\(\int \frac {(b+2 c x) (d+e x)^{5/2}}{a+b x+c x^2} \, dx\) [585]

Optimal result

Integrand size = 28, antiderivative size = 258 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{a+b x+c x^2} \, dx=\frac {2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \sqrt {d+e x}}{c^2}+\frac {2 (2 c d-b e) (d+e x)^{3/2}}{3 c}+\frac {4}{5} (d+e x)^{5/2}-\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} c^{5/2}}-\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} c^{5/2}} \] Output:

2*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))*(e*x+d)^(1/2)/c^2+2/3*(-b*e+2*c*d)*( 
e*x+d)^(3/2)/c+4/5*(e*x+d)^(5/2)-1/4*(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(5/2 
)*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^(5/2)-1/4*(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(5/2)*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^(5/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.11 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.53 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{a+b x+c x^2} \, dx=\frac {2 \sqrt {c} \sqrt {d+e x} \left (15 b^2 e^2-5 c e (7 b d+6 a e+b e x)+c^2 \left (46 d^2+22 d e x+6 e^2 x^2\right )\right )-\frac {15 \left (b^3 \left (i b+\sqrt {-b^2+4 a c}\right ) e^3-2 c^3 d^2 \left (\sqrt {-b^2+4 a c} d+6 i a e\right )-b c e^2 \left (3 i b^2 d+3 b \sqrt {-b^2+4 a c} d+5 i a b e+3 a \sqrt {-b^2+4 a c} e\right )+c^2 e \left (3 i b^2 d^2+2 a e \left (3 \sqrt {-b^2+4 a c} d+2 i a e\right )+3 b d \left (\sqrt {-b^2+4 a c} d+4 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {15 \left (b^3 \left (-i b+\sqrt {-b^2+4 a c}\right ) e^3-2 c^3 d^2 \left (\sqrt {-b^2+4 a c} d-6 i a e\right )+b c e^2 \left (3 i b^2 d-3 b \sqrt {-b^2+4 a c} d+5 i a b e-3 a \sqrt {-b^2+4 a c} e\right )+c^2 e \left (-3 i b^2 d^2+2 a e \left (3 \sqrt {-b^2+4 a c} d-2 i a e\right )+3 b d \left (\sqrt {-b^2+4 a c} d-4 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{15 c^{5/2}} \] Input:

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

Output:

(2*Sqrt[c]*Sqrt[d + e*x]*(15*b^2*e^2 - 5*c*e*(7*b*d + 6*a*e + b*e*x) + c^2 
*(46*d^2 + 22*d*e*x + 6*e^2*x^2)) - (15*(b^3*(I*b + Sqrt[-b^2 + 4*a*c])*e^ 
3 - 2*c^3*d^2*(Sqrt[-b^2 + 4*a*c]*d + (6*I)*a*e) - b*c*e^2*((3*I)*b^2*d + 
3*b*Sqrt[-b^2 + 4*a*c]*d + (5*I)*a*b*e + 3*a*Sqrt[-b^2 + 4*a*c]*e) + c^2*e 
*((3*I)*b^2*d^2 + 2*a*e*(3*Sqrt[-b^2 + 4*a*c]*d + (2*I)*a*e) + 3*b*d*(Sqrt 
[-b^2 + 4*a*c]*d + (4*I)*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqr 
t[-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]) - (15*(b^3*((-I)*b + Sqrt[-b^2 + 4*a 
*c])*e^3 - 2*c^3*d^2*(Sqrt[-b^2 + 4*a*c]*d - (6*I)*a*e) + b*c*e^2*((3*I)*b 
^2*d - 3*b*Sqrt[-b^2 + 4*a*c]*d + (5*I)*a*b*e - 3*a*Sqrt[-b^2 + 4*a*c]*e) 
+ c^2*e*((-3*I)*b^2*d^2 + 2*a*e*(3*Sqrt[-b^2 + 4*a*c]*d - (2*I)*a*e) + 3*b 
*d*(Sqrt[-b^2 + 4*a*c]*d - (4*I)*a*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]))/(15*c^(5/2))
 

