\(\int \frac {(d+e x)^{5/2}}{(a+b x+c x^2)^3} \, dx\) [546]

Optimal result

Integrand size = 22, antiderivative size = 548 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \sqrt {d+e x} \left (3 b^2 d e+4 a c d e-4 b \left (c d^2+a e^2\right )-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (32 c^3 d^3-b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d-4 b \sqrt {b^2-4 a c} d-6 a b e+2 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \left (\frac {(2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )}{\sqrt {b^2-4 a c}}-e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^2 \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

-1/2*(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)^2 
-3/4*(e*x+d)^(1/2)*(3*b^2*d*e+4*a*c*d*e-4*b*(a*e^2+c*d^2)-(8*c^2*d^2+b^2*e 
^2-4*c*e*(-a*e+2*b*d))*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)-3/8*(32*c^3*d^3-b^2 
*(b-(-4*a*c+b^2)^(1/2))*e^3-8*c^2*d*e*(6*b*d-(-4*a*c+b^2)^(1/2)*d-3*a*e)+2 
*c*e^2*(9*b^2*d-4*b*(-4*a*c+b^2)^(1/2)*d-6*a*b*e+2*a*(-4*a*c+b^2)^(1/2)*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))*2^(1/2)/c^(1/2)/(-4*a*c+b^2)^(5/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^ 
(1/2)+3/8*((-b*e+2*c*d)*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4*b*d))/(-4*a*c+ 
b^2)^(1/2)-e*(8*c^2*d^2+b^2*e^2-4*c*e*(-a*e+2*b*d)))*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^(1/2)/( 
-4*a*c+b^2)^2/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2)
 

Mathematica [A] (verified)

Time = 13.52 (sec) , antiderivative size = 506, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\sqrt {d+e x} \left (b^3 \left (-2 d^2-9 d e x+5 e^2 x^2\right )+b^2 \left (a e (-5 d+19 e x)+c x \left (8 d^2-37 d e x+3 e^2 x^2\right )\right )+4 c \left (6 c^2 d^2 x^3-a^2 e (7 d+e x)+a c x \left (10 d^2+d e x+3 e^2 x^2\right )\right )+4 b \left (3 a^2 e^2+3 c^2 d x^2 (3 d-2 e x)+a c \left (5 d^2-9 d e x+4 e^2 x^2\right )\right )\right )}{4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {3 \sqrt {4 c d+2 \left (-b+\sqrt {b^2-4 a c}\right ) e} \left (16 c^2 d^2+b \left (3 b+2 \sqrt {b^2-4 a c}\right ) e^2+4 c e \left (-4 b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{8 \sqrt {c} \left (b^2-4 a c\right )^{5/2}}+\frac {3 \sqrt {4 c d-2 \left (b+\sqrt {b^2-4 a c}\right ) e} \left (16 c^2 d^2+b \left (3 b-2 \sqrt {b^2-4 a c}\right ) e^2+4 c e \left (-4 b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{8 \sqrt {c} \left (b^2-4 a c\right )^{5/2}} \] Input:

Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^3,x]
 

Output:

(Sqrt[d + e*x]*(b^3*(-2*d^2 - 9*d*e*x + 5*e^2*x^2) + b^2*(a*e*(-5*d + 19*e 
*x) + c*x*(8*d^2 - 37*d*e*x + 3*e^2*x^2)) + 4*c*(6*c^2*d^2*x^3 - a^2*e*(7* 
d + e*x) + a*c*x*(10*d^2 + d*e*x + 3*e^2*x^2)) + 4*b*(3*a^2*e^2 + 3*c^2*d* 
x^2*(3*d - 2*e*x) + a*c*(5*d^2 - 9*d*e*x + 4*e^2*x^2))))/(4*(b^2 - 4*a*c)^ 
2*(a + x*(b + c*x))^2) - (3*Sqrt[4*c*d + 2*(-b + Sqrt[b^2 - 4*a*c])*e]*(16 
*c^2*d^2 + b*(3*b + 2*Sqrt[b^2 - 4*a*c])*e^2 + 4*c*e*(-4*b*d - Sqrt[b^2 - 
4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e 
+ Sqrt[b^2 - 4*a*c]*e]])/(8*Sqrt[c]*(b^2 - 4*a*c)^(5/2)) + (3*Sqrt[4*c*d - 
 2*(b + Sqrt[b^2 - 4*a*c])*e]*(16*c^2*d^2 + b*(3*b - 2*Sqrt[b^2 - 4*a*c])* 
e^2 + 4*c*e*(-4*b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c] 
*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(8*Sqrt[c]*(b^2 
- 4*a*c)^(5/2))
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 517, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1164, 27, 1234, 27, 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.

