\(\int \frac {(d+e x)^{5/2}}{(a+b x+c x^2)^2} \, dx\) [540]

Optimal result

Integrand size = 22, antiderivative size = 445 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )+\frac {(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right ) \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )-\frac {(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right ) \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.69 (sec) , antiderivative size = 796, normalized size of antiderivative = 1.79 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {c} \sqrt {d+e x} \left (a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 \sqrt {2} e^2 \left (6 c d+\left (-3 b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 i c^3 d^3-b^2 \left (5 i b+\sqrt {-b^2+4 a c}\right ) e^3-2 c^2 d e \left (6 i b d+\sqrt {-b^2+4 a c} d+16 i a e\right )+2 c e^2 \left (7 i b^2 d+b \sqrt {-b^2+4 a c} d+8 i a b e+a \sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\left (-b^2+4 a c\right )^{3/2} \sqrt {-c d+\frac {1}{2} \left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {\left (-8 i c^3 d^3-b^2 \left (-5 i b+\sqrt {-b^2+4 a c}\right ) e^3+2 c^2 d e \left (6 i b d-\sqrt {-b^2+4 a c} d+16 i a e\right )+2 c e^2 \left (-7 i b^2 d+b \sqrt {-b^2+4 a c} d-8 i a b e+a \sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\left (-b^2+4 a c\right )^{3/2} \sqrt {-c d+\frac {1}{2} \left (b+i \sqrt {-b^2+4 a c}\right ) e}}+\frac {2 \sqrt {2} e^2 \left (-6 c d+\left (3 b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{2 c^{3/2}} \] Input:

Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x]
 

Output:

((-2*Sqrt[c]*Sqrt[d + e*x]*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 
 2*e*x) - 2*a*c*e*(2*d + e*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*Sqr 
t[2]*e^2*(6*c*d + (-3*b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sq 
rt[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]) - (((8*I)*c^3*d^3 - b^2*((5*I)* 
b + Sqrt[-b^2 + 4*a*c])*e^3 - 2*c^2*d*e*((6*I)*b*d + Sqrt[-b^2 + 4*a*c]*d 
+ (16*I)*a*e) + 2*c*e^2*((7*I)*b^2*d + b*Sqrt[-b^2 + 4*a*c]*d + (8*I)*a*b* 
e + a*Sqrt[-b^2 + 4*a*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]])/((-b^2 + 4*a*c)^(3/2)*Sqrt[-(c*d) 
+ ((b - I*Sqrt[-b^2 + 4*a*c])*e)/2]) - (((-8*I)*c^3*d^3 - b^2*((-5*I)*b + 
Sqrt[-b^2 + 4*a*c])*e^3 + 2*c^2*d*e*((6*I)*b*d - Sqrt[-b^2 + 4*a*c]*d + (1 
6*I)*a*e) + 2*c*e^2*((-7*I)*b^2*d + b*Sqrt[-b^2 + 4*a*c]*d - (8*I)*a*b*e + 
 a*Sqrt[-b^2 + 4*a*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]])/((-b^2 + 4*a*c)^(3/2)*Sqrt[-(c*d) + ( 
(b + I*Sqrt[-b^2 + 4*a*c])*e)/2]) + (2*Sqrt[2]*e^2*(-6*c*d + (3*b + Sqrt[b 
^2 - 4*a*c])*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]))/(2*c^(3/2))
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 440, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1164, 27, 1196, 1197, 27, 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.

\(\displaystyle \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx\)

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {\int \frac {\sqrt {d+e x} \left (4 c d^2-5 b e d+6 a e^2-e (2 c d-b e) x\right )}{2 \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\sqrt {d+e x} \left (4 c d^2-e (5 b d-6 a e)-e (2 c d-b e) x\right )}{c x^2+b x+a}dx}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1196

\(\displaystyle -\frac {\frac {\int \frac {4 c^2 d^3-c e (5 b d-8 a e) d-a b e^3+e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {\frac {2 \int \frac {e \left ((2 c d-b e) \left (c d^2-b e d+a e^2\right )+\left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {2 e \int \frac {(2 c d-b e) \left (c d^2-b e d+a e^2\right )+\left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {\frac {2 e \left (\frac {1}{2} \left (\frac {(2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {1}{2} \left (-\frac {(2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {2 e \left (-\frac {\left (\frac {(2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-\frac {(2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c}-\frac {2 e \sqrt {d+e x} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