Rubi [A] (verified)

Time = 2.00 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.65, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1196, 27, 1196, 1196, 25, 1197, 1480, 221}

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

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

\(\Big \downarrow \) 1196

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1196

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

\(\Big \downarrow \) 1196

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {2 \sqrt {d+e x} \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{c}-\frac {2 \left (\frac {1}{2} \left (e \sqrt {b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )-(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}-\frac {1}{2} \left (e \sqrt {b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{c}}{c}+\frac {2 (d+e x)^{3/2} (2 c d-b e)}{3 c}+\frac {4}{5} (d+e x)^{5/2}\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

(2*(2*c*d - b*e)*(d + e*x)^(3/2))/(3*c) + (4*(d + e*x)^(5/2))/5 + ((2*(2*c 
^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*Sqrt[d + e*x])/c - (2*(((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)))*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]) - ((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))) 
*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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(525\) vs. \(2(208)=416\).

Time = 1.95 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.04

method	result	size
pseudoelliptic	\(-\frac {4 \left (\sqrt {2}\, \left (\frac {3 \left (-\frac {c^{2} d^{2}}{3}+e \left (a e +\frac {b d}{3}\right ) c -\frac {b^{2} e^{2}}{3}\right ) \left (b e -2 c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{4}+e^{2} \left (-3 c^{2} d^{2}+\left (a \,e^{2}+3 b d e \right ) c -b^{2} e^{2}\right ) \left (a c -\frac {b^{2}}{4}\right )\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\sqrt {2}\, \left (-\frac {3 \left (-\frac {c^{2} d^{2}}{3}+e \left (a e +\frac {b d}{3}\right ) c -\frac {b^{2} e^{2}}{3}\right ) \left (b e -2 c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{4}+e^{2} \left (-3 c^{2} d^{2}+\left (a \,e^{2}+3 b d e \right ) c -b^{2} e^{2}\right ) \left (a c -\frac {b^{2}}{4}\right )\right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (-\frac {23}{15} d^{2}-\frac {11}{15} d e x -\frac {1}{5} e^{2} x^{2}\right ) c^{2}+e \left (\left (\frac {b x}{6}+a \right ) e +\frac {7 b d}{6}\right ) c -\frac {b^{2} e^{2}}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )\right )}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, c^{2}}\)	\(526\)
risch	\(-\frac {2 \left (-6 c^{2} e^{2} x^{2}+5 e^{2} x b c -22 c^{2} d e x +30 a c \,e^{2}-15 b^{2} e^{2}+35 b c d e -46 c^{2} d^{2}\right ) \sqrt {e x +d}}{15 c^{2}}+\frac {\frac {\left (-4 e^{4} a^{2} c^{2}+5 a \,b^{2} c \,e^{4}-12 a b \,c^{2} d \,e^{3}+12 d^{2} e^{2} a \,c^{3}-b^{4} e^{4}+3 d \,e^{3} b^{3} c -3 d^{2} e^{2} b^{2} c^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 e^{4} a^{2} c^{2}-5 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-3 d \,e^{3} b^{3} c +3 d^{2} e^{2} b^{2} c^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c}\)	\(736\)
derivativedivides	\(-\frac {2 \left (-\frac {2 \left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {2 c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}+2 b c d e \sqrt {e x +d}-2 c^{2} d^{2} \sqrt {e x +d}\right )}{c^{2}}+\frac {\frac {\left (-4 e^{4} a^{2} c^{2}+5 a \,b^{2} c \,e^{4}-12 a b \,c^{2} d \,e^{3}+12 d^{2} e^{2} a \,c^{3}-b^{4} e^{4}+3 d \,e^{3} b^{3} c -3 d^{2} e^{2} b^{2} c^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\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 )}{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 (4 e^{4} a^{2} c^{2}-5 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-3 d \,e^{3} b^{3} c +3 d^{2} e^{2} b^{2} c^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\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 )}{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}\)	\(767\)
default	\(-\frac {2 \left (-\frac {2 \left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {2 c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}+2 b c d e \sqrt {e x +d}-2 c^{2} d^{2} \sqrt {e x +d}\right )}{c^{2}}+\frac {\frac {\left (-4 e^{4} a^{2} c^{2}+5 a \,b^{2} c \,e^{4}-12 a b \,c^{2} d \,e^{3}+12 d^{2} e^{2} a \,c^{3}-b^{4} e^{4}+3 d \,e^{3} b^{3} c -3 d^{2} e^{2} b^{2} c^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\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 )}{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 (4 e^{4} a^{2} c^{2}-5 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-3 d \,e^{3} b^{3} c +3 d^{2} e^{2} b^{2} c^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\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 )}{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}\)	\(767\)

