\(\int \frac {(d+e x)^3}{(a+b x+c x^2)^{5/4}} \, dx\) [754]

Optimal result

Integrand size = 22, antiderivative size = 255 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/4}} \, dx=-\frac {4 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac {2 e \left (24 c^2 d^2+7 b^2 e^2-2 c e (9 b d+8 a e)+6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4}}{3 c^2 \left (b^2-4 a c\right )}+\frac {(2 c d-b e) \left (4 c^2 d^2+7 b^2 e^2-4 c e (b d+6 a e)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )}{\sqrt {2} c^3 \sqrt {b^2-4 a c} \sqrt [4]{a+b x+c x^2}} \] Output:

-4*(e*x+d)^2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/4)+2 
/3*e*(24*c^2*d^2+7*b^2*e^2-2*c*e*(8*a*e+9*b*d)+6*c*e*(-b*e+2*c*d)*x)*(c*x^ 
2+b*x+a)^(3/4)/c^2/(-4*a*c+b^2)+1/2*(-b*e+2*c*d)*(4*c^2*d^2+7*b^2*e^2-4*c* 
e*(6*a*e+b*d))*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*EllipticE(sin(1/2*arc 
sin((2*c*x+b)/(-4*a*c+b^2)^(1/2))),2^(1/2))*2^(1/2)/c^3/(-4*a*c+b^2)^(1/2) 
/(c*x^2+b*x+a)^(1/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.83 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/4}} \, dx=\frac {-8 c \left (-7 b^3 e^3 x-b^2 e^2 (7 a e+c x (-18 d+e x))+2 b c \left (3 c d^2 (d-3 e x)+a e^2 (9 d+11 e x)\right )+4 c \left (4 a^2 e^3+3 c^2 d^3 x+a c e \left (-9 d^2-9 d e x+e^2 x^2\right )\right )\right )+3 \sqrt {2} (2 c d-b e) \left (4 c^2 d^2+7 b^2 e^2-4 c e (b d+6 a e)\right ) (b+2 c x) \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{12 c^3 \left (b^2-4 a c\right ) \sqrt [4]{a+x (b+c x)}} \] Input:

Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(5/4),x]
 

Output:

(-8*c*(-7*b^3*e^3*x - b^2*e^2*(7*a*e + c*x*(-18*d + e*x)) + 2*b*c*(3*c*d^2 
*(d - 3*e*x) + a*e^2*(9*d + 11*e*x)) + 4*c*(4*a^2*e^3 + 3*c^2*d^3*x + a*c* 
e*(-9*d^2 - 9*d*e*x + e^2*x^2))) + 3*Sqrt[2]*(2*c*d - b*e)*(4*c^2*d^2 + 7* 
b^2*e^2 - 4*c*e*(b*d + 6*a*e))*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 
4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c) 
])/(12*c^3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/4))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(708\) vs. \(2(255)=510\).

Time = 0.71 (sec) , antiderivative size = 708, normalized size of antiderivative = 2.78, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1164, 27, 1225, 1094, 834, 761, 1510}

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.

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

\(\Big \downarrow \) 1164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {2 \left (\frac {(2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right ) \int \frac {1}{\sqrt [4]{c x^2+b x+a}}dx}{4 c^2}+\frac {e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (8 a e+9 b d)+7 b^2 e^2+6 c e x (2 c d-b e)+24 c^2 d^2\right )}{3 c^2}\right )}{b^2-4 a c}-\frac {4 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {2 \left (\frac {\sqrt {(b+2 c x)^2} (2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right ) \int \frac {\sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{c^2 (b+2 c x)}+\frac {e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (8 a e+9 b d)+7 b^2 e^2+6 c e x (2 c d-b e)+24 c^2 d^2\right )}{3 c^2}\right )}{b^2-4 a c}-\frac {4 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}\)

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 761

\(\displaystyle \frac {2 \left (\frac {\sqrt {(b+2 c x)^2} (2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right ) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{c^2 (b+2 c x)}+\frac {e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (8 a e+9 b d)+7 b^2 e^2+6 c e x (2 c d-b e)+24 c^2 d^2\right )}{3 c^2}\right )}{b^2-4 a c}-\frac {4 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {2 \left (\frac {\sqrt {(b+2 c x)^2} (2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right ) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \left (\frac {\sqrt [4]{b^2-4 a c} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt [4]{a+b x+c x^2} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}\right )}{2 \sqrt {c}}\right )}{c^2 (b+2 c x)}+\frac {e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (8 a e+9 b d)+7 b^2 e^2+6 c e x (2 c d-b e)+24 c^2 d^2\right )}{3 c^2}\right )}{b^2-4 a c}-\frac {4 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}\)

