\(\int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx\) [533]

Optimal result

Integrand size = 22, antiderivative size = 415 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\frac {2 e (2 c d-b e) \sqrt {d+e x}}{c^2}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\sqrt {2} \left (e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (2 c^3 d^3-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d+\sqrt {b^2-4 a c} d+2 a e\right )+c e^2 \left (3 b^2 d+a \sqrt {b^2-4 a c} e+3 b \left (\sqrt {b^2-4 a c} d+a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

2*e*(-b*e+2*c*d)*(e*x+d)^(1/2)/c^2+2/3*e*(e*x+d)^(3/2)/c-2^(1/2)*(e*(3*c^2 
*d^2+b^2*e^2-c*e*(a*e+3*b*d))+(-b*e+2*c*d)*(c^2*d^2+b^2*e^2-c*e*(3*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))/c^(5/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)+ 
2^(1/2)*(2*c^3*d^3-b^2*(b+(-4*a*c+b^2)^(1/2))*e^3-3*c^2*d*e*(b*d+(-4*a*c+b 
^2)^(1/2)*d+2*a*e)+c*e^2*(3*b^2*d+a*(-4*a*c+b^2)^(1/2)*e+3*b*((-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))/c^(5/2)/(-4*a*c+b^2)^(1/2)/(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 = 1.94 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\frac {2 \sqrt {c} e \sqrt {d+e x} (7 c d-3 b e+c e x)+\frac {3 \left (-2 i c^3 d^3+b^2 \left (i b+\sqrt {-b^2+4 a c}\right ) e^3+3 c^2 d e \left (i b d+\sqrt {-b^2+4 a c} d+2 i a e\right )-c e^2 \left (3 i b^2 d+3 b \sqrt {-b^2+4 a c} d+3 i a b e+a \sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {3 \left (2 i c^3 d^3+b^2 \left (-i b+\sqrt {-b^2+4 a c}\right ) e^3+3 c^2 d e \left (-i b d+\sqrt {-b^2+4 a c} d-2 i a e\right )+c e^2 \left (3 i b^2 d-3 b \sqrt {-b^2+4 a c} d+3 i a b e-a \sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{3 c^{5/2}} \] Input:

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

Output:

(2*Sqrt[c]*e*Sqrt[d + e*x]*(7*c*d - 3*b*e + c*e*x) + (3*((-2*I)*c^3*d^3 + 
b^2*(I*b + Sqrt[-b^2 + 4*a*c])*e^3 + 3*c^2*d*e*(I*b*d + Sqrt[-b^2 + 4*a*c] 
*d + (2*I)*a*e) - c*e^2*((3*I)*b^2*d + 3*b*Sqrt[-b^2 + 4*a*c]*d + (3*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]])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2* 
c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (3*((2*I)*c^3*d^3 + b^2*((-I)*b + S 
qrt[-b^2 + 4*a*c])*e^3 + 3*c^2*d*e*((-I)*b*d + Sqrt[-b^2 + 4*a*c]*d - (2*I 
)*a*e) + c*e^2*((3*I)*b^2*d - 3*b*Sqrt[-b^2 + 4*a*c]*d + (3*I)*a*b*e - a*S 
qrt[-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]])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b 
+ I*Sqrt[-b^2 + 4*a*c])*e]))/(3*c^(5/2))
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1146, 1196, 1197, 25, 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.

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

\(\Big \downarrow \) 1146

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

\(\Big \downarrow \) 1196

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(2*e*(d + e*x)^(3/2))/(3*c) + ((2*e*(2*c*d - b*e)*Sqrt[d + e*x])/c - (2*e* 
(((3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e) + ((2*c*d - b*e)*(c^2*d^2 + b^2 
*e^2 - c*e*(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]) + ((3*c^2*d^2 + b^2*e^2 - c*e*(3 
*b*d + a*e) - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))/(Sqr 
t[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)/c
 

Defintions of rubi rules used

rule 25

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

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 1146

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

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 = 2.62 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(-\frac {2 e \left (-\frac {\left (\left (-3 c^{2} d^{2}+\left (a \,e^{2}+3 b d e \right ) c -b^{2} e^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-3 \left (-\frac {c^{2} d^{2}}{3}+e \left (a e +\frac {b d}{3}\right ) c -\frac {b^{2} e^{2}}{3}\right ) \left (b e -2 c d \right )\right ) \sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )}{2}+\left (\frac {\sqrt {2}\, \left (\left (-3 c^{2} d^{2}+\left (a \,e^{2}+3 b d e \right ) c -b^{2} e^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+3 \left (-\frac {c^{2} d^{2}}{3}+e \left (a e +\frac {b d}{3}\right ) c -\frac {b^{2} e^{2}}{3}\right ) \left (b e -2 c d \right )\right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )}{2}+\sqrt {e x +d}\, \left (\frac {\left (-e x -7 d \right ) c}{3}+b e \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, c^{2}}\)	\(472\)
risch	\(-\frac {2 e \left (-c e x +3 b e -7 c d \right ) \sqrt {e x +d}}{3 c^{2}}-\frac {8 e \left (-\frac {\left (-3 a b c \,e^{3}+6 d \,e^{2} a \,c^{2}+b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-2 d^{3} c^{3}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}-b^{3} e^{3}+3 d \,e^{2} b^{2} c -3 d^{2} e b \,c^{2}+2 d^{3} c^{3}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}\)	\(531\)
derivativedivides	\(2 e \left (-\frac {-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+\sqrt {e x +d}\, b e -2 c d \sqrt {e x +d}}{c^{2}}+\frac {-\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}-b^{3} e^{3}+3 d \,e^{2} b^{2} c -3 d^{2} e b \,c^{2}+2 d^{3} c^{3}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-3 a b c \,e^{3}+6 d \,e^{2} a \,c^{2}+b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-2 d^{3} c^{3}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\right )\)	\(543\)
default	\(2 e \left (-\frac {-\frac {c \left (e x +d \right )^{\frac {3}{2}}}{3}+\sqrt {e x +d}\, b e -2 c d \sqrt {e x +d}}{c^{2}}+\frac {-\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}-b^{3} e^{3}+3 d \,e^{2} b^{2} c -3 d^{2} e b \,c^{2}+2 d^{3} c^{3}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-3 a b c \,e^{3}+6 d \,e^{2} a \,c^{2}+b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-2 d^{3} c^{3}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\right )\)	\(543\)

