\(\int \frac {x^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx\) [12]

Optimal result

Integrand size = 25, antiderivative size = 362 \[ \int \frac {x^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=-\frac {\left (64 c^3 d^3-16 b c^2 d^2 e-5 b^3 e^3-4 b c e^2 (2 b d-a e)-2 c e \left (16 c^2 d^2+5 b^2 e^2+4 c e (2 b d-a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}-\frac {(14 c d+5 b e) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (128 c^4 d^4-5 b^4 e^4-8 b^2 c e^3 (b d-3 a e)-64 c^3 d^2 e (b d-a e)-16 c^2 e^2 (b d-a e)^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2} e^5}-\frac {d^3 \sqrt {c d^2-b d e+a e^2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^5} \] Output:

-1/64*(64*c^3*d^3-16*b*c^2*d^2*e-5*b^3*e^3-4*b*c*e^2*(-a*e+2*b*d)-2*c*e*(1 
6*c^2*d^2+5*b^2*e^2+4*c*e*(-a*e+2*b*d))*x)*(c*x^2+b*x+a)^(1/2)/c^3/e^4-1/2 
4*(5*b*e+14*c*d)*(c*x^2+b*x+a)^(3/2)/c^2/e^2+1/4*(e*x+d)*(c*x^2+b*x+a)^(3/ 
2)/c/e^2+1/128*(128*c^4*d^4-5*b^4*e^4-8*b^2*c*e^3*(-3*a*e+b*d)-64*c^3*d^2* 
e*(-a*e+b*d)-16*c^2*e^2*(-a*e+b*d)^2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2 
+b*x+a)^(1/2))/c^(7/2)/e^5-d^3*(a*e^2-b*d*e+c*d^2)^(1/2)*arctanh(1/2*(b*d- 
2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/e^5
 

Mathematica [A] (verified)

Time = 2.50 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {2 \sqrt {c} e \sqrt {a+x (b+c x)} \left (15 b^3 e^3-2 b c e^2 (-12 b d+26 a e+5 b e x)-16 c^3 \left (12 d^3-6 d^2 e x+4 d e^2 x^2-3 e^3 x^3\right )+8 c^2 e \left (a e (-8 d+3 e x)+b \left (6 d^2-2 d e x+e^2 x^2\right )\right )\right )-768 c^{7/2} d^3 \sqrt {-c d^2+b d e-a e^2} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )-3 \left (128 c^4 d^4-5 b^4 e^4-8 b^2 c e^3 (b d-3 a e)-64 c^3 d^2 e (b d-a e)-16 c^2 e^2 (b d-a e)^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{384 c^{7/2} e^5} \] Input:

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

Output:

(2*Sqrt[c]*e*Sqrt[a + x*(b + c*x)]*(15*b^3*e^3 - 2*b*c*e^2*(-12*b*d + 26*a 
*e + 5*b*e*x) - 16*c^3*(12*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 - 3*e^3*x^3) + 8* 
c^2*e*(a*e*(-8*d + 3*e*x) + b*(6*d^2 - 2*d*e*x + e^2*x^2))) - 768*c^(7/2)* 
d^3*Sqrt[-(c*d^2) + b*d*e - a*e^2]*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + 
x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]] - 3*(128*c^4*d^4 - 5*b^4*e^4 
 - 8*b^2*c*e^3*(b*d - 3*a*e) - 64*c^3*d^2*e*(b*d - a*e) - 16*c^2*e^2*(b*d 
- a*e)^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(384*c^(7/2)*e 
^5)
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1267, 27, 2184, 27, 1231, 27, 1269, 1092, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1267