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

\(\Big \downarrow \) 1164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1234

\(\displaystyle -\frac {3 \left (\frac {\sqrt {d+e x} \left (-x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 b \left (a e^2+c d^2\right )+4 a c d e+3 b^2 d e\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {16 c^2 d^3-4 c e (5 b d-3 a e) d+b e^2 (5 b d-4 a e)+e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{b^2-4 a c}\right )}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (\frac {\sqrt {d+e x} \left (-x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 b \left (a e^2+c d^2\right )+4 a c d e+3 b^2 d e\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {16 c^2 d^3-4 c e (5 b d-3 a e) d+b e^2 (5 b d-4 a e)+e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right )}\right )}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {3 \left (\frac {\sqrt {d+e x} \left (-x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 b \left (a e^2+c d^2\right )+4 a c d e+3 b^2 d e\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {e \left (\frac {1}{2} \left (\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {1}{2} \left (-\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{b^2-4 a c}\right )}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {3 \left (\frac {\sqrt {d+e x} \left (-x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 b \left (a e^2+c d^2\right )+4 a c d e+3 b^2 d e\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {e \left (-\frac {\left (\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{b^2-4 a c}\right )}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

Input:

Int[(d + e*x)^(5/2)/(a + b*x + c*x^2)^3,x]
 

Output:

-1/2*((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) - (3*((Sqrt[d + e*x]*(3*b^2*d*e + 4*a*c*d*e - 4*b*(c*d^2 
+ a*e^2) - (8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x))/((b^2 - 4*a*c)* 
(a + b*x + c*x^2)) - (e*(-(((8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e) + ( 
(2*c*d - b*e)*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*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 
])) - ((8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e) - ((2*c*d - b*e)*(16*c^2 
*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e)))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(S 
qrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(S 
qrt[2]*Sqrt[c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/(b^2 - 4*a*c)))/ 
(4*(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 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 1234

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 
 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g 
*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* 
(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 
] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 = 2.26 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.17

