\(\int \frac {b+2 c x}{(d+e x)^{3/2} (a+b x+c x^2)} \, dx\) [589]

Optimal result

Integrand size = 28, antiderivative size = 354 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\frac {2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {c} \left (b \left (b+\sqrt {b^2-4 a c}\right ) e-2 c \left (\sqrt {b^2-4 a c} d+2 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {2} \sqrt {c} \left (b \left (b-\sqrt {b^2-4 a c}\right ) e+2 c \left (\sqrt {b^2-4 a c} d-2 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \] Output:

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

Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.99 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\frac {4 c d-2 b e}{\left (c d^2+e (-b d+a e)\right ) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {c} \left (b \left (b+\sqrt {b^2-4 a c}\right ) e-2 c \left (\sqrt {b^2-4 a c} d+2 a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e} \left (-c d^2+e (b d-a e)\right )}+\frac {\sqrt {2} \sqrt {c} \left (b \left (-b+\sqrt {b^2-4 a c}\right ) e+c \left (-2 \sqrt {b^2-4 a c} d+4 a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e} \left (-c d^2+e (b d-a e)\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1198, 1197, 25, 1480, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx\)

\(\Big \downarrow \) 1198

\(\displaystyle \frac {\int \frac {-e b^2+c d b+2 a c e+c (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{a e^2-b d e+c d^2}+\frac {2 (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {2 \int -\frac {2 c^2 d^2+b^2 e^2-2 c e (b d+a e)-c (2 c d-b e) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{a e^2-b d e+c d^2}+\frac {2 (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {2 c^2 d^2+b^2 e^2-2 c e (b d+a e)-c (2 c d-b e) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {2 \left (\frac {c \left (2 c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt {b^2-4 a c}\right )\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 \sqrt {b^2-4 a c}}-\frac {c \left (b e \left (\sqrt {b^2-4 a c}+b\right )-2 c \left (d \sqrt {b^2-4 a c}+2 a e\right )\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 \sqrt {b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac {2 (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (\frac {\sqrt {c} \left (b e \left (\sqrt {b^2-4 a c}+b\right )-2 c \left (d \sqrt {b^2-4 a c}+2 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {c} \left (2 c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt {b^2-4 a c}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{a e^2-b d e+c d^2}+\frac {2 (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 221

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 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[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] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1198

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

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(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]
 

Maple [A] (verified)

Time = 1.69 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.02

method	result	size
derivativedivides	\(\frac {8 c \left (\frac {\left (-4 a c \,e^{2}+b^{2} e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}-b^{2} e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \,e^{2}-b d e +c \,d^{2}}-\frac {2 \left (b e -2 c d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}\)	\(362\)
default	\(\frac {8 c \left (\frac {\left (-4 a c \,e^{2}+b^{2} e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}-b^{2} e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \,e^{2}-b d e +c \,d^{2}}-\frac {2 \left (b e -2 c d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}\)	\(362\)
pseudoelliptic	\(-\frac {2 \left (2 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, c \sqrt {e x +d}\, \left (\left (-\frac {b e}{4}+\frac {c d}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e^{2} \left (a c -\frac {b^{2}}{4}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (2 \sqrt {2}\, c \sqrt {e x +d}\, \left (\left (\frac {b e}{4}-\frac {c d}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e^{2} \left (a c -\frac {b^{2}}{4}\right )\right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (b e -2 c d \right )\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (a \,e^{2}-b d e +c \,d^{2}\right )}\)	\(403\)

Input:

int((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
 

Output:

8/(a*e^2-b*d*e+c*d^2)*c*(1/8*(-4*a*c*e^2+b^2*e^2-(-e^2*(4*a*c-b^2))^(1/2)* 
b*e+2*(-e^2*(4*a*c-b^2))^(1/2)*c*d)/(-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((e*x+d)^(1/2)*c*2^(1/2)/( 
(b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))-1/8*(4*a*c*e^2-b^2*e^2-(-e^ 
2*(4*a*c-b^2))^(1/2)*b*e+2*(-e^2*(4*a*c-b^2))^(1/2)*c*d)/(-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((e 
*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))-2* 
(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8557 vs. \(2 (305) = 610\).

Time = 0.36 (sec) , antiderivative size = 8557, normalized size of antiderivative = 24.17 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((2*c*x+b)/(e*x+d)**(3/2)/(c*x**2+b*x+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\int { \frac {2 \, c x + b}{{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

integrate((2*c*x + b)/((c*x^2 + b*x + a)*(e*x + d)^(3/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1398 vs. \(2 (305) = 610\).

