\(\int \frac {(A+B x) (b x-c x^2)^{3/2}}{(d+e x)^{5/2}} \, dx\) [145]

Optimal result

Integrand size = 29, antiderivative size = 516 \[ \int \frac {(A+B x) \left (b x-c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=-\frac {2 (A e (5 c d+2 b e)-B d (8 c d+5 b e)) x^2 \sqrt {b x-c x^2}}{3 d^2 e^2 \sqrt {d+e x}}-\frac {2 (5 A e (8 c d+3 b e)-4 B d (16 c d+9 b e)) \sqrt {d+e x} \sqrt {b x-c x^2}}{15 d e^4}+\frac {2 (10 A e (3 c d+b e)-B d (48 c d+25 b e)) x \sqrt {d+e x} \sqrt {b x-c x^2}}{15 d^2 e^3}-\frac {2 (B d-A e) x \left (b x-c x^2\right )^{3/2}}{3 d e (d+e x)^{3/2}}-\frac {2 \sqrt {b} \left (40 A c e (2 c d+b e)-B \left (128 c^2 d^2+88 b c d e+3 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{15 \sqrt {c} e^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x-c x^2}}+\frac {2 \sqrt {b} \left (5 A e \left (16 c^2 d^2+16 b c d e+3 b^2 e^2\right )-B d \left (128 c^2 d^2+152 b c d e+39 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{15 \sqrt {c} e^5 \sqrt {d+e x} \sqrt {b x-c x^2}} \] Output:

-2/3*(A*e*(2*b*e+5*c*d)-B*d*(5*b*e+8*c*d))*x^2*(-c*x^2+b*x)^(1/2)/d^2/e^2/ 
(e*x+d)^(1/2)-2/15*(5*A*e*(3*b*e+8*c*d)-4*B*d*(9*b*e+16*c*d))*(e*x+d)^(1/2 
)*(-c*x^2+b*x)^(1/2)/d/e^4+2/15*(10*A*e*(b*e+3*c*d)-B*d*(25*b*e+48*c*d))*x 
*(e*x+d)^(1/2)*(-c*x^2+b*x)^(1/2)/d^2/e^3-2/3*(-A*e+B*d)*x*(-c*x^2+b*x)^(3 
/2)/d/e/(e*x+d)^(3/2)-2/15*b^(1/2)*(40*A*c*e*(b*e+2*c*d)-B*(3*b^2*e^2+88*b 
*c*d*e+128*c^2*d^2))*x^(1/2)*(1-c*x/b)^(1/2)*(e*x+d)^(1/2)*EllipticE(c^(1/ 
2)*x^(1/2)/b^(1/2),(-b*e/c/d)^(1/2))/c^(1/2)/e^5/(1+e*x/d)^(1/2)/(-c*x^2+b 
*x)^(1/2)+2/15*b^(1/2)*(5*A*e*(3*b^2*e^2+16*b*c*d*e+16*c^2*d^2)-B*d*(39*b^ 
2*e^2+152*b*c*d*e+128*c^2*d^2))*x^(1/2)*(1-c*x/b)^(1/2)*(1+e*x/d)^(1/2)*El 
lipticF(c^(1/2)*x^(1/2)/b^(1/2),(-b*e/c/d)^(1/2))/c^(1/2)/e^5/(e*x+d)^(1/2 
)/(-c*x^2+b*x)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.12 (sec) , antiderivative size = 454, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (b x-c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 (x (b-c x))^{3/2} \left (\frac {e x (-b+c x) \left (5 A e \left (b e (3 d+4 e x)+c \left (8 d^2+10 d e x+e^2 x^2\right )\right )-B \left (b e \left (36 d^2+47 d e x+6 e^2 x^2\right )+c \left (64 d^3+80 d^2 e x+8 d e^2 x^2-3 e^3 x^3\right )\right )\right )}{d+e x}+\frac {\sqrt {-\frac {b}{c}} \left (40 A c e (2 c d+b e)-B \left (128 c^2 d^2+88 b c d e+3 b^2 e^2\right )\right ) (b-c x) (d+e x)+i b e \left (40 A c e (2 c d+b e)-B \left (128 c^2 d^2+88 b c d e+3 b^2 e^2\right )\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right )|-\frac {c d}{b e}\right )-i b e \left (5 A c e (8 c d+5 b e)-B \left (64 c^2 d^2+52 b c d e+3 b^2 e^2\right )\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right ),-\frac {c d}{b e}\right )}{\sqrt {-\frac {b}{c}} c}\right )}{15 e^5 x^2 (b-c x)^2 \sqrt {d+e x}} \] Input:

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

Output:

(2*(x*(b - c*x))^(3/2)*((e*x*(-b + c*x)*(5*A*e*(b*e*(3*d + 4*e*x) + c*(8*d 
^2 + 10*d*e*x + e^2*x^2)) - B*(b*e*(36*d^2 + 47*d*e*x + 6*e^2*x^2) + c*(64 
*d^3 + 80*d^2*e*x + 8*d*e^2*x^2 - 3*e^3*x^3))))/(d + e*x) + (Sqrt[-(b/c)]* 
(40*A*c*e*(2*c*d + b*e) - B*(128*c^2*d^2 + 88*b*c*d*e + 3*b^2*e^2))*(b - c 
*x)*(d + e*x) + I*b*e*(40*A*c*e*(2*c*d + b*e) - B*(128*c^2*d^2 + 88*b*c*d* 
e + 3*b^2*e^2))*Sqrt[1 - b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*Ar 
cSinh[Sqrt[-(b/c)]/Sqrt[x]], -((c*d)/(b*e))] - I*b*e*(5*A*c*e*(8*c*d + 5*b 
*e) - B*(64*c^2*d^2 + 52*b*c*d*e + 3*b^2*e^2))*Sqrt[1 - b/(c*x)]*Sqrt[1 + 
d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-(b/c)]/Sqrt[x]], -((c*d)/(b*e)) 
])/(Sqrt[-(b/c)]*c)))/(15*e^5*x^2*(b - c*x)^2*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 425, normalized size of antiderivative = 0.82, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {1230, 27, 1230, 27, 1269, 1169, 122, 120, 127, 126}

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.

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

\(\Big \downarrow \) 1230

\(\displaystyle \frac {2 \left (b x-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {2 \int \frac {(b (8 B d-5 A e)-(16 B c d+3 b B e-10 A c e) x) \sqrt {b x-c x^2}}{2 (d+e x)^{3/2}}dx}{5 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (b x-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\int \frac {(b (8 B d-5 A e)-(16 B c d+3 b B e-10 A c e) x) \sqrt {b x-c x^2}}{(d+e x)^{3/2}}dx}{5 e^2}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {2 \left (b x-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x-c x^2} (-e x (-10 A c e+3 b B e+16 B c d)+5 A e (3 b e+8 c d)-4 B d (9 b e+16 c d))}{3 e^2 \sqrt {d+e x}}-\frac {2 \int \frac {b (5 A e (8 c d+3 b e)-4 B d (16 c d+9 b e))-\left (40 A c e (2 c d+b e)-B \left (128 c^2 d^2+88 b c e d+3 b^2 e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {b x-c x^2}}dx}{3 e^2}}{5 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (b x-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x-c x^2} (-e x (-10 A c e+3 b B e+16 B c d)+5 A e (3 b e+8 c d)-4 B d (9 b e+16 c d))}{3 e^2 \sqrt {d+e x}}-\frac {\int \frac {b (5 A e (8 c d+3 b e)-4 B d (16 c d+9 b e))-\left (40 A c e (2 c d+b e)-B \left (128 c^2 d^2+88 b c e d+3 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{3 e^2}}{5 e^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 \left (b x-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x-c x^2} (-e x (-10 A c e+3 b B e+16 B c d)+5 A e (3 b e+8 c d)-4 B d (9 b e+16 c d))}{3 e^2 \sqrt {d+e x}}-\frac {\frac {\left (5 A e \left (3 b^2 e^2+16 b c d e+16 c^2 d^2\right )-B d \left (39 b^2 e^2+152 b c d e+128 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{e}-\frac {\left (40 A c e (b e+2 c d)-B \left (3 b^2 e^2+88 b c d e+128 c^2 d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x-c x^2}}dx}{e}}{3 e^2}}{5 e^2}\)

