\(\int \frac {x^3 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx\) [88]

Optimal result

Integrand size = 27, antiderivative size = 599 \[ \int \frac {x^3 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \left (8 c^2 d^2+24 b^2 e^2+c e (13 b d-25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3 e^2}-\frac {6 (3 c d+2 b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2 e^2}+\frac {2 (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e^2}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 c^3 d^3-48 b^3 e^3+c^2 d e (9 b d-19 a e)+8 b c e^2 (2 b d+13 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 e^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (8 c^2 d^2+24 b^2 e^2+c e (13 b d-25 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2/105*(8*c^2*d^2+24*b^2*e^2+c*e*(-25*a*e+13*b*d))*(e*x+d)^(1/2)*(c*x^2+b*x 
+a)^(1/2)/c^3/e^2-6/35*(2*b*e+3*c*d)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^2 
/e^2+2/7*(e*x+d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c/e^2+1/105*2^(1/2)*(-4*a*c+b^2 
)^(1/2)*(8*c^3*d^3-48*b^3*e^3+c^2*d*e*(-19*a*e+9*b*d)+8*b*c*e^2*(13*a*e+2* 
b*d))*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticE(1/2*(1 
+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c 
*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c^4/e^3/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+ 
b^2)^(1/2))*e))^(1/2)/(c*x^2+b*x+a)^(1/2)-2/105*2^(1/2)*(-4*a*c+b^2)^(1/2) 
*(a*e^2-b*d*e+c*d^2)*(8*c^2*d^2+24*b^2*e^2+c*e*(-25*a*e+13*b*d))*(c*(e*x+d 
)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^ 
(1/2)*EllipticF(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4 
*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c^4/e^3/(e*x+d) 
^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.12 (sec) , antiderivative size = 5357, normalized size of antiderivative = 8.94 \[ \int \frac {x^3 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1283, 2184, 27, 2184, 27, 1269, 1172, 321, 327}

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.

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

\(\Big \downarrow \) 1283

\(\displaystyle \frac {2 x^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\int \frac {x \left (-\left ((c d-6 b e) x^2\right )+5 (b d+a e) x+4 a d\right )}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{7 c}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {2 x^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\frac {2 \int \frac {e^2 \left (7 c^2 d^2-24 b^2 e^2-c e (13 b d-25 a e)\right ) x^2+e \left (2 c^2 d^3-c e (7 b d-23 a e) d-6 b e^2 (5 b d+3 a e)\right ) x+d e (b d+3 a e) (c d-6 b e)}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{5 c e^3}-\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (c d-6 b e)}{5 c e^2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\frac {\int \frac {e^2 \left (7 c^2 d^2-24 b^2 e^2-c e (13 b d-25 a e)\right ) x^2+e \left (2 c^2 d^3-c e (7 b d-23 a e) d-6 b e^2 (5 b d+3 a e)\right ) x+d e (b d+3 a e) (c d-6 b e)}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{5 c e^3}-\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (c d-6 b e)}{5 c e^2}}{7 c}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {2 x^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\frac {\frac {2 \int \frac {e^3 \left (24 d e^2 b^3-\left (5 c d^2 e-24 a e^3\right ) b^2-2 c d \left (2 c d^2+33 a e^2\right ) b+a c e \left (2 c d^2-25 a e^2\right )-\left (8 c^3 d^3+c^2 e (9 b d-19 a e) d-48 b^3 e^3+8 b c e^2 (2 b d+13 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c e^2}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (13 b d-25 a e)-24 b^2 e^2+7 c^2 d^2\right )}{3 c}}{5 c e^3}-\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (c d-6 b e)}{5 c e^2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\frac {\frac {e \int \frac {24 d e^2 b^3-\left (5 c d^2 e-24 a e^3\right ) b^2-2 c d \left (2 c d^2+33 a e^2\right ) b+a c e \left (2 c d^2-25 a e^2\right )-\left (8 c^3 d^3+c^2 e (9 b d-19 a e) d-48 b^3 e^3+8 b c e^2 (2 b d+13 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (13 b d-25 a e)-24 b^2 e^2+7 c^2 d^2\right )}{3 c}}{5 c e^3}-\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (c d-6 b e)}{5 c e^2}}{7 c}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 x^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\frac {\frac {e \left (\frac {\left (a e^2-b d e+c d^2\right ) \left (-25 a c e^2+24 b^2 e^2+13 b c d e+8 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (c^2 d e (9 b d-19 a e)+8 b c e^2 (13 a e+2 b d)-48 b^3 e^3+8 c^3 d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}\right )}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (13 b d-25 a e)-24 b^2 e^2+7 c^2 d^2\right )}{3 c}}{5 c e^3}-\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (c d-6 b e)}{5 c e^2}}{7 c}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2 x^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\frac {\frac {e \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-25 a c e^2+24 b^2 e^2+13 b c d e+8 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 d e (9 b d-19 a e)+8 b c e^2 (13 a e+2 b d)-48 b^3 e^3+8 c^3 d^3\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (13 b d-25 a e)-24 b^2 e^2+7 c^2 d^2\right )}{3 c}}{5 c e^3}-\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (c d-6 b e)}{5 c e^2}}{7 c}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 x^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\frac {\frac {e \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-25 a c e^2+24 b^2 e^2+13 b c d e+8 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 d e (9 b d-19 a e)+8 b c e^2 (13 a e+2 b d)-48 b^3 e^3+8 c^3 d^3\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (13 b d-25 a e)-24 b^2 e^2+7 c^2 d^2\right )}{3 c}}{5 c e^3}-\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (c d-6 b e)}{5 c e^2}}{7 c}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 x^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{7 c}-\frac {\frac {\frac {e \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-25 a c e^2+24 b^2 e^2+13 b c d e+8 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 d e (9 b d-19 a e)+8 b c e^2 (13 a e+2 b d)-48 b^3 e^3+8 c^3 d^3\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (13 b d-25 a e)-24 b^2 e^2+7 c^2 d^2\right )}{3 c}}{5 c e^3}-\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (c d-6 b e)}{5 c e^2}}{7 c}\)