\(\displaystyle \frac {\int -\frac {\sqrt {c x^2+b x+a} \left (e^2 (14 c d+5 b e) x^2+2 e \left (3 c d^2+e (4 b d+a e)\right ) x+d e (3 b d+2 a e)\right )}{2 (d+e x)}dx}{4 c e^3}+\frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\int \frac {\sqrt {c x^2+b x+a} \left (e^2 (14 c d+5 b e) x^2+2 e \left (3 c d^2+e (4 b d+a e)\right ) x+d e (3 b d+2 a e)\right )}{d+e x}dx}{8 c e^3}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\frac {\int -\frac {3 e^3 \left (d \left (5 e b^2+8 c d b-4 a c e\right )+\left (16 c^2 d^2+5 b^2 e^2+4 c e (2 b d-a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)}dx}{3 c e^2}+\frac {e \left (a+b x+c x^2\right )^{3/2} (5 b e+14 c d)}{3 c}}{8 c e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{3/2} (5 b e+14 c d)}{3 c}-\frac {e \int \frac {\left (d \left (5 e b^2+8 c d b-4 a c e\right )+\left (16 c^2 d^2+5 b^2 e^2+4 c e (2 b d-a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{2 c}}{8 c e^3}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{3/2} (5 b e+14 c d)}{3 c}-\frac {e \left (-\frac {\int \frac {d \left (5 e^3 b^4+8 c d e^2 b^3+8 c e \left (2 c d^2-3 a e^2\right ) b^2-32 c^2 d \left (2 c d^2+a e^2\right ) b+16 a c^2 e \left (4 c d^2+a e^2\right )\right )-\left (128 c^4 d^4-64 c^3 e (b d-a e) d^2-5 b^4 e^4-16 c^2 e^2 (b d-a e)^2-8 b^2 c e^3 (b d-3 a e)\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (4 c e (2 b d-a e)+5 b^2 e^2+16 c^2 d^2\right )-4 b c e^2 (2 b d-a e)-5 b^3 e^3-16 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}\right )}{2 c}}{8 c e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{3/2} (5 b e+14 c d)}{3 c}-\frac {e \left (-\frac {\int \frac {d \left (5 e^3 b^4+8 c d e^2 b^3+8 c e \left (2 c d^2-3 a e^2\right ) b^2-32 c^2 d \left (2 c d^2+a e^2\right ) b+16 a c^2 e \left (4 c d^2+a e^2\right )\right )-\left (128 c^4 d^4-64 c^3 e (b d-a e) d^2-5 b^4 e^4-16 c^2 e^2 (b d-a e)^2-8 b^2 c e^3 (b d-3 a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (4 c e (2 b d-a e)+5 b^2 e^2+16 c^2 d^2\right )-4 b c e^2 (2 b d-a e)-5 b^3 e^3-16 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}\right )}{2 c}}{8 c e^3}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{3/2} (5 b e+14 c d)}{3 c}-\frac {e \left (-\frac {\frac {128 c^3 d^3 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (-8 b^2 c e^3 (b d-3 a e)-64 c^3 d^2 e (b d-a e)-16 c^2 e^2 (b d-a e)^2-5 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (4 c e (2 b d-a e)+5 b^2 e^2+16 c^2 d^2\right )-4 b c e^2 (2 b d-a e)-5 b^3 e^3-16 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}\right )}{2 c}}{8 c e^3}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{3/2} (5 b e+14 c d)}{3 c}-\frac {e \left (-\frac {\frac {128 c^3 d^3 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \left (-8 b^2 c e^3 (b d-3 a e)-64 c^3 d^2 e (b d-a e)-16 c^2 e^2 (b d-a e)^2-5 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (4 c e (2 b d-a e)+5 b^2 e^2+16 c^2 d^2\right )-4 b c e^2 (2 b d-a e)-5 b^3 e^3-16 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}\right )}{2 c}}{8 c e^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{3/2} (5 b e+14 c d)}{3 c}-\frac {e \left (-\frac {\frac {128 c^3 d^3 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 b^2 c e^3 (b d-3 a e)-64 c^3 d^2 e (b d-a e)-16 c^2 e^2 (b d-a e)^2-5 b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (4 c e (2 b d-a e)+5 b^2 e^2+16 c^2 d^2\right )-4 b c e^2 (2 b d-a e)-5 b^3 e^3-16 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}\right )}{2 c}}{8 c e^3}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{3/2} (5 b e+14 c d)}{3 c}-\frac {e \left (-\frac {-\frac {256 c^3 d^3 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 b^2 c e^3 (b d-3 a e)-64 c^3 d^2 e (b d-a e)-16 c^2 e^2 (b d-a e)^2-5 b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (4 c e (2 b d-a e)+5 b^2 e^2+16 c^2 d^2\right )-4 b c e^2 (2 b d-a e)-5 b^3 e^3-16 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}\right )}{2 c}}{8 c e^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{3/2} (5 b e+14 c d)}{3 c}-\frac {e \left (-\frac {\frac {128 c^3 d^3 \sqrt {a e^2-b d e+c d^2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 b^2 c e^3 (b d-3 a e)-64 c^3 d^2 e (b d-a e)-16 c^2 e^2 (b d-a e)^2-5 b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (4 c e (2 b d-a e)+5 b^2 e^2+16 c^2 d^2\right )-4 b c e^2 (2 b d-a e)-5 b^3 e^3-16 b c^2 d^2 e+64 c^3 d^3\right )}{4 c e^2}\right )}{2 c}}{8 c e^3}\)

