\(\int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx\) [636]

Optimal result

Integrand size = 24, antiderivative size = 561 \[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\frac {2 \sqrt {d+e x} \left (3 c^2 d^2-4 b^2 e^2+c e (9 b d-5 a e)+12 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2/105*(e*x+d)^(1/2)*(3*c^2*d^2-4*b^2*e^2+c*e*(-5*a*e+9*b*d)+12*c*e*(-b*e+2 
*c*d)*x)*(c*x^2+b*x+a)^(1/2)/c^2/e+2/7*e*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(3/2) 
/c-1/105*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-b*e+2*c*d)*(3*c^2*d^2+8*b^2*e^2-c*e* 
(29*a*e+3*b*d))*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*Ellipt 
icE(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^3/e^2/(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)+4/105*2^(1/2)*(-4*a*c+ 
b^2)^(1/2)*(a*e^2-b*d*e+c*d^2)*(3*c^2*d^2+2*b^2*e^2-c*e*(5*a*e+3*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^3/e^2/( 
e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.49 (sec) , antiderivative size = 1289, normalized size of antiderivative = 2.30 \[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

Sqrt[d + e*x]*((2*(3*c^2*d^2 + 9*b*c*d*e - 4*b^2*e^2 + 10*a*c*e^2))/(105*c 
^2*e) + (2*(8*c*d + b*e)*x)/(35*c) + (2*e*x^2)/7)*Sqrt[a + x*(b + c*x)] + 
((d + e*x)^(3/2)*Sqrt[a + x*(b + c*x)]*(4*(-2*c*d + b*e)*Sqrt[(c*d^2 + e*( 
-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(3*c^2*d^2 + 8*b^ 
2*e^2 - c*e*(3*b*d + 29*a*e))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + 
 e*x) + (a*e)/(d + e*x)))/(d + e*x)) - (I*Sqrt[2]*(-2*c*d + b*e)*(2*c*d - 
b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(3*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*a* 
e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d 
 + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e 
^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/ 
(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c) 
*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d 
+ b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[ 
(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e* 
x] + (I*Sqrt[2]*(-8*b^4*e^4 + b^3*e^3*(27*c*d + 8*Sqrt[(b^2 - 4*a*c)*e^2]) 
 - b^2*c*e^2*(27*c*d^2 - 37*a*e^2 + 19*d*Sqrt[(b^2 - 4*a*c)*e^2]) + 2*c^2* 
(-10*a^2*e^4 - 3*c*d^3*Sqrt[(b^2 - 4*a*c)*e^2] + a*d*e^2*(54*c*d + 29*Sqrt 
[(b^2 - 4*a*c)*e^2])) + b*c*e*(9*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] - a*e^2*(10 
8*c*d + 29*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a 
*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x))...
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1166, 27, 1231, 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.

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

\(\Big \downarrow \) 1166

\(\displaystyle \frac {2 \int \frac {\left (7 c d^2-e (3 b d+a e)+4 e (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{2 \sqrt {d+e x}}dx}{7 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (7 c d^2-e (3 b d+a e)+4 e (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{\sqrt {d+e x}}dx}{7 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{15 c e}-\frac {2 \int -\frac {e \left (4 d e^2 b^3-\left (9 c d^2 e-4 a e^3\right ) b^2-c d \left (3 c d^2+25 a e^2\right ) b+2 a c e \left (27 c d^2-5 a e^2\right )-(2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{7 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {4 d e^2 b^3-\left (9 c d^2 e-4 a e^3\right ) b^2-c d \left (3 c d^2+25 a e^2\right ) b+2 a c e \left (27 c d^2-5 a e^2\right )-(2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{15 c e}}{7 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {2 \left (a e^2-b d e+c d^2\right ) \left (-5 a c e^2+2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}}{15 c e}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{15 c e}}{7 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {\frac {\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-5 a c e^2+2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{15 c e}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{15 c e}}{7 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-5 a c e^2+2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{15 c e}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{15 c e}}{7 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-5 a c e^2+2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{15 c e}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{15 c e}}{7 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}\)