\(\Big \downarrow \) 1169

\(\displaystyle \frac {2 \left (b x-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x-c x^2} (-e x (-10 A c e+3 b B e+16 B c d)+5 A e (3 b e+8 c d)-4 B d (9 b e+16 c d))}{3 e^2 \sqrt {d+e x}}-\frac {\frac {\sqrt {x} \sqrt {b-c x} \left (5 A e \left (3 b^2 e^2+16 b c d e+16 c^2 d^2\right )-B d \left (39 b^2 e^2+152 b c d e+128 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {\sqrt {x} \sqrt {b-c x} \left (40 A c e (b e+2 c d)-B \left (3 b^2 e^2+88 b c d e+128 c^2 d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b-c x}}dx}{e \sqrt {b x-c x^2}}}{3 e^2}}{5 e^2}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {2 \left (b x-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x-c x^2} (-e x (-10 A c e+3 b B e+16 B c d)+5 A e (3 b e+8 c d)-4 B d (9 b e+16 c d))}{3 e^2 \sqrt {d+e x}}-\frac {\frac {\sqrt {x} \sqrt {b-c x} \left (5 A e \left (3 b^2 e^2+16 b c d e+16 c^2 d^2\right )-B d \left (39 b^2 e^2+152 b c d e+128 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (40 A c e (b e+2 c d)-B \left (3 b^2 e^2+88 b c d e+128 c^2 d^2\right )\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {1-\frac {c x}{b}}}dx}{e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}}{3 e^2}}{5 e^2}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {2 \left (b x-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x-c x^2} (-e x (-10 A c e+3 b B e+16 B c d)+5 A e (3 b e+8 c d)-4 B d (9 b e+16 c d))}{3 e^2 \sqrt {d+e x}}-\frac {\frac {\sqrt {x} \sqrt {b-c x} \left (5 A e \left (3 b^2 e^2+16 b c d e+16 c^2 d^2\right )-B d \left (39 b^2 e^2+152 b c d e+128 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (40 A c e (b e+2 c d)-B \left (3 b^2 e^2+88 b c d e+128 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}}{3 e^2}}{5 e^2}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {2 \left (b x-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x-c x^2} (-e x (-10 A c e+3 b B e+16 B c d)+5 A e (3 b e+8 c d)-4 B d (9 b e+16 c d))}{3 e^2 \sqrt {d+e x}}-\frac {\frac {\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} \left (5 A e \left (3 b^2 e^2+16 b c d e+16 c^2 d^2\right )-B d \left (39 b^2 e^2+152 b c d e+128 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (40 A c e (b e+2 c d)-B \left (3 b^2 e^2+88 b c d e+128 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}}{3 e^2}}{5 e^2}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {2 \left (b x-c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {b x-c x^2} (-e x (-10 A c e+3 b B e+16 B c d)+5 A e (3 b e+8 c d)-4 B d (9 b e+16 c d))}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} \left (5 A e \left (3 b^2 e^2+16 b c d e+16 c^2 d^2\right )-B d \left (39 b^2 e^2+152 b c d e+128 c^2 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (40 A c e (b e+2 c d)-B \left (3 b^2 e^2+88 b c d e+128 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}}{3 e^2}}{5 e^2}\)

Input:

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

Output:

(2*(8*B*d - 5*A*e + 3*B*e*x)*(b*x - c*x^2)^(3/2))/(15*e^2*(d + e*x)^(3/2)) 
 - ((2*(5*A*e*(8*c*d + 3*b*e) - 4*B*d*(16*c*d + 9*b*e) - e*(16*B*c*d + 3*b 
*B*e - 10*A*c*e)*x)*Sqrt[b*x - c*x^2])/(3*e^2*Sqrt[d + e*x]) - ((-2*Sqrt[b 
]*(40*A*c*e*(2*c*d + b*e) - B*(128*c^2*d^2 + 88*b*c*d*e + 3*b^2*e^2))*Sqrt 
[x]*Sqrt[1 - (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqr 
t[b]], -((b*e)/(c*d))])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x - c*x^2]) + 
(2*Sqrt[b]*(5*A*e*(16*c^2*d^2 + 16*b*c*d*e + 3*b^2*e^2) - B*d*(128*c^2*d^2 
 + 152*b*c*d*e + 39*b^2*e^2))*Sqrt[x]*Sqrt[1 - (c*x)/b]*Sqrt[1 + (e*x)/d]* 
EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[b]], -((b*e)/(c*d))])/(Sqrt[c]*e*S 
qrt[d + e*x]*Sqrt[b*x - c*x^2]))/(3*e^2))/(5*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 120

