Integrand size = 20, antiderivative size = 120 \[ \int \frac {d+e x}{\sqrt [4]{a+b x+c x^2}} \, dx=\frac {2 e \left (a+b x+c x^2\right )^{3/4}}{3 c}+\frac {\sqrt {b^2-4 a c} (2 c d-b e) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )}{\sqrt {2} c^2 \sqrt [4]{a+b x+c x^2}} \] Output:
2/3*e*(c*x^2+b*x+a)^(3/4)/c+1/2*(-4*a*c+b^2)^(1/2)*(-b*e+2*c*d)*(-c*(c*x^2 +b*x+a)/(-4*a*c+b^2))^(1/4)*EllipticE(sin(1/2*arcsin((2*c*x+b)/(-4*a*c+b^2 )^(1/2))),2^(1/2))*2^(1/2)/c^2/(c*x^2+b*x+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.93 \[ \int \frac {d+e x}{\sqrt [4]{a+b x+c x^2}} \, dx=\frac {8 c e (a+x (b+c x))+3 \sqrt {2} (2 c d-b e) (b+2 c x) \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{12 c^2 \sqrt [4]{a+x (b+c x)}} \] Input:
Integrate[(d + e*x)/(a + b*x + c*x^2)^(1/4),x]
Output:
(8*c*e*(a + x*(b + c*x)) + 3*Sqrt[2]*(2*c*d - b*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^2*(a + x*(b + c*x))^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(571\) vs. \(2(120)=240\).
Time = 0.50 (sec) , antiderivative size = 571, normalized size of antiderivative = 4.76, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1160, 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.
\(\displaystyle \int \frac {d+e x}{\sqrt [4]{a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {(2 c d-b e) \int \frac {1}{\sqrt [4]{c x^2+b x+a}}dx}{2 c}+\frac {2 e \left (a+b x+c x^2\right )^{3/4}}{3 c}\) |
\(\Big \downarrow \) 1094 |
\(\displaystyle \frac {2 \sqrt {(b+2 c x)^2} (2 c d-b e) \int \frac {\sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{c (b+2 c x)}+\frac {2 e \left (a+b x+c x^2\right )^{3/4}}{3 c}\) |
\(\Big \downarrow \) 834 |
\(\displaystyle \frac {2 \sqrt {(b+2 c x)^2} (2 c d-b e) \left (\frac {\sqrt {b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{c (b+2 c x)}+\frac {2 e \left (a+b x+c x^2\right )^{3/4}}{3 c}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {2 \sqrt {(b+2 c x)^2} (2 c d-b e) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{c (b+2 c x)}+\frac {2 e \left (a+b x+c x^2\right )^{3/4}}{3 c}\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle \frac {2 \sqrt {(b+2 c x)^2} (2 c d-b e) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \left (\frac {\sqrt [4]{b^2-4 a c} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt [4]{a+b x+c x^2} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}\right )}{2 \sqrt {c}}\right )}{c (b+2 c x)}+\frac {2 e \left (a+b x+c x^2\right )^{3/4}}{3 c}\) |
Input:
Int[(d + e*x)/(a + b*x + c*x^2)^(1/4),x]
Output:
(2*e*(a + b*x + c*x^2)^(3/4))/(3*c) + (2*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2] *(-1/2*(Sqrt[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 )]*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])/(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)]*Ellipti cF[2*ArcTan[(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 *(b + 2*c*x))
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]
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]
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]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
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]
\[\int \frac {e x +d}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x\]
Input:
int((e*x+d)/(c*x^2+b*x+a)^(1/4),x)
Output:
int((e*x+d)/(c*x^2+b*x+a)^(1/4),x)
\[ \int \frac {d+e x}{\sqrt [4]{a+b x+c x^2}} \, dx=\int { \frac {e x + d}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((e*x+d)/(c*x^2+b*x+a)^(1/4),x, algorithm="fricas")
Output:
integral((e*x + d)/(c*x^2 + b*x + a)^(1/4), x)
\[ \int \frac {d+e x}{\sqrt [4]{a+b x+c x^2}} \, dx=\int \frac {d + e x}{\sqrt [4]{a + b x + c x^{2}}}\, dx \] Input:
integrate((e*x+d)/(c*x**2+b*x+a)**(1/4),x)
Output:
Integral((d + e*x)/(a + b*x + c*x**2)**(1/4), x)
\[ \int \frac {d+e x}{\sqrt [4]{a+b x+c x^2}} \, dx=\int { \frac {e x + d}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((e*x+d)/(c*x^2+b*x+a)^(1/4),x, algorithm="maxima")
Output:
integrate((e*x + d)/(c*x^2 + b*x + a)^(1/4), x)
\[ \int \frac {d+e x}{\sqrt [4]{a+b x+c x^2}} \, dx=\int { \frac {e x + d}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((e*x+d)/(c*x^2+b*x+a)^(1/4),x, algorithm="giac")
Output:
integrate((e*x + d)/(c*x^2 + b*x + a)^(1/4), x)
Timed out. \[ \int \frac {d+e x}{\sqrt [4]{a+b x+c x^2}} \, dx=\int \frac {d+e\,x}{{\left (c\,x^2+b\,x+a\right )}^{1/4}} \,d x \] Input:
int((d + e*x)/(a + b*x + c*x^2)^(1/4),x)
Output:
int((d + e*x)/(a + b*x + c*x^2)^(1/4), x)
\[ \int \frac {d+e x}{\sqrt [4]{a+b x+c x^2}} \, dx=\left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) e +\left (\int \frac {1}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) d \] Input:
int((e*x+d)/(c*x^2+b*x+a)^(1/4),x)
Output:
int(x/(a + b*x + c*x**2)**(1/4),x)*e + int(1/(a + b*x + c*x**2)**(1/4),x)* d