Input:

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

Output:

(2*e*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2))/(7*c) + ((2*Sqrt[d + e*x]*(3*c 
^2*d^2 - 4*b^2*e^2 + c*e*(9*b*d - 5*a*e) + 12*c*e*(2*c*d - b*e)*x)*Sqrt[a 
+ b*x + c*x^2])/(15*c*e) + (-((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(3* 
c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*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 
 + Sqrt[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])) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^ 
2 - b*d*e + a*e^2)*(3*c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2 - 5*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)/ 
Sqrt[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]))/(15*c*e))/(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 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

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 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[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. \(1211\) vs. \(2(501)=1002\).

Time = 3.25 (sec) , antiderivative size = 1212, normalized size of antiderivative = 2.16

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1212\)
risch	\(\text {Expression too large to display}\)	\(2640\)
default	\(\text {Expression too large to display}\)	\(6516\)

Input:

int((e*x+d)^(3/2)*(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*e*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*(b*e^2+2*d*e*c-2/7*e* 
(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*( 
a*e^2+2*b*d*e+c*d^2-2/7*e*(5/2*a*e+5/2*b*d)-2/5*(b*e^2+2*d*e*c-2/7*e*(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*(a*d^2-2/5*(b*e^2+2*d*e*c-2/7*e*(3*b*e+3*c*d))/c/e*a*d-2/3*(a*e^2 
+2*b*d*e+c*d^2-2/7*e*(5/2*a*e+5/2*b*d)-2/5*(b*e^2+2*d*e*c-2/7*e*(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*(10/7*a*d*e+b*d^2-2/5*(b*e^2+2*d*e*c 
-2/7*e*(3*b*e+3*c*d))/c/e*(3/2*a*e+3/2*b*d)-2/3*(a*e^2+2*b*d*e+c*d^2-2/7*e 
*(5/2*a*e+5/2*b*d)-2/5*(b*e^2+2*d*e*c-2/7*e*(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...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.05 \[ \int (d+e x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\frac {2 \, {\left ({\left (6 \, c^{4} d^{4} - 12 \, b c^{3} d^{3} e - {\left (17 \, b^{2} c^{2} - 104 \, a c^{3}\right )} d^{2} e^{2} + {\left (23 \, b^{3} c - 104 \, a b c^{2}\right )} d e^{3} - {\left (8 \, b^{4} - 41 \, a b^{2} c + 30 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (6 \, c^{4} d^{3} e - 9 \, b c^{3} d^{2} e^{2} + {\left (19 \, b^{2} c^{2} - 58 \, a c^{3}\right )} d e^{3} - {\left (8 \, b^{3} c - 29 \, a b c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (15 \, c^{4} e^{4} x^{2} + 3 \, c^{4} d^{2} e^{2} + 9 \, b c^{3} d e^{3} - 2 \, {\left (2 \, b^{2} c^{2} - 5 \, a c^{3}\right )} e^{4} + 3 \, {\left (8 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{315 \, c^{4} e^{3}} \] Input:

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

Output:

2/315*((6*c^4*d^4 - 12*b*c^3*d^3*e - (17*b^2*c^2 - 104*a*c^3)*d^2*e^2 + (2 
3*b^3*c - 104*a*b*c^2)*d*e^3 - (8*b^4 - 41*a*b^2*c + 30*a^2*c^2)*e^4)*sqrt 
(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*(6*c^4* 
d^3*e - 9*b*c^3*d^2*e^2 + (19*b^2*c^2 - 58*a*c^3)*d*e^3 - (8*b^3*c - 29*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 + 3*c^4*d^2*e^2 + 9*b*c^3*d* 
e^3 - 2*(2*b^2*c^2 - 5*a*c^3)*e^4 + 3*(8*c^4*d*e^3 + b*c^3*e^4)*x)*sqrt(c* 
x^2 + b*x + a)*sqrt(e*x + d))/(c^4*e^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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