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- 
b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt 
Q[e, 0] &&  !LtQ[-b/d, 0]
 

rule 122

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) 
)   Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b 
, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 126

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* 
Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & 
& GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x 
]))   Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free 
Q[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 1169

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> 
 Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2])   Int[(d + e*x)^m/(Sqrt[x]* 
Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq 
Q[m^2, 1/4]
 

rule 1230

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

rule 1269

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(938\) vs. \(2(450)=900\).

Time = 3.00 (sec) , antiderivative size = 939, normalized size of antiderivative = 1.82

method	result	size
elliptic	\(\frac {\sqrt {x \left (-c x +b \right )}\, \sqrt {\left (-c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 d \left (A b \,e^{2}+A c d e -B b d e -B c \,d^{2}\right ) \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{3 e^{6} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (-c e \,x^{2}+b e x \right ) \left (4 A b \,e^{2}+8 A c d e -7 B b d e -11 B c \,d^{2}\right )}{3 e^{5} \sqrt {\left (x +\frac {d}{e}\right ) \left (-c e \,x^{2}+b e x \right )}}-\frac {2 B c x \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{5 e^{3}}-\frac {2 \left (\frac {c \left (A c e -2 B b e -2 B c d \right )}{e^{3}}+\frac {2 B c \left (2 b e -2 c d \right )}{5 e^{3}}\right ) \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{3 c e}+\frac {2 \left (\frac {A \,b^{2} e^{3}+4 A b c d \,e^{2}+3 A \,c^{2} d^{2} e -2 B \,b^{2} d \,e^{2}-6 B b c \,d^{2} e -4 B \,c^{2} d^{3}}{e^{5}}-\frac {d \left (A b \,e^{2}+A c d e -B b d e -B c \,d^{2}\right ) c}{3 e^{5}}-\frac {\left (4 A b \,e^{2}+8 A c d e -7 B b d e -11 B c \,d^{2}\right ) \left (b e +c d \right )}{3 e^{5}}+\frac {b \left (4 A b \,e^{2}+8 A c d e -7 B b d e -11 B c \,d^{2}\right )}{3 e^{4}}+\frac {\left (\frac {c \left (A c e -2 B b e -2 B c d \right )}{e^{3}}+\frac {2 B c \left (2 b e -2 c d \right )}{5 e^{3}}\right ) b d}{3 c e}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}+\frac {2 \left (-\frac {2 A b c \,e^{2}+2 A \,c^{2} d e -B \,e^{2} b^{2}-4 B b c d e -3 B \,c^{2} d^{2}}{e^{4}}-\frac {\left (4 A b \,e^{2}+8 A c d e -7 B b d e -11 B c \,d^{2}\right ) c}{3 e^{4}}+\frac {3 B c b d}{5 e^{3}}+\frac {2 \left (\frac {c \left (A c e -2 B b e -2 B c d \right )}{e^{3}}+\frac {2 B c \left (2 b e -2 c d \right )}{5 e^{3}}\right ) \left (b e -c d \right )}{3 c e}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )+\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (-c x +b \right )}\)	\(939\)
default	\(\text {Expression too large to display}\)	\(2154\)

Input:

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

Output:

1/(e*x+d)^(1/2)*(x*(-c*x+b))^(1/2)*((-c*x+b)*x*(e*x+d))^(1/2)/x/(-c*x+b)*( 
2/3*d*(A*b*e^2+A*c*d*e-B*b*d*e-B*c*d^2)/e^6*(-c*e*x^3+b*e*x^2-c*d*x^2+b*d* 
x)^(1/2)/(x+d/e)^2-2/3*(-c*e*x^2+b*e*x)*(4*A*b*e^2+8*A*c*d*e-7*B*b*d*e-11* 
B*c*d^2)/e^5/((x+d/e)*(-c*e*x^2+b*e*x))^(1/2)-2/5*B*c/e^3*x*(-c*e*x^3+b*e* 
x^2-c*d*x^2+b*d*x)^(1/2)-2/3*(c/e^3*(A*c*e-2*B*b*e-2*B*c*d)+2/5*B*c/e^3*(2 
*b*e-2*c*d))/c/e*(-c*e*x^3+b*e*x^2-c*d*x^2+b*d*x)^(1/2)+2*((A*b^2*e^3+4*A* 
b*c*d*e^2+3*A*c^2*d^2*e-2*B*b^2*d*e^2-6*B*b*c*d^2*e-4*B*c^2*d^3)/e^5-1/3*d 
*(A*b*e^2+A*c*d*e-B*b*d*e-B*c*d^2)/e^5*c-1/3*(4*A*b*e^2+8*A*c*d*e-7*B*b*d* 
e-11*B*c*d^2)/e^5*(b*e+c*d)+1/3*b/e^4*(4*A*b*e^2+8*A*c*d*e-7*B*b*d*e-11*B* 
c*d^2)+1/3*(c/e^3*(A*c*e-2*B*b*e-2*B*c*d)+2/5*B*c/e^3*(2*b*e-2*c*d))/c/e*b 
*d)*d/e*((x+d/e)/d*e)^(1/2)*((x-b/c)/(-d/e-b/c))^(1/2)*(-e*x/d)^(1/2)/(-c* 
e*x^3+b*e*x^2-c*d*x^2+b*d*x)^(1/2)*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d 
/e-b/c))^(1/2))+2*(-1/e^4*(2*A*b*c*e^2+2*A*c^2*d*e-B*b^2*e^2-4*B*b*c*d*e-3 
*B*c^2*d^2)-1/3*(4*A*b*e^2+8*A*c*d*e-7*B*b*d*e-11*B*c*d^2)/e^4*c+3/5*B*c/e 
^3*b*d+2/3*(c/e^3*(A*c*e-2*B*b*e-2*B*c*d)+2/5*B*c/e^3*(2*b*e-2*c*d))/c/e*( 
b*e-c*d))*d/e*((x+d/e)/d*e)^(1/2)*((x-b/c)/(-d/e-b/c))^(1/2)*(-e*x/d)^(1/2 
)/(-c*e*x^3+b*e*x^2-c*d*x^2+b*d*x)^(1/2)*((-d/e-b/c)*EllipticE(((x+d/e)/d* 
e)^(1/2),(-d/e/(-d/e-b/c))^(1/2))+b/c*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/ 
(-d/e-b/c))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 872, normalized size of antiderivative = 1.69 \[ \int \frac {(A+B x) \left (b x-c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/45*((128*B*c^3*d^5 + 8*(19*B*b*c^2 - 10*A*c^3)*d^4*e + (23*B*b^2*c - 80* 
A*b*c^2)*d^3*e^2 - (3*B*b^3 + 5*A*b^2*c)*d^2*e^3 + (128*B*c^3*d^3*e^2 + 8* 
(19*B*b*c^2 - 10*A*c^3)*d^2*e^3 + (23*B*b^2*c - 80*A*b*c^2)*d*e^4 - (3*B*b 
^3 + 5*A*b^2*c)*e^5)*x^2 + 2*(128*B*c^3*d^4*e + 8*(19*B*b*c^2 - 10*A*c^3)* 
d^3*e^2 + (23*B*b^2*c - 80*A*b*c^2)*d^2*e^3 - (3*B*b^3 + 5*A*b^2*c)*d*e^4) 
*x)*sqrt(-c*e)*weierstrassPInverse(4/3*(c^2*d^2 + b*c*d*e + b^2*e^2)/(c^2* 
e^2), -4/27*(2*c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 2*b^3*e^3)/(c^3*e 
^3), 1/3*(3*c*e*x + c*d - b*e)/(c*e)) + 3*(128*B*c^3*d^4*e + 8*(11*B*b*c^2 
 - 10*A*c^3)*d^3*e^2 + (3*B*b^2*c - 40*A*b*c^2)*d^2*e^3 + (128*B*c^3*d^2*e 
^3 + 8*(11*B*b*c^2 - 10*A*c^3)*d*e^4 + (3*B*b^2*c - 40*A*b*c^2)*e^5)*x^2 + 
 2*(128*B*c^3*d^3*e^2 + 8*(11*B*b*c^2 - 10*A*c^3)*d^2*e^3 + (3*B*b^2*c - 4 
0*A*b*c^2)*d*e^4)*x)*sqrt(-c*e)*weierstrassZeta(4/3*(c^2*d^2 + b*c*d*e + b 
^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 2*b^ 
3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 + b*c*d*e + b^2*e^2)/(c 
^2*e^2), -4/27*(2*c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 2*b^3*e^3)/(c^ 
3*e^3), 1/3*(3*c*e*x + c*d - b*e)/(c*e))) - 3*(3*B*c^3*e^5*x^3 - 64*B*c^3* 
d^3*e^2 + 15*A*b*c^2*d*e^4 - 4*(9*B*b*c^2 - 10*A*c^3)*d^2*e^3 - (8*B*c^3*d 
*e^4 + (6*B*b*c^2 - 5*A*c^3)*e^5)*x^2 - (80*B*c^3*d^2*e^3 - 20*A*b*c^2*e^5 
 + (47*B*b*c^2 - 50*A*c^3)*d*e^4)*x)*sqrt(-c*x^2 + b*x)*sqrt(e*x + d))/(c^ 
2*e^8*x^2 + 2*c^2*d*e^7*x + c^2*d^2*e^6)
 

