left (-\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\right )\)	\(252\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*c*x + b)*sqrt(x*e + d)/(c*x^2 + b*x + a)^2, x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2655 vs. \(2 (190) = 380\).
time = 2.38, size = 2655, normalized size = 11.91 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*
b^2 - 4*a^2*c)*e^2)*e^3/sqrt((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*
c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c
)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(2*sqrt(x*e + d)*c*e^4 + sqrt(1/2)*((b^2 - 4*a*c)*e^4 - (2*(b^2*c^2 - 4*a*c
^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3)*e^3/sqr
t((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4
*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*sqrt((2*c*d*e^2 - b*e^3 + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)
*d*e + (a*b^2 - 4*a^2*c)*e^2)*e^3/sqrt((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*
c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))/((b^2*c - 4*a*c^2)*d^2 - (b^3
- 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))) - sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + ((b^2*c - 4
*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*e^3/sqrt((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b
*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))/
((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(2*sqrt(x*e + d)*c*e^4 - sqrt(1/2)*(
(b^2 - 4*a*c)*e^4 - (2*(b^2*c^2 - 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d
*e^2 - (a*b^3 - 4*a^2*b*c)*e^3)*e^3/sqrt((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^
2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*sqrt((2*c*d*e^2 - b*e^3 + (
(b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*e^3/sqrt((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3
*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3
*c)*e^4))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))) + sqrt(1/2)*(c*x^2 + b*x + a
)*sqrt((2*c*d*e^2 - b*e^3 - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*e^3/sqrt((b^
2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*
b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*
log(2*sqrt(x*e + d)*c*e^4 + sqrt(1/2)*((b^2 - 4*a*c)*e^4 + (2*(b^2*c^2 - 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)*
d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3)*e^3/sqrt((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b
^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a
^3*c)*e^4))*sqrt((2*c*d*e^2 - b*e^3 - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*e^
3/sqrt((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^
3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2
*c)*e^2))) - sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*
e + (a*b^2 - 4*a^2*c)*e^2)*e^3/sqrt((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c -
8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))/((b^2*c - 4*a*c^2)*d^2 - (b^3 -
4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(2*sqrt(x*e + d)*c*e^4 - sqrt(1/2)*((b^2 - 4*a*c)*e^4 + (2*(b^2*c^2
- 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3)*
e^3/sqrt((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*
b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*sqrt((2*c*d*e^2 - b*e^3 - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4
*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*e^3/sqrt((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2
*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))/((b^2*c - 4*a*c^2)*d^2
- (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))) + 2*sqrt(x*e + d))/(c*x^2 + b*x + a)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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((2*c*x+b)*(e*x+d)**(1/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 2.47, size = 351, normalized size = 1.57 \begin {gather*} -\frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right ) e}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c d + {\left (b^{2} - 4 \, a c - \sqrt {b^{2} - 4 \, a c} b\right )} e\right )} {\left | c \right |}} + \frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right ) e}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c d - {\left (b^{2} - 4 \, a c + \sqrt {b^{2} - 4 \, a c} b\right )} e\right )} {\left | c \right |}} - \frac {\sqrt {x e + d} e^{2}}{{\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 + a e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c*d - b*e + sqrt
((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))*e/((2*sqrt(b^2 - 4*a*c)*c*d + (b^2 - 4*a*c - sqrt(b^2 - 4
*a*c)*b)*e)*abs(c)) + sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt
(-(2*c*d - b*e - sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))*e/((2*sqrt(b^2 - 4*a*c)*c*d - (b^2 -
4*a*c + sqrt(b^2 - 4*a*c)*b)*e)*abs(c)) - sqrt(x*e + d)*e^2/((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e +
d)*b*e - b*d*e + a*e^2)

________________________________________________________________________________________

Mupad [B]
time = 2.93, size = 2500, normalized size = 11.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________