Input:

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

Output:

((d + e*x)*(a + b*x + c*x^2)^(3/2))/(4*c*e^2) - ((e*(14*c*d + 5*b*e)*(a + 
b*x + c*x^2)^(3/2))/(3*c) - (e*(-1/4*((64*c^3*d^3 - 16*b*c^2*d^2*e - 5*b^3 
*e^3 - 4*b*c*e^2*(2*b*d - a*e) - 2*c*e*(16*c^2*d^2 + 5*b^2*e^2 + 4*c*e*(2* 
b*d - a*e))*x)*Sqrt[a + b*x + c*x^2])/(c*e^2) - (-(((128*c^4*d^4 - 5*b^4*e 
^4 - 8*b^2*c*e^3*(b*d - 3*a*e) - 64*c^3*d^2*e*(b*d - a*e) - 16*c^2*e^2*(b* 
d - a*e)^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[ 
c]*e)) + (128*c^3*d^3*Sqrt[c*d^2 - b*d*e + a*e^2]*ArcTanh[(b*d - 2*a*e + ( 
2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e) 
/(8*c*e^2)))/(2*c))/(8*c*e^3)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

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

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 1267

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

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 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 [A] (verified)

Time = 1.49 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.29

method	result	size
risch	\(-\frac {\left (-48 c^{3} x^{3} e^{3}-8 b \,c^{2} e^{3} x^{2}+64 c^{3} d \,e^{2} x^{2}-24 a \,c^{2} e^{3} x +10 x \,b^{2} c \,e^{3}+16 b \,c^{2} d \,e^{2} x -96 d^{2} e \,c^{3} x +52 a b c \,e^{3}+64 d \,e^{2} a \,c^{2}-15 b^{3} e^{3}-24 d \,e^{2} b^{2} c -48 d^{2} e b \,c^{2}+192 d^{3} c^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{192 c^{3} e^{4}}-\frac {\frac {\left (16 e^{4} a^{2} c^{2}-24 a \,b^{2} c \,e^{4}-32 a b \,c^{2} d \,e^{3}-64 d^{2} e^{2} a \,c^{3}+5 b^{4} e^{4}+8 d \,e^{3} b^{3} c +16 d^{2} e^{2} b^{2} c^{2}+64 d^{3} e b \,c^{3}-128 d^{4} c^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}-\frac {128 d^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{3} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{128 e^{4} c^{3}}\)	\(467\)
default	\(\frac {d^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{e^{3}}+\frac {\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}}{e}-\frac {d \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{e^{2}}-\frac {d^{3} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{4}}\)	\(686\)

Input:

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

Output:

-1/192*(-48*c^3*e^3*x^3-8*b*c^2*e^3*x^2+64*c^3*d*e^2*x^2-24*a*c^2*e^3*x+10 
*b^2*c*e^3*x+16*b*c^2*d*e^2*x-96*c^3*d^2*e*x+52*a*b*c*e^3+64*a*c^2*d*e^2-1 
5*b^3*e^3-24*b^2*c*d*e^2-48*b*c^2*d^2*e+192*c^3*d^3)/c^3*(c*x^2+b*x+a)^(1/ 
2)/e^4-1/128/e^4/c^3*((16*a^2*c^2*e^4-24*a*b^2*c*e^4-32*a*b*c^2*d*e^3-64*a 
*c^3*d^2*e^2+5*b^4*e^4+8*b^3*c*d*e^3+16*b^2*c^2*d^2*e^2+64*b*c^3*d^3*e-128 
*c^4*d^4)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-128*d^3*(a 
*e^2-b*d*e+c*d^2)*c^3/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d 
*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*( 
x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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