Sympy [F]

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

integrate((B*x+A)*(-c*x**2+b*x)**(3/2)/(e*x+d)**(5/2),x)
 

Output:

Integral((-x*(-b + c*x))**(3/2)*(A + B*x)/(d + e*x)**(5/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(A+B x) \left (b x-c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\text {too large to display} \] Input:

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

Output:

(120*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*a*b**2*c*d*e**2 + 80*sqrt(x)*sqrt 
(d + e*x)*sqrt(b - c*x)*a*b**2*c*e**3*x + 180*sqrt(x)*sqrt(d + e*x)*sqrt(b 
 - c*x)*a*b*c**2*d**2*e + 200*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*a*b*c**2 
*d*e**2*x - 20*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*a*b*c**2*e**3*x**2 + 12 
0*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*a*c**3*d**2*e*x - 20*sqrt(x)*sqrt(d 
+ e*x)*sqrt(b - c*x)*a*c**3*d*e**2*x**2 - 18*sqrt(x)*sqrt(d + e*x)*sqrt(b 
- c*x)*b**4*d*e**2 - 12*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*b**4*e**3*x - 
246*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*b**3*c*d**2*e - 176*sqrt(x)*sqrt(d 
 + e*x)*sqrt(b - c*x)*b**3*c*d*e**2*x + 24*sqrt(x)*sqrt(d + e*x)*sqrt(b - 
c*x)*b**3*c*e**3*x**2 - 288*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*b**2*c**2* 
d**3 - 356*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*b**2*c**2*d**2*e*x + 56*sqr 
t(x)*sqrt(d + e*x)*sqrt(b - c*x)*b**2*c**2*d*e**2*x**2 - 12*sqrt(x)*sqrt(d 
 + e*x)*sqrt(b - c*x)*b**2*c**2*e**3*x**3 - 192*sqrt(x)*sqrt(d + e*x)*sqrt 
(b - c*x)*b*c**3*d**3*x + 32*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*b*c**3*d* 
*2*e*x**2 - 12*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*b*c**3*d*e**2*x**3 - 60 
*int((sqrt(d + e*x)*sqrt(b - c*x))/(sqrt(x)*b**2*d**3*e + 3*sqrt(x)*b**2*d 
**2*e**2*x + 3*sqrt(x)*b**2*d*e**3*x**2 + sqrt(x)*b**2*e**4*x**3 + sqrt(x) 
*b*c*d**4 + 2*sqrt(x)*b*c*d**3*e*x - 2*sqrt(x)*b*c*d*e**3*x**3 - sqrt(x)*b 
*c*e**4*x**4 - sqrt(x)*c**2*d**4*x - 3*sqrt(x)*c**2*d**3*e*x**2 - 3*sqrt(x 
)*c**2*d**2*e**2*x**3 - sqrt(x)*c**2*d*e**3*x**4),x)*a*b**4*c*d**4*e**3...