Input:

Int[(d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1196

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

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 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 = 1.62 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.22

method	result	size
pseudoelliptic	\(\frac {-3 \left (\left (\frac {c^{2} d^{2}}{3}+e \left (a e -\frac {b d}{3}\right ) c -\frac {b^{2} e^{2}}{6}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-\frac {4 \left (b e -2 c d \right ) \left (\frac {c^{2} d^{2}}{2}+e \left (a e -\frac {b d}{2}\right ) c -\frac {b^{2} e^{2}}{8}\right )}{3}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, e \sqrt {2}\, \left (c \,x^{2}+b x +a \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (3 e \sqrt {2}\, \left (\left (\frac {c^{2} d^{2}}{3}+e \left (a e -\frac {b d}{3}\right ) c -\frac {b^{2} e^{2}}{6}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\frac {4 \left (b e -2 c d \right ) \left (\frac {c^{2} d^{2}}{2}+e \left (a e -\frac {b d}{2}\right ) c -\frac {b^{2} e^{2}}{8}\right )}{3}\right ) \left (c \,x^{2}+b x +a \right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {e x +d}\, \left (2 c^{2} d^{2} x +\left (-2 a \,e^{2} x -4 d \left (\frac {b x}{2}+a \right ) e +b \,d^{2}\right ) c +b \,e^{2} \left (b x +a \right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{4 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (c \,x^{2}+b x +a \right ) c \left (a c -\frac {b^{2}}{4}\right )}\)	\(542\)
derivativedivides	\(2 e^{3} \left (\frac {-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}+\frac {\left (a \,e^{3} b -2 a d \,e^{2} c -d \,e^{2} b^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) \sqrt {e x +d}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {-\frac {\left (-8 a b c \,e^{3}+16 d \,e^{2} a \,c^{2}+b^{3} e^{3}+2 d \,e^{2} b^{2} c -12 d^{2} e b \,c^{2}+8 d^{3} c^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (8 a b c \,e^{3}-16 d \,e^{2} a \,c^{2}-b^{3} e^{3}-2 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(692\)
default	\(2 e^{3} \left (\frac {-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}+\frac {\left (a \,e^{3} b -2 a d \,e^{2} c -d \,e^{2} b^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) \sqrt {e x +d}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {-\frac {\left (-8 a b c \,e^{3}+16 d \,e^{2} a \,c^{2}+b^{3} e^{3}+2 d \,e^{2} b^{2} c -12 d^{2} e b \,c^{2}+8 d^{3} c^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (8 a b c \,e^{3}-16 d \,e^{2} a \,c^{2}-b^{3} e^{3}-2 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(692\)

Input:

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/4/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)/((-b*e+2*c*d+(-4*e^ 
2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(-3*((1/3*c^2*d^2+e*(a*e-1/3*b*d)*c-1/6*b 
^2*e^2)*(-4*e^2*(a*c-1/4*b^2))^(1/2)-4/3*(b*e-2*c*d)*(1/2*c^2*d^2+e*(a*e-1 
/2*b*d)*c-1/8*b^2*e^2))*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2) 
*e*2^(1/2)*(c*x^2+b*x+a)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-4* 
e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1 
/2))*c)^(1/2)*(3*e*2^(1/2)*((1/3*c^2*d^2+e*(a*e-1/3*b*d)*c-1/6*b^2*e^2)*(- 
4*e^2*(a*c-1/4*b^2))^(1/2)+4/3*(b*e-2*c*d)*(1/2*c^2*d^2+e*(a*e-1/2*b*d)*c- 
1/8*b^2*e^2))*(c*x^2+b*x+a)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-4 
*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1 
/2))*c)^(1/2)*(e*x+d)^(1/2)*(2*c^2*d^2*x+(-2*a*e^2*x-4*d*(1/2*b*x+a)*e+b*d 
^2)*c+b*e^2*(b*x+a))*(-4*e^2*(a*c-1/4*b^2))^(1/2)))/(-4*e^2*(a*c-1/4*b^2)) 
^(1/2)/(c*x^2+b*x+a)/c/(a*c-1/4*b^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5256 vs. \(2 (403) = 806\).