Input:

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

Output:

(2*x^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(7*c) - ((-2*(c*d - 6*b*e)*(d 
+ e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(5*c*e^2) + ((2*e*(7*c^2*d^2 - 24*b^2* 
e^2 - c*e*(13*b*d - 25*a*e))*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c) + 
(e*(-((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8*c^3*d^3 - 48*b^3*e^3 + c^2*d*e*(9*b*d 
- 19*a*e) + 8*b*c*e^2*(2*b*d + 13*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + 
 c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2* 
c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + S 
qrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4* 
a*c])*e)]*Sqrt[a + b*x + c*x^2])) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - 
b*d*e + a*e^2)*(8*c^2*d^2 + 13*b*c*d*e + 24*b^2*e^2 - 25*a*c*e^2)*Sqrt[(c* 
(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2 
))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/S 
qrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^ 
2 - 4*a*c])*e)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])))/(3*c))/(5*c*e 
^3))/(7*c)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

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]
 

rule 1283

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_.) + (b_.) 
*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2*e*(d + e*x)^(m - 1)*Sqrt[f + g*x 
]*(Sqrt[a + b*x + c*x^2]/(c*(2*m + 1))), x] - Simp[1/(c*(2*m + 1))   Int[(( 
d + e*x)^(m - 2)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[e*(b*d*f + a*( 
d*g + 2*e*f*(m - 1))) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f 
*m + d*g*(2*m + 1)) + b*e*(2*d*g + e*f*(2*m - 1)))*x + e*(2*b*e*g*m - c*(e* 
f + d*g*(4*m - 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && I 
ntegerQ[2*m] && GtQ[m, 1]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1124\) vs. \(2(533)=1066\).