method	result	size
pseudoelliptic	\(\frac {-\frac {3 e \left (\left (2 c^{2} d^{2}+\left (a \,e^{2}-2 b d e \right ) c +\frac {b^{2} e^{2}}{4}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-3 \left (\frac {4 c^{2} d^{2}}{3}+e \left (a e -\frac {4 b d}{3}\right ) c +\frac {b^{2} e^{2}}{12}\right ) \left (b e -2 c d \right )\right ) \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \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 )}{32}+\frac {3 \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\frac {\left (\left (2 c^{2} d^{2}+\left (a \,e^{2}-2 b d e \right ) c +\frac {b^{2} e^{2}}{4}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+3 \left (\frac {4 c^{2} d^{2}}{3}+e \left (a e -\frac {4 b d}{3}\right ) c +\frac {b^{2} e^{2}}{12}\right ) \left (b e -2 c d \right )\right ) e \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} \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 )}{2}+\left (2 c^{3} d^{2} x^{3}+\frac {10 x \left (\frac {3 a \,e^{2} x^{2}}{10}+\frac {d x \left (-6 b x +a \right ) e}{10}+d^{2} \left (\frac {9 b x}{10}+a \right )\right ) c^{2}}{3}+\frac {\left (-x \left (-\frac {3}{4} b^{2} x^{2}-4 a b x +a^{2}\right ) e^{2}-7 d \left (\frac {37}{28} b^{2} x^{2}+\frac {9}{7} a b x +a^{2}\right ) e +5 \left (\frac {2 b x}{5}+a \right ) d^{2} b \right ) c}{3}+\left (\left (\frac {b x}{3}+a \right ) e -\frac {2 b d}{3}\right ) b \left (\left (\frac {5 b x}{4}+a \right ) e +\frac {b d}{4}\right )\right ) \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 )}\right )}{16}}{\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 )^{2} \left (a c -\frac {b^{2}}{4}\right )^{2}}\)	\(639\)
derivativedivides	\(2 e^{5} \left (\frac {\frac {3 c \left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}+\frac {\left (16 a b c \,e^{3}-32 d \,e^{2} a \,c^{2}+5 b^{3} e^{3}-46 d \,e^{2} b^{2} c +108 d^{2} e b \,c^{2}-72 d^{3} c^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}-\frac {\left (4 e^{4} a^{2} c -19 a \,b^{2} e^{4}+68 a b c d \,e^{3}-68 d^{2} e^{2} a \,c^{2}+19 b^{3} d \,e^{3}-91 b^{2} c \,d^{2} e^{2}+144 b \,c^{2} d^{3} e -72 d^{4} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}+\frac {3 \left (a^{2} b \,e^{5}-2 d \,e^{4} a^{2} c -2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 d^{3} e^{2} a \,c^{2}+b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 d^{5} c^{3}\right ) \sqrt {e x +d}}{2 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{\left (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}\right )^{2}}+\frac {3 c \left (-\frac {\left (-12 a b c \,e^{3}+24 d \,e^{2} a \,c^{2}-b^{3} e^{3}+18 d \,e^{2} b^{2} c -48 d^{2} e b \,c^{2}+32 d^{3} c^{3}+4 \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}-8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +8 \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 )}{8 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 (12 a b c \,e^{3}-24 d \,e^{2} a \,c^{2}+b^{3} e^{3}-18 d \,e^{2} b^{2} c +48 d^{2} e b \,c^{2}-32 d^{3} c^{3}+4 \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}-8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +8 \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 )}{8 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}}\right )}{2 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\right )\)	\(964\)
default	\(2 e^{5} \left (\frac {\frac {3 c \left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}+\frac {\left (16 a b c \,e^{3}-32 d \,e^{2} a \,c^{2}+5 b^{3} e^{3}-46 d \,e^{2} b^{2} c +108 d^{2} e b \,c^{2}-72 d^{3} c^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}-\frac {\left (4 e^{4} a^{2} c -19 a \,b^{2} e^{4}+68 a b c d \,e^{3}-68 d^{2} e^{2} a \,c^{2}+19 b^{3} d \,e^{3}-91 b^{2} c \,d^{2} e^{2}+144 b \,c^{2} d^{3} e -72 d^{4} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}+\frac {3 \left (a^{2} b \,e^{5}-2 d \,e^{4} a^{2} c -2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 d^{3} e^{2} a \,c^{2}+b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 d^{5} c^{3}\right ) \sqrt {e x +d}}{2 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{\left (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}\right )^{2}}+\frac {3 c \left (-\frac {\left (-12 a b c \,e^{3}+24 d \,e^{2} a \,c^{2}-b^{3} e^{3}+18 d \,e^{2} b^{2} c -48 d^{2} e b \,c^{2}+32 d^{3} c^{3}+4 \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}-8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +8 \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 )}{8 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 (12 a b c \,e^{3}-24 d \,e^{2} a \,c^{2}+b^{3} e^{3}-18 d \,e^{2} b^{2} c +48 d^{2} e b \,c^{2}-32 d^{3} c^{3}+4 \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}-8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +8 \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 )}{8 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}}\right )}{2 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\right )\)	\(964\)

Input:

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

3/16/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(-1/2*e*((2*c^2*d^ 
2+(a*e^2-2*b*d*e)*c+1/4*b^2*e^2)*(-4*e^2*(a*c-1/4*b^2))^(1/2)-3*(4/3*c^2*d 
^2+e*(a*e-4/3*b*d)*c+1/12*b^2*e^2)*(b*e-2*c*d))*2^(1/2)*(c*x^2+b*x+a)^2*(( 
b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*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)*(1/2*((2*c^2*d^2+(a*e^2-2*b*d*e)*c+1 
/4*b^2*e^2)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+3*(4/3*c^2*d^2+e*(a*e-4/3*b*d)*c+ 
1/12*b^2*e^2)*(b*e-2*c*d))*e*2^(1/2)*(c*x^2+b*x+a)^2*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))+(2*c^3*d^2*x 
^3+10/3*x*(3/10*a*e^2*x^2+1/10*d*x*(-6*b*x+a)*e+d^2*(9/10*b*x+a))*c^2+1/3* 
(-x*(-3/4*b^2*x^2-4*a*b*x+a^2)*e^2-7*d*(37/28*b^2*x^2+9/7*a*b*x+a^2)*e+5*( 
2/5*b*x+a)*d^2*b)*c+((1/3*b*x+a)*e-2/3*b*d)*b*((5/4*b*x+a)*e+1/4*b*d))*(e* 
x+d)^(1/2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(-4*e^2*(a*c 
-1/4*b^2))^(1/2)))/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)/(-4 
*e^2*(a*c-1/4*b^2))^(1/2)/(c*x^2+b*x+a)^2/(a*c-1/4*b^2)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5124 vs. \(2 (488) = 976\).

Time = 0.32 (sec) , antiderivative size = 5124, normalized size of antiderivative = 9.35 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(5/2)/(c*x**2+b*x+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")
 

Output:

integrate((e*x + d)^(5/2)/(c*x^2 + b*x + a)^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2512 vs. \(2 (488) = 976\).

Time = 0.97 (sec) , antiderivative size = 2512, normalized size of antiderivative = 4.58 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

3/32*((b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)^2*(8*c^2*d^2*e - 8*b*c*d*e^2 + 
(b^2 + 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e) + 8*(2 
*(b^2*c^3 - 4*a*c^4)*sqrt(b^2 - 4*a*c)*d^3*e - 3*(b^3*c^2 - 4*a*b*c^3)*sqr 
t(b^2 - 4*a*c)*d^2*e^2 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*sqrt(b^2 - 4*a* 
c)*d*e^3 - (a*b^3*c - 4*a^2*b*c^2)*sqrt(b^2 - 4*a*c)*e^4)*sqrt(-4*c^2*d + 
2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e) - 
 (64*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^4*e - 128*(b 
^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^2 + 12*(7*b^8 
*c^2 - 80*a*b^6*c^3 + 288*a^2*b^4*c^4 - 256*a^3*b^2*c^5 - 256*a^4*c^6)*d^2 
*e^3 - 4*(5*b^9*c - 48*a*b^7*c^2 + 96*a^2*b^5*c^3 + 256*a^3*b^3*c^4 - 768* 
a^4*b*c^5)*d*e^4 + (b^10 - 96*a^2*b^6*c^2 + 512*a^3*b^4*c^3 - 768*a^4*b^2* 
c^4)*e^5)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt( 
1/2)*sqrt(e*x + d)/sqrt(-(2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5* 
e + 8*a*b^3*c*e - 16*a^2*b*c^2*e + sqrt((2*b^4*c*d - 16*a*b^2*c^2*d + 32*a 
^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2*b*c^2*e)^2 - 4*(b^4*c*d^2 - 8*a*b^ 
2*c^2*d^2 + 16*a^2*c^3*d^2 - b^5*d*e + 8*a*b^3*c*d*e - 16*a^2*b*c^2*d*e + 
a*b^4*e^2 - 8*a^2*b^2*c*e^2 + 16*a^3*c^2*e^2)*(b^4*c - 8*a*b^2*c^2 + 16*a^ 
2*c^3)))/(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))/(((b^6*c^2 - 12*a*b^4*c^3 + 
48*a^2*b^2*c^4 - 64*a^3*c^5)*sqrt(b^2 - 4*a*c)*d^2 - (b^7*c - 12*a*b^5*c^2 
 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*sqrt(b^2 - 4*a*c)*d*e + (a*b^6*c - 12...
 