Time = 0.50 (sec) , antiderivative size = 5256, normalized size of antiderivative = 11.81 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(5/2)/(c*x**2+b*x+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}} \,d x } \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")
 

Output:

integrate((e*x + d)^(5/2)/(c*x^2 + b*x + a)^2, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1398 vs. \(2 (403) = 806\).

Time = 0.79 (sec) , antiderivative size = 1398, normalized size of antiderivative = 3.14 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")
 

Output:

-(2*(e*x + d)^(3/2)*c^2*d^2*e - 2*sqrt(e*x + d)*c^2*d^3*e - 2*(e*x + d)^(3 
/2)*b*c*d*e^2 + 3*sqrt(e*x + d)*b*c*d^2*e^2 + (e*x + d)^(3/2)*b^2*e^3 - 2* 
(e*x + d)^(3/2)*a*c*e^3 - sqrt(e*x + d)*b^2*d*e^3 - 2*sqrt(e*x + d)*a*c*d* 
e^3 + sqrt(e*x + d)*a*b*e^4)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + ( 
e*x + d)*b*e - b*d*e + a*e^2)*(b^2*c - 4*a*c^2)) - (16*(b^2*c^6 - 4*a*c^7) 
*d^4*e - 32*(b^3*c^5 - 4*a*b*c^6)*d^3*e^2 + 16*(b^4*c^4 - 2*a*b^2*c^5 - 8* 
a^2*c^6)*d^2*e^3 - 32*(a*b^3*c^4 - 4*a^2*b*c^5)*d*e^4 - (b^6*c^2 - 12*a*b^ 
4*c^3 + 32*a^2*b^2*c^4)*e^5 - (2*c^2*d^2*e - 2*b*c*d*e^2 - (b^2 - 6*a*c)*e 
^3)*(b^2*c*e - 4*a*c^2*e)^2 - 2*(2*sqrt(b^2 - 4*a*c)*c^4*d^3*e - 3*sqrt(b^ 
2 - 4*a*c)*b*c^3*d^2*e^2 - sqrt(b^2 - 4*a*c)*a*b*c^2*e^4 + (b^2*c^2 + 2*a* 
c^3)*sqrt(b^2 - 4*a*c)*d*e^3)*abs(-b^2*c*e + 4*a*c^2*e))*arctan(2*sqrt(1/2 
)*sqrt(e*x + d)/sqrt(-(2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c^2*e + s 
qrt((2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c^2*e)^2 - 4*(b^2*c^2*d^2 - 
 4*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*e^2)*(b 
^2*c^2 - 4*a*c^3)))/(b^2*c^2 - 4*a*c^3)))/(sqrt(-4*c^2*d + 2*(b*c + sqrt(b 
^2 - 4*a*c)*c)*e)*(2*(b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d + (b^4*c - 8* 
a*b^2*c^2 + 16*a^2*c^3 - (b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c))*e)*abs(-b^ 
2*c*e + 4*a*c^2*e)*abs(c)) + (16*(b^2*c^6 - 4*a*c^7)*d^4*e - 32*(b^3*c^5 - 
 4*a*b*c^6)*d^3*e^2 + 16*(b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^2*e^3 - 32* 
(a*b^3*c^4 - 4*a^2*b*c^5)*d*e^4 - (b^6*c^2 - 12*a*b^4*c^3 + 32*a^2*b^2*...
 

Mupad [B] (verification not implemented)

Time = 8.39 (sec) , antiderivative size = 21160, normalized size of antiderivative = 47.55 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int((d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x)
 

Output:

atan(((((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c 
^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 
+ 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a 
^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5 
*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2* 
c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) - (2*(d + 
e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a 
*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6 
*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1 
504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8 
*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) 
+ 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5 
*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b 
^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960 
*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c 
^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6* 
c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2 
*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 
- 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c 
^3*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b...
 

Reduce [B] (verification not implemented)

Time = 13.34 (sec) , antiderivative size = 14820, normalized size of antiderivative = 33.30 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x)
 

Output:

( - 24*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))*a**2*c**2*e**2 + 2*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))*a*b* 
*2*c*e**2 + 16*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))*a*b*c**2*d*e - 24*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)*ata 
n((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*b*c**2*e**2*x - 16*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))*a*c**3*d**2 - 24*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) ...