\(\int (d+e x)^3 (a+b x+c x^2)^{3/4} \, dx\) [2516]

Optimal result

Integrand size = 22, antiderivative size = 703 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/4} \, dx=\frac {(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac {e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac {\sqrt {b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{80 c^{9/2} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}+\frac {\left (b^2-4 a c\right )^{7/4} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{80 \sqrt {2} c^{19/4} (b+2 c x)}-\frac {\left (b^2-4 a c\right )^{7/4} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{160 \sqrt {2} c^{19/4} (b+2 c x)} \]

[Out]

1/120*(-b*e+2*c*d)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(2*a*e+3*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(3/4)/c^4+2/11*e*(e*x+d)
^2*(c*x^2+b*x+a)^(7/4)/c+1/462*e*(312*c^2*d^2+55*b^2*e^2-2*c*e*(24*a*e+121*b*d)+70*c*e*(-b*e+2*c*d)*x)*(c*x^2+
b*x+a)^(7/4)/c^3-1/80*(-b*e+2*c*d)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(2*a*e+3*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/4)*(-
4*a*c+b^2)^(1/2)/c^(9/2)/(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))+1/160*(-4*a*c+b^2)^(7/4)*(-b*e+2
*c*d)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(2*a*e+3*b*d))*(cos(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2
)^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))*EllipticE(sin(2*arcta
n(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4))),1/2*2^(1/2))*(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a
*c+b^2)^(1/2))*((2*c*x+b)^2/(-4*a*c+b^2)/(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))^2)^(1/2)/c^(19/4
)/(2*c*x+b)*2^(1/2)-1/320*(-4*a*c+b^2)^(7/4)*(-b*e+2*c*d)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(2*a*e+3*b*d))*(cos(2*ar
ctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4
)*2^(1/2)/(-4*a*c+b^2)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))
,1/2*2^(1/2))*(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))*((2*c*x+b)^2/(-4*a*c+b^2)/(1+2*c^(1/2)*(c*x
^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))^2)^(1/2)/c^(19/4)/(2*c*x+b)*2^(1/2)

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {756, 793, 626, 637, 311, 226, 1210} \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/4} \, dx=-\frac {\left (b^2-4 a c\right )^{7/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{160 \sqrt {2} c^{19/4} (b+2 c x)}+\frac {\left (b^2-4 a c\right )^{7/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{80 \sqrt {2} c^{19/4} (b+2 c x)}-\frac {\sqrt {b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{80 c^{9/2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{120 c^4}+\frac {e \left (a+b x+c x^2\right )^{7/4} \left (-2 c e (24 a e+121 b d)+55 b^2 e^2+70 c e x (2 c d-b e)+312 c^2 d^2\right )}{462 c^3}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c} \]

[In]

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

[Out]

((2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4))/(120*c^4)
+ (2*e*(d + e*x)^2*(a + b*x + c*x^2)^(7/4))/(11*c) + (e*(312*c^2*d^2 + 55*b^2*e^2 - 2*c*e*(121*b*d + 24*a*e)
+ 70*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/4))/(462*c^3) - (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(12*c^2*d^2 +
5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(80*c^(9/2)*(1 + (2*Sqrt[c]*Sqrt[a + b
*x + c*x^2])/Sqrt[b^2 - 4*a*c])) + ((b^2 - 4*a*c)^(7/4)*(2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d +
2*a*e))*Sqrt[(b + 2*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)]*(1 +
(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/
4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(80*Sqrt[2]*c^(19/4)*(b + 2*c*x)) - ((b^2 - 4*a*c)^(7/4)*(2*c*d - b*e)*(12*c^2
*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*Sqrt[(b + 2*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)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt
[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(160*Sqrt[2]*c^(19/4)*(b + 2*c*x))

Rule 226

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 311

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

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 637

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[d*(Sqrt[(b + 2*c*x)
^2]/(b + 2*c*x)), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 756

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 793

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] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 1210

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
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps \begin{align*} \text {integral}& = \frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac {2 \int (d+e x) \left (\frac {1}{4} \left (22 c d^2-7 b d e-8 a e^2\right )+\frac {15}{4} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4} \, dx}{11 c} \\ & = \frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac {e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}+\frac {\left ((2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/4} \, dx}{24 c^3} \\ & = \frac {(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac {e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right )\right ) \int \frac {1}{\sqrt [4]{a+b x+c x^2}} \, dx}{160 c^4} \\ & = \frac {(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac {e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{40 c^4 (b+2 c x)} \\ & = \frac {(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac {e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac {\left (\left (b^2-4 a c\right )^{3/2} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{80 c^{9/2} (b+2 c x)}+\frac {\left (\left (b^2-4 a c\right )^{3/2} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {1-\frac {2 \sqrt {c} x^2}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{80 c^{9/2} (b+2 c x)} \\ & = \frac {(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac {2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac {e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac {\sqrt {b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{80 c^{9/2} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}+\frac {\left (b^2-4 a c\right )^{7/4} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{80 \sqrt {2} c^{19/4} (b+2 c x)}-\frac {\left (b^2-4 a c\right )^{7/4} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{160 \sqrt {2} c^{19/4} (b+2 c x)} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.45 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.33 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/4} \, dx=\frac {13440 c^4 e (d+e x)^2 (a+x (b+c x))^2+160 c^2 e (a+x (b+c x))^2 \left (55 b^2 e^2+4 c^2 d (78 d+35 e x)-2 c e (121 b d+24 a e+35 b e x)\right )+77 (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (8 c (a+x (b+c x))-3 \sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{73920 c^5 \sqrt [4]{a+x (b+c x)}} \]

[In]

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

[Out]

(13440*c^4*e*(d + e*x)^2*(a + x*(b + c*x))^2 + 160*c^2*e*(a + x*(b + c*x))^2*(55*b^2*e^2 + 4*c^2*d*(78*d + 35*
e*x) - 2*c*e*(121*b*d + 24*a*e + 35*b*e*x)) + 77*(2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e)
)*(b + 2*c*x)*(8*c*(a + x*(b + c*x)) - 3*Sqrt[2]*(b^2 - 4*a*c)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hy
pergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(73920*c^5*(a + x*(b + c*x))^(1/4))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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