Time = 4.64 (sec) , antiderivative size = 1125, normalized size of antiderivative = 1.88

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1125\)
risch	\(\text {Expression too large to display}\)	\(2640\)
default	\(\text {Expression too large to display}\)	\(6517\)

Input:

int(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/7/c*x^2 
*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/5*(d-2/7/c*(3*b*e+3*c*d 
))/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(-2/7/c*(5/2* 
a*e+5/2*b*d)-2/5*(d-2/7/c*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(c*e*x^3+b 
*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(-2/5*(d-2/7/c*(3*b*e+3*c*d))/c/e* 
a*d-2/3*(-2/7/c*(5/2*a*e+5/2*b*d)-2/5*(d-2/7/c*(3*b*e+3*c*d))/c/e*(2*b*e+2 
*c*d))/c/e*(1/2*a*e+1/2*b*d))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/ 
(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2) 
))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/ 
2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2 
+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2)) 
/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^ 
2)^(1/2))))^(1/2))+2*(-4/7*a/c*d-2/5*(d-2/7/c*(3*b*e+3*c*d))/c/e*(3/2*a*e+ 
3/2*b*d)-2/3*(-2/7/c*(5/2*a*e+5/2*b*d)-2/5*(d-2/7/c*(3*b*e+3*c*d))/c/e*(2* 
b*e+2*c*d))/c/e*(b*e+c*d))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/ 
e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/ 
(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2)) 
/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a* 
e*x+b*d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+d 
/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 588, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=-\frac {2 \, {\left ({\left (8 \, c^{4} d^{4} + 5 \, b c^{3} d^{3} e + {\left (10 \, b^{2} c^{2} - 13 \, a c^{3}\right )} d^{2} e^{2} + {\left (40 \, b^{3} c - 113 \, a b c^{2}\right )} d e^{3} - {\left (48 \, b^{4} - 176 \, a b^{2} c + 75 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (8 \, c^{4} d^{3} e + 9 \, b c^{3} d^{2} e^{2} + {\left (16 \, b^{2} c^{2} - 19 \, a c^{3}\right )} d e^{3} - 8 \, {\left (6 \, b^{3} c - 13 \, a b c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (15 \, c^{4} e^{4} x^{2} - 4 \, c^{4} d^{2} e^{2} - 5 \, b c^{3} d e^{3} + {\left (24 \, b^{2} c^{2} - 25 \, a c^{3}\right )} e^{4} + 3 \, {\left (c^{4} d e^{3} - 6 \, b c^{3} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{315 \, c^{5} e^{4}} \] Input:

integrate(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

-2/315*((8*c^4*d^4 + 5*b*c^3*d^3*e + (10*b^2*c^2 - 13*a*c^3)*d^2*e^2 + (40 
*b^3*c - 113*a*b*c^2)*d*e^3 - (48*b^4 - 176*a*b^2*c + 75*a^2*c^2)*e^4)*sqr 
t(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^ 
2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2* 
b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*(8*c^4 
*d^3*e + 9*b*c^3*d^2*e^2 + (16*b^2*c^2 - 19*a*c^3)*d*e^3 - 8*(6*b^3*c - 13 
*a*b*c^2)*e^4)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3 
*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^ 
2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2* 
d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d 
^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*( 
3*c*e*x + c*d + b*e)/(c*e))) - 3*(15*c^4*e^4*x^2 - 4*c^4*d^2*e^2 - 5*b*c^3 
*d*e^3 + (24*b^2*c^2 - 25*a*c^3)*e^4 + 3*(c^4*d*e^3 - 6*b*c^3*e^4)*x)*sqrt 
(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^5*e^4)
 

Sympy [F]

\[ \int \frac {x^3 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {x^{3} \sqrt {d + e x}}{\sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate(x**3*(e*x+d)**(1/2)/(c*x**2+b*x+a)**(1/2),x)
 

Output:

Integral(x**3*sqrt(d + e*x)/sqrt(a + b*x + c*x**2), x)
 

Maxima [F]

\[ \int \frac {x^3 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d} x^{3}}{\sqrt {c x^{2} + b x + a}} \,d x } \] Input:

integrate(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^3 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d} x^{3}}{\sqrt {c x^{2} + b x + a}} \,d x } \] Input:

integrate(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {x^3\,\sqrt {d+e\,x}}{\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int((x^3*(d + e*x)^(1/2))/(a + b*x + c*x^2)^(1/2),x)
 

Output:

int((x^3*(d + e*x)^(1/2))/(a + b*x + c*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^3 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {x^{3} \sqrt {e x +d}}{\sqrt {c \,x^{2}+b x +a}}d x \] Input:

int(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x)
 

Output:

int(x^3*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x)