\(\int \frac {1}{(d+e x)^{5/2} (a+b x+c x^2)} \, dx\) [538]

Optimal result

Integrand size = 22, antiderivative size = 414 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )} \, dx=-\frac {2 e}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 e (2 c d-b e)}{\left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {c} \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}+\frac {\sqrt {2} \sqrt {c} \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )} \, dx=\frac {\frac {2 e (-c d (7 d+6 e x)+e (4 b d-a e+3 b e x))}{(d+e x)^{3/2}}+\frac {3 \sqrt {2} \sqrt {c} \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \sqrt {2} \sqrt {c} \left (-2 c^2 d^2+b \left (-b+\sqrt {b^2-4 a c}\right ) e^2+2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{3 \left (c d^2+e (-b d+a e)\right )^2} \] Input:

Integrate[1/((d + e*x)^(5/2)*(a + b*x + c*x^2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1147, 1198, 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.

\(\displaystyle \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )} \, dx\)

\(\Big \downarrow \) 1147

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

\(\Big \downarrow \) 1198

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

Int[1/((d + e*x)^(5/2)*(a + b*x + c*x^2)),x]
 

Output:

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

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 1147

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

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 = 1.49 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(2 e \left (\frac {4 c \left (-\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {1}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {-b e +2 c d}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}\right )\)	\(419\)
default	\(2 e \left (\frac {4 c \left (-\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {1}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {-b e +2 c d}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}\right )\)	\(419\)
pseudoelliptic	\(\frac {2 e \left (\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, c \left (\left (c d -\frac {b e}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-c^{2} d^{2}+\left (a \,e^{2}+b d e \right ) c -\frac {b^{2} e^{2}}{2}\right ) \left (e x +d \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (\left (\frac {b e}{2}-c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-c^{2} d^{2}+\left (a \,e^{2}+b d e \right ) c -\frac {b^{2} e^{2}}{2}\right ) \sqrt {2}\, c \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-\frac {\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (\left (6 d e x +7 d^{2}\right ) c +e \left (\left (-3 b x +a \right ) e -4 b d \right )\right )}{3}\right )\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}\)	\(453\)

Input:

int(1/(e*x+d)^(5/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
 

Output:

2*e*(4/(a*e^2-b*d*e+c*d^2)^2*c*(-1/8*(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d 
^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)*arc 
tanh((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*(2*a*c*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^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)))-1/3/(a*e^2-b*d*e+c*d^2)/(e*x 
+d)^(3/2)-1/(a*e^2-b*d*e+c*d^2)^2*(-b*e+2*c*d)/(e*x+d)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25912 vs. \(2 (360) = 720\).

Time = 3.86 (sec) , antiderivative size = 25912, normalized size of antiderivative = 62.59 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

integrate(1/((c*x^2 + b*x + a)*(e*x + d)^(5/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3302 vs. \(2 (360) = 720\).

Time = 0.57 (sec) , antiderivative size = 3302, normalized size of antiderivative = 7.98 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x+a),x, algorithm="giac")
 

Output:

-1/4*((c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3 + 2*a*c*d^2*e^3 - 2*a*b*d*e 
^4 + a^2*e^5)^2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^2*c 
 - 4*a*c^2)*d*e - (b^3 - 4*a*b*c)*e^2) - 2*(3*sqrt(b^2 - 4*a*c)*c^4*d^6*e 
- 9*sqrt(b^2 - 4*a*c)*b*c^3*d^5*e^2 + 5*(2*b^2*c^2 + a*c^3)*sqrt(b^2 - 4*a 
*c)*d^4*e^3 - 5*(b^3*c + 2*a*b*c^2)*sqrt(b^2 - 4*a*c)*d^3*e^4 + (b^4 + 7*a 
*b^2*c + a^2*c^2)*sqrt(b^2 - 4*a*c)*d^2*e^5 - (2*a*b^3 + a^2*b*c)*sqrt(b^2 
 - 4*a*c)*d*e^6 + (a^2*b^2 - a^3*c)*sqrt(b^2 - 4*a*c)*e^7)*sqrt(-4*c^2*d + 
 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2* 
e^3 + 2*a*c*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5) + (4*c^7*d^11*e - 22*b*c^6*d^ 
10*e^2 + 4*(13*b^2*c^5 + 3*a*c^6)*d^9*e^3 - 3*(23*b^3*c^4 + 18*a*b*c^5)*d^ 
8*e^4 + 8*(7*b^4*c^3 + 13*a*b^2*c^4 + a^2*c^5)*d^7*e^5 - 28*(b^5*c^2 + 4*a 
*b^3*c^3 + a^2*b*c^4)*d^6*e^6 + 8*(b^6*c + 9*a*b^4*c^2 + 6*a^2*b^2*c^3 - a 
^3*c^4)*d^5*e^7 - (b^7 + 26*a*b^5*c + 50*a^2*b^3*c^2 - 20*a^3*b*c^3)*d^4*e 
^8 + 4*(a*b^6 + 7*a^2*b^4*c - 2*a^3*b^2*c^2 - 3*a^4*c^3)*d^3*e^9 - 2*(3*a^ 
2*b^5 + 4*a^3*b^3*c - 9*a^4*b*c^2)*d^2*e^10 + 4*(a^3*b^4 - a^4*b^2*c - a^5 
*c^2)*d*e^11 - (a^4*b^3 - 2*a^5*b*c)*e^12)*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*c^3*d^5 - 5*b 
*c^2*d^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2 
*e^3 + 2*a*b^2*d*e^4 + 2*a^2*c*d*e^4 - a^2*b*e^5 + sqrt((2*c^3*d^5 - 5*b*c 
^2*d^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^...
 