Input:

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

Output:

-4*(2^(1/2)*(3/4*(-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)* 
(-4*e^2*(a*c-1/4*b^2))^(1/2)+e^2*(-3*c^2*d^2+(a*e^2+3*b*d*e)*c-b^2*e^2)*(a 
*c-1/4*b^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)*(2^(1/2)*(-3/4*(-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)*(-4*e^2*(a*c-1/4*b^2))^ 
(1/2)+e^2*(-3*c^2*d^2+(a*e^2+3*b*d*e)*c-b^2*e^2)*(a*c-1/4*b^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))+( 
e*x+d)^(1/2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*((-23/15*d 
^2-11/15*d*e*x-1/5*e^2*x^2)*c^2+e*((1/6*b*x+a)*e+7/6*b*d)*c-1/2*b^2*e^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)/((-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. 7259 vs. \(2 (208) = 416\).

Time = 1.30 (sec) , antiderivative size = 7259, normalized size of antiderivative = 28.14 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((2*c*x + b)*(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. 1071 vs. \(2 (208) = 416\).

Time = 0.26 (sec) , antiderivative size = 1071, normalized size of antiderivative = 4.15 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*((2*sqrt(b^2 - 4*a*c)*c^3*d^3 - 3*sqrt(b^2 - 4*a*c)*b*c^2*d^2*e + 3*(b 
^2*c - 2*a*c^2)*sqrt(b^2 - 4*a*c)*d*e^2 - (b^3 - 3*a*b*c)*sqrt(b^2 - 4*a*c 
)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^2*e^2 - 2*(2*c^5 
*d^4 - 4*b*c^4*d^3*e + 3*b^2*c^3*d^2*e^2 - b^3*c^2*d*e^3 + (a*b^2*c^2 - 2* 
a^2*c^3)*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs( 
e) - (6*sqrt(b^2 - 4*a*c)*c^5*d^3*e^2 - 9*sqrt(b^2 - 4*a*c)*b*c^4*d^2*e^3 
+ (5*b^2*c^3 - 2*a*c^4)*sqrt(b^2 - 4*a*c)*d*e^4 - (b^3*c^2 - a*b*c^3)*sqrt 
(b^2 - 4*a*c)*e^5)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arcta 
n(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c^6*d - b*c^5*e + sqrt(-4*(c^6*d^2 - 
b*c^5*d*e + a*c^5*e^2)*c^6 + (2*c^6*d - b*c^5*e)^2))/c^6))/((c^5*d^2 - b*c 
^4*d*e + a*c^4*e^2)*c^2*abs(e)) - 1/4*((2*sqrt(b^2 - 4*a*c)*c^3*d^3 - 3*sq 
rt(b^2 - 4*a*c)*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*sqrt(b^2 - 4*a*c)*d*e^2 
- (b^3 - 3*a*b*c)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 
 - 4*a*c)*c)*e)*c^2*e^2 + 2*(2*c^5*d^4 - 4*b*c^4*d^3*e + 3*b^2*c^3*d^2*e^2 
 - b^3*c^2*d*e^3 + (a*b^2*c^2 - 2*a^2*c^3)*e^4)*sqrt(-4*c^2*d + 2*(b*c + s 
qrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) - (6*sqrt(b^2 - 4*a*c)*c^5*d^3*e^2 - 
9*sqrt(b^2 - 4*a*c)*b*c^4*d^2*e^3 + (5*b^2*c^3 - 2*a*c^4)*sqrt(b^2 - 4*a*c 
)*d*e^4 - (b^3*c^2 - a*b*c^3)*sqrt(b^2 - 4*a*c)*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^6 
*d - b*c^5*e - sqrt(-4*(c^6*d^2 - b*c^5*d*e + a*c^5*e^2)*c^6 + (2*c^6*d...
 

