\(\int \frac {d+e x}{(a+b x+c x^2)^{4/3}} \, dx\) [692]

Optimal result

Integrand size = 20, antiderivative size = 117 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{4/3}} \, dx=-\frac {3 e}{2 c \sqrt [3]{a+b x+c x^2}}+\frac {2^{2/3} (2 c d-b e) (b+2 c x) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c^2 \left (a+b x+c x^2\right )^{4/3}} \] Output:

-3/2*e/c/(c*x^2+b*x+a)^(1/3)+2^(2/3)*(-b*e+2*c*d)*(2*c*x+b)*(-c*(c*x^2+b*x 
+a)/(-4*a*c+b^2))^(4/3)*hypergeom([1/2, 4/3],[3/2],(2*c*x+b)^2/(-4*a*c+b^2 
))/c^2/(c*x^2+b*x+a)^(4/3)
 

Mathematica [A] (verified)

Time = 11.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.43 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{4/3}} \, dx=-\frac {3 \left (8 c (-2 a e+2 c d x+b (d-e x))+2^{2/3} (-2 c d+b e) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt [3]{\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {-b+\sqrt {b^2-4 a c}-2 c x}{2 \sqrt {b^2-4 a c}}\right )\right )}{8 c \left (b^2-4 a c\right ) \sqrt [3]{a+x (b+c x)}} \] Input:

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

Output:

(-3*(8*c*(-2*a*e + 2*c*d*x + b*(d - e*x)) + 2^(2/3)*(-2*c*d + b*e)*(b - Sq 
rt[b^2 - 4*a*c] + 2*c*x)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c 
])^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x) 
/(2*Sqrt[b^2 - 4*a*c])]))/(8*c*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/3))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1083\) vs. \(2(117)=234\).

Time = 0.83 (sec) , antiderivative size = 1083, normalized size of antiderivative = 9.26, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1159, 1095, 832, 759, 2416}

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.

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

\(\Big \downarrow \) 1159

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

\(\Big \downarrow \) 1095

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

\(\Big \downarrow \) 832

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

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 2416

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

Input:

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

Output:

(-3*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^(1/3 
)) + (3*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2]*(((2^(1/3)*Sqrt[b^2 - 4*a*c + 4* 
c*(a + b*x + c*x^2)])/(c^(1/3)*((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3 
)*c^(1/3)*(a + b*x + c*x^2)^(1/3))) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*(b^2 - 4* 
a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3)) 
*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x 
+ c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/3))/((1 + Sqrt[3]) 
*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*Ellipti 
cE[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + 
c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b* 
x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2^(2/3)*c^(1/3)*Sqrt[((b^2 - 4*a*c)^ 
(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3)))/((1 
 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3)) 
^2]*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)]))/(2^(2/3)*c^(1/3)) - ((1 - 
Sqrt[3])*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2 
/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c 
^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a 
+ b*x + c*x^2)^(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3) 
*(a + b*x + c*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c) 
^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 -...
 

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 832

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && PosQ[a]
 

rule 1095

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[3*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(3*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^3], x], x, (a + b*x + c*x^2)^(1/3)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[3*p]
 

rule 1159

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

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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