Mupad [B] (verification not implemented)

Time = 10.35 (sec) , antiderivative size = 58096, normalized size of antiderivative = 140.33 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

int(1/((d + e*x)^(5/2)*(a + b*x + c*x^2)),x)
 

Output:

- atan((((d + e*x)^(1/2)*(128*b*c^12*d^15*e^3 - 16*c^13*d^16*e^2 - 16*a^8* 
c^5*e^18 - 8*a^6*b^4*c^3*e^18 + 32*a^7*b^2*c^4*e^18 + 320*a^2*c^11*d^12*e^ 
6 + 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 
 + 320*a^6*c^7*d^4*e^14 - 480*b^2*c^11*d^14*e^4 + 1120*b^3*c^10*d^13*e^5 - 
 1800*b^4*c^9*d^12*e^6 + 2064*b^5*c^8*d^11*e^7 - 1688*b^6*c^7*d^10*e^8 + 9 
60*b^7*c^6*d^9*e^9 - 360*b^8*c^5*d^8*e^10 + 80*b^9*c^4*d^7*e^11 - 8*b^10*c 
^3*d^6*e^12 + 4512*a^2*b^2*c^9*d^10*e^8 - 4960*a^2*b^3*c^8*d^9*e^9 + 1800* 
a^2*b^4*c^7*d^8*e^10 + 1440*a^2*b^5*c^6*d^7*e^11 - 1840*a^2*b^6*c^5*d^6*e^ 
12 + 768*a^2*b^7*c^4*d^5*e^13 - 120*a^2*b^8*c^3*d^4*e^14 + 10080*a^3*b^2*c 
^8*d^8*e^10 - 9600*a^3*b^3*c^7*d^7*e^11 + 4000*a^3*b^4*c^6*d^6*e^12 + 96*a 
^3*b^5*c^5*d^5*e^13 - 640*a^3*b^6*c^4*d^4*e^14 + 160*a^3*b^7*c^3*d^3*e^15 
+ 8800*a^4*b^2*c^7*d^6*e^12 - 6240*a^4*b^3*c^6*d^5*e^13 + 1800*a^4*b^4*c^5 
*d^4*e^14 + 80*a^4*b^5*c^4*d^3*e^15 - 120*a^4*b^6*c^3*d^2*e^16 + 3360*a^5* 
b^2*c^6*d^4*e^14 - 1600*a^5*b^3*c^5*d^3*e^15 + 240*a^5*b^4*c^4*d^2*e^16 + 
480*a^6*b^2*c^5*d^2*e^16 - 160*a*b^2*c^10*d^12*e^6 + 960*a*b^3*c^9*d^11*e^ 
7 - 2448*a*b^4*c^8*d^10*e^8 + 3440*a*b^5*c^7*d^9*e^9 - 2880*a*b^6*c^6*d^8* 
e^10 + 1440*a*b^7*c^5*d^7*e^11 - 400*a*b^8*c^4*d^6*e^12 + 48*a*b^9*c^3*d^5 
*e^13 - 1920*a^2*b*c^10*d^11*e^7 - 5120*a^3*b*c^9*d^9*e^9 - 5760*a^4*b*c^8 
*d^7*e^11 - 3072*a^5*b*c^7*d^5*e^13 + 48*a^5*b^5*c^3*d*e^17 - 640*a^6*b*c^ 
6*d^3*e^15 - 160*a^6*b^3*c^4*d*e^17) + (-(b^7*e^5 + 8*a*c^6*d^5 - 2*b^2...
 

Reduce [B] (verification not implemented)

Time = 25.60 (sec) , antiderivative size = 12955, normalized size of antiderivative = 31.29 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)^(5/2)/(c*x^2+b*x+a),x)
 

Output:

( - 18*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)*sqr 
t(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*a*b*c*d*e**3 - 18*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))*a*b*c*e**4*x + 36*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)*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*c**2*d**2*e**2 + 36*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))*a*c**2*d*e**3 
*x + 6*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)*sqr 
t(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*b**3*d*e**3 + 6*sqrt(d + e*x)*s 
qrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e**2 ...