Input:

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

Output:

-2/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*e/((-b*e+2*c*d+(-4*e 
^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(-1/2*((-3*c^2*d^2+(a*e^2+3*b*d*e)*c-b^2 
*e^2)*(-4*e^2*(a*c-1/4*b^2))^(1/2)-3*(-1/3*c^2*d^2+e*(a*e+1/3*b*d)*c-1/3*b 
^2*e^2)*(b*e-2*c*d))*2^(1/2)*((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))+(1/2*2^(1/2)*((-3*c^2*d^2+(a*e^2+3*b*d*e)*c-b^2*e^2)*(-4* 
e^2*(a*c-1/4*b^2))^(1/2)+3*(-1/3*c^2*d^2+e*(a*e+1/3*b*d)*c-1/3*b^2*e^2)*(b 
*e-2*c*d))*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))+(e*x+d)^(1/2)*(1/3*(-e*x-7*d)*c+b*e)*(-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))*((-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^ 
2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6701 vs. \(2 (361) = 722\).

Time = 2.00 (sec) , antiderivative size = 6701, normalized size of antiderivative = 16.15 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (361) = 722\).

Time = 0.42 (sec) , antiderivative size = 1030, normalized size of antiderivative = 2.48 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 7.75 (sec) , antiderivative size = 16475, normalized size of antiderivative = 39.70 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*e*(d + e*x)^(3/2))/(3*c) - atan(((((8*(a*b^3*c^3*e^6 - 4*a^2*b*c^4*e^6 
+ 8*a*c^6*d^3*e^3 + 8*a^2*c^5*d*e^5 - b^4*c^3*d*e^5 - 2*b^2*c^5*d^3*e^3 + 
3*b^3*c^4*d^2*e^4 - 12*a*b*c^5*d^2*e^4 + 2*a*b^2*c^4*d*e^5))/c^3 - (8*(d + 
 e*x)^(1/2)*((2*b^2*c^5*d^5 - 8*a*c^6*d^5 - b^7*e^5 + b^4*e^5*(-(4*a*c - b 
^2)^3)^(1/2) + 20*a^3*b*c^3*e^5 - 40*a^3*c^4*d*e^4 - 5*b^3*c^4*d^4*e + 5*c 
^4*d^4*e*(-(4*a*c - b^2)^3)^(1/2) - 25*a^2*b^3*c^2*e^5 + a^2*c^2*e^5*(-(4* 
a*c - b^2)^3)^(1/2) + 80*a^2*c^5*d^3*e^2 + 10*b^4*c^3*d^3*e^2 - 10*b^5*c^2 
*d^2*e^3 + 9*a*b^5*c*e^5 + 5*b^6*c*d*e^4 + 20*a*b*c^5*d^4*e + 10*b^2*c^2*d 
^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 
 40*a*b^4*c^2*d*e^4 - 5*b^3*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 60*a*b^2*c^ 
4*d^3*e^2 + 70*a*b^3*c^3*d^2*e^3 - 120*a^2*b*c^4*d^2*e^3 + 90*a^2*b^2*c^3* 
d*e^4 - 10*a*c^3*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 10*b*c^3*d^3*e^2*(-(4* 
a*c - b^2)^3)^(1/2) + 10*a*b*c^2*d*e^4*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^ 
2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2)*(b^3*c^5*e^3 - 2*b^2*c^6*d*e^2 - 4* 
a*b*c^6*e^3 + 8*a*c^7*d*e^2))/c^3)*((2*b^2*c^5*d^5 - 8*a*c^6*d^5 - b^7*e^5 
 + b^4*e^5*(-(4*a*c - b^2)^3)^(1/2) + 20*a^3*b*c^3*e^5 - 40*a^3*c^4*d*e^4 
- 5*b^3*c^4*d^4*e + 5*c^4*d^4*e*(-(4*a*c - b^2)^3)^(1/2) - 25*a^2*b^3*c^2* 
e^5 + a^2*c^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 80*a^2*c^5*d^3*e^2 + 10*b^4*c 
^3*d^3*e^2 - 10*b^5*c^2*d^2*e^3 + 9*a*b^5*c*e^5 + 5*b^6*c*d*e^4 + 20*a*b*c 
^5*d^4*e + 10*b^2*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e^5*...
 

Reduce [B] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 4862, normalized size of antiderivative = 11.72 \[ \int \frac {(d+e x)^{5/2}}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(12*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*e**2 - 6*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*sqr 
t(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**2*c*e** 
2 + 12*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*c**2*d*e - 12*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))*c**3*d* 
*2 - 18*sqrt(c)*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*sq 
rt(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**3 + 36*sqrt(c)*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...