Time = 0.32 (sec) , antiderivative size = 1398, normalized size of antiderivative = 3.95 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="giac")
 

Output:

2*(2*c*d - b*e)/((c*d^2 - b*d*e + a*e^2)*sqrt(e*x + d)) + 1/4*((c*d^2*e - 
b*d*e^2 + a*e^3)^2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*sqr 
t(b^2 - 4*a*c)*c*d - sqrt(b^2 - 4*a*c)*b*e) - 2*(2*c^3*d^4 - 4*b*c^2*d^3*e 
 + 3*b^2*c*d^2*e^2 - b^3*d*e^3 + (a*b^2 - 2*a^2*c)*e^4)*sqrt(-4*c^2*d + 2* 
(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c*d^2*e - b*d*e^2 + a*e^3) + (2*sqrt(b^ 
2 - 4*a*c)*c^3*d^5*e^2 - 5*sqrt(b^2 - 4*a*c)*b*c^2*d^4*e^3 - sqrt(b^2 - 4* 
a*c)*a^2*b*e^7 + 4*(b^2*c + a*c^2)*sqrt(b^2 - 4*a*c)*d^3*e^4 - (b^3 + 6*a* 
b*c)*sqrt(b^2 - 4*a*c)*d^2*e^5 + 2*(a*b^2 + a^2*c)*sqrt(b^2 - 4*a*c)*d*e^6 
)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqr 
t(e*x + d)/sqrt(-(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b* 
e^3 + sqrt((2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3)^2 
 - 4*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + 
a^2*e^4)*(c^2*d^2 - b*c*d*e + a*c*e^2)))/(c^2*d^2 - b*c*d*e + a*c*e^2)))/( 
(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4 
*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*abs(c*d^2*e - 
b*d*e^2 + a*e^3)*abs(c)) - 1/4*((c*d^2*e - b*d*e^2 + a*e^3)^2*sqrt(-4*c^2* 
d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*sqrt(b^2 - 4*a*c)*c*d - sqrt(b^2 - 
 4*a*c)*b*e) + 2*(2*c^3*d^4 - 4*b*c^2*d^3*e + 3*b^2*c*d^2*e^2 - b^3*d*e^3 
+ (a*b^2 - 2*a^2*c)*e^4)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)* 
abs(c*d^2*e - b*d*e^2 + a*e^3) + (2*sqrt(b^2 - 4*a*c)*c^3*d^5*e^2 - 5*s...
 

Mupad [B] (verification not implemented)

Time = 19.27 (sec) , antiderivative size = 33147, normalized size of antiderivative = 93.64 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

int((b + 2*c*x)/((d + e*x)^(3/2)*(a + b*x + c*x^2)),x)
 

Output:

(4*c*d - 2*b*e + 2^(1/2)*a*e^2*atan(-((2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^ 
2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6* 
a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1 
/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 
3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3* 
a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64 
*a*c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^1 
0 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^ 
3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4* 
e^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^ 
7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 
 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 + 4 
0*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^ 
2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e^9) + (2^(1/2) 
*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c* 
e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 
3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^ 
6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^ 
2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3) 
)^(1/2)*(64*a*c^9*d^10*e^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*...
 

Reduce [B] (verification not implemented)

Time = 2.42 (sec) , antiderivative size = 2515, normalized size of antiderivative = 7.10 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:

int((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x)
 

Output:

(2*sqrt(d + e*x)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c* 
d)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e 
+ c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a* 
e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*b*e - 4*sqrt(d + e*x)*sqrt(2*sqrt(c 
)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e**2 - b*d*e + c*d** 
2)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*s 
qrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 
 2*c*d))*c*d - 2*sqrt(c)*sqrt(d + e*x)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e 
+ c*d**2) + b*e - 2*c*d)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2 
) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b 
*d*e + c*d**2) + b*e - 2*c*d))*a*e**2 + 2*sqrt(c)*sqrt(d + e*x)*sqrt(2*sqr 
t(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*atan((sqrt(2*sqrt(c)*sqr 
t(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt( 
2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*b*d*e - 2*sqrt(c)* 
sqrt(d + e*x)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)* 
atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt 
(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2* 
c*d))*c*d**2 - 2*sqrt(d + e*x)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2 
) + b*e - 2*c*d)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt(c)*sqrt(a 
*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) + 2*sqrt(c)*sqrt(d + e*x))/sqrt(...