Input:

Int[(d + e*x)^3/(a + b*x + c*x^2)^(5/4),x]
 

Output:

(-4*(d + e*x)^2*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + 
 c*x^2)^(1/4)) + (2*((e*(24*c^2*d^2 + 7*b^2*e^2 - 2*c*e*(9*b*d + 8*a*e) + 
6*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/4))/(3*c^2) + ((2*c*d - b*e)*( 
4*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(b*d + 6*a*e))*Sqrt[(b + 2*c*x)^2]*(-1/2*(Sq 
rt[b^2 - 4*a*c]*(-(((a + b*x + c*x^2)^(1/4)*Sqrt[b^2 - 4*a*c + 4*c*(a + b* 
x + c*x^2)])/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^ 
2 - 4*a*c]))) + ((b^2 - 4*a*c)^(1/4)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2] 
)/Sqrt[b^2 - 4*a*c])*Sqrt[(b^2 - 4*a*c + 4*c*(a + b*x + c*x^2))/((b^2 - 4* 
a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*Ellipti 
cE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)] 
, 1/2])/(Sqrt[2]*c^(1/4)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])))/Sqrt 
[c] + ((b^2 - 4*a*c)^(3/4)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 
 - 4*a*c])*Sqrt[(b^2 - 4*a*c + 4*c*(a + b*x + c*x^2))/((b^2 - 4*a*c)*(1 + 
(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*EllipticF[2*ArcTa 
n[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(4 
*Sqrt[2]*c^(3/4)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])))/(c^2*(b + 2* 
c*x))))/(b^2 - 4*a*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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 834

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S 
imp[1/q   Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q   Int[(1 - q*x^2)/Sqrt[a 
 + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1094

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

rule 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

Maple [F]

\[\int \frac {\left (e x +d \right )^{3}}{\left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/4}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate((e*x+d)^3/(c*x^2+b*x+a)^(5/4),x, algorithm="fricas")
 

Output:

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(c*x^2 + b*x + a)^(3/4) 
/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2), x)
 

Sympy [F]

\[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/4}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\left (a + b x + c x^{2}\right )^{\frac {5}{4}}}\, dx \] Input:

integrate((e*x+d)**3/(c*x**2+b*x+a)**(5/4),x)
 

Output:

Integral((d + e*x)**3/(a + b*x + c*x**2)**(5/4), x)
 

Maxima [F]

\[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/4}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate((e*x+d)^3/(c*x^2+b*x+a)^(5/4),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/4}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate((e*x+d)^3/(c*x^2+b*x+a)^(5/4),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/4}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{{\left (c\,x^2+b\,x+a\right )}^{5/4}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/4}} \, dx=\left (\int \frac {x^{3}}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b x +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} c \,x^{2}}d x \right ) e^{3}+3 \left (\int \frac {x^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b x +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} c \,x^{2}}d x \right ) d \,e^{2}+3 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b x +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} c \,x^{2}}d x \right ) d^{2} e +\left (\int \frac {1}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b x +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} c \,x^{2}}d x \right ) d^{3} \] Input:

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

Output:

int(x**3/((a + b*x + c*x**2)**(1/4)*a + (a + b*x + c*x**2)**(1/4)*b*x + (a 
 + b*x + c*x**2)**(1/4)*c*x**2),x)*e**3 + 3*int(x**2/((a + b*x + c*x**2)** 
(1/4)*a + (a + b*x + c*x**2)**(1/4)*b*x + (a + b*x + c*x**2)**(1/4)*c*x**2 
),x)*d*e**2 + 3*int(x/((a + b*x + c*x**2)**(1/4)*a + (a + b*x + c*x**2)**( 
1/4)*b*x + (a + b*x + c*x**2)**(1/4)*c*x**2),x)*d**2*e + int(1/((a + b*x + 
 c*x**2)**(1/4)*a + (a + b*x + c*x**2)**(1/4)*b*x + (a + b*x + c*x**2)**(1 
/4)*c*x**2),x)*d**3