Mupad [B] (verification not implemented)

Time = 1.73 (sec) , antiderivative size = 13848, normalized size of antiderivative = 53.67 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((8*(8*a^3*c^5*e^6 + a*b^4*c^3*e^6 - 8*a*c^7*d^4*e^2 - b^5*c^3*d*e^ 
5 - 6*a^2*b^2*c^4*e^6 + 2*b^2*c^6*d^4*e^2 - 4*b^3*c^5*d^3*e^3 + 3*b^4*c^4* 
d^2*e^4 + 16*a*b*c^6*d^3*e^3 + 4*a*b^3*c^4*d*e^5 - 12*a*b^2*c^5*d^2*e^4))/ 
c^3 - (8*(d + e*x)^(1/2)*(-(b^5*e^5 - 2*c^5*d^5 + b^4*e^5*(b^2 - 4*a*c)^(1 
/2) + 5*a^2*b*c^2*e^5 + 20*a*c^4*d^3*e^2 - 10*a^2*c^3*d*e^4 + a^2*c^2*e^5* 
(b^2 - 4*a*c)^(1/2) - 10*b^2*c^3*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 5*a*b^3*c* 
e^5 + 5*b*c^4*d^4*e - 5*b^4*c*d*e^4 + 5*c^4*d^4*e*(b^2 - 4*a*c)^(1/2) + 10 
*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2) - 
 5*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) - 30*a*b*c^3*d^2*e^3 + 20*a*b^2*c^2*d*e 
^4 - 10*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 10*b*c^3*d^3*e^2*(b^2 - 4*a*c) 
^(1/2) + 10*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(1/2))/(2*c^5))^(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)*(-(b^5*e^5 - 2*c 
^5*d^5 + b^4*e^5*(b^2 - 4*a*c)^(1/2) + 5*a^2*b*c^2*e^5 + 20*a*c^4*d^3*e^2 
- 10*a^2*c^3*d*e^4 + a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2) - 10*b^2*c^3*d^3*e^2 
+ 10*b^3*c^2*d^2*e^3 - 5*a*b^3*c*e^5 + 5*b*c^4*d^4*e - 5*b^4*c*d*e^4 + 5*c 
^4*d^4*e*(b^2 - 4*a*c)^(1/2) + 10*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 3* 
a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2) - 5*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) - 30*a 
*b*c^3*d^2*e^3 + 20*a*b^2*c^2*d*e^4 - 10*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) 
 - 10*b*c^3*d^3*e^2*(b^2 - 4*a*c)^(1/2) + 10*a*b*c^2*d*e^4*(b^2 - 4*a*c)^( 
1/2))/(2*c^5))^(1/2) - (8*(d + e*x)^(1/2)*(b^8*e^8 + 8*a^4*c^4*e^8 - 8*...
 

Reduce [B] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 2881, normalized size of antiderivative = 11.17 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 30*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*e + 60*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**2*d - 30* 
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)*sq 
rt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))* 
a*c*e**2 + 30*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 + c*d**2) + 
b*e - 2*c*d))*b**2*e**2 - 90*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 + c*d**2) + b*e - 2*c*d))*b*c*d*e + 90*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 + c*d**2) + b*e - 2*c*d))*c**2*d**2 + 30*sqrt(2*sqrt(c)...