Mupad [B] (verification not implemented)

Time = 16.15 (sec) , antiderivative size = 13637, normalized size of antiderivative = 24.89 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((d + e*x)^(5/2)/(a + b*x + c*x^2)^3,x)
 

Output:

log((3*2^(1/2)*((3*2^(1/2)*((12*c^2*e^3*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d 
*e))/(4*a*c - b^2) + (3*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e 
*x)^(1/2)*(-(b^15*e^5 + e^5*(-(4*a*c - b^2)^15)^(1/2) + 524288*a^5*c^10*d^ 
5 - 512*b^10*c^5*d^5 + 10240*a*b^8*c^6*d^5 - 81920*a^7*b*c^7*e^5 + 163840* 
a^7*c^8*d*e^4 + 1280*b^11*c^4*d^4*e - 81920*a^2*b^6*c^7*d^5 + 327680*a^3*b 
^4*c^8*d^5 - 655360*a^4*b^2*c^9*d^5 - 560*a^2*b^11*c^2*e^5 + 4160*a^3*b^9* 
c^3*e^5 - 11520*a^4*b^7*c^4*e^5 - 1024*a^5*b^5*c^5*e^5 + 61440*a^6*b^3*c^6 
*e^5 + 655360*a^6*c^9*d^3*e^2 - 1120*b^12*c^3*d^3*e^2 + 400*b^13*c^2*d^2*e 
^3 + 20*a*b^13*c*e^5 - 50*b^14*c*d*e^4 - 166400*a^2*b^8*c^5*d^3*e^2 + 4480 
0*a^2*b^9*c^4*d^2*e^3 + 614400*a^3*b^6*c^6*d^3*e^2 - 102400*a^3*b^7*c^5*d^ 
2*e^3 - 1024000*a^4*b^4*c^7*d^3*e^2 - 102400*a^4*b^5*c^6*d^2*e^3 + 327680* 
a^5*b^2*c^8*d^3*e^2 + 819200*a^5*b^3*c^7*d^2*e^3 - 25600*a*b^9*c^5*d^4*e + 
 600*a*b^12*c^2*d*e^4 - 1310720*a^5*b*c^9*d^4*e + 21760*a*b^10*c^4*d^3*e^2 
 - 7040*a*b^11*c^3*d^2*e^3 + 204800*a^2*b^7*c^6*d^4*e - 160*a^2*b^10*c^3*d 
*e^4 - 819200*a^3*b^5*c^7*d^4*e - 28800*a^3*b^8*c^4*d*e^4 + 1638400*a^4*b^ 
3*c^8*d^4*e + 166400*a^4*b^6*c^5*d*e^4 - 358400*a^5*b^4*c^6*d*e^4 - 983040 
*a^6*b*c^8*d^2*e^3 + 204800*a^6*b^2*c^7*d*e^4)/(c*(4*a*c - b^2)^10))^(1/2) 
)/2)*(-(b^15*e^5 + e^5*(-(4*a*c - b^2)^15)^(1/2) + 524288*a^5*c^10*d^5 - 5 
12*b^10*c^5*d^5 + 10240*a*b^8*c^6*d^5 - 81920*a^7*b*c^7*e^5 + 163840*a^7*c 
^8*d*e^4 + 1280*b^11*c^4*d^4*e - 81920*a^2*b^6*c^7*d^5 + 327680*a^3*b^4...
 

Reduce [F]

\[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx=\int \frac {\left (e x +d \right )^{\frac {5}{2}}}{\left (c \,x^{2}+b x +a \right )^{3}}d x \] Input:

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^3,x)
 

Output:

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^3,x)