\(\int (d+e x) (a+b x+c x^2)^{5/4} \, dx\) [2524]

Optimal result

Integrand size = 20, antiderivative size = 285 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/4} \, dx=-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac {5 \left (b^2-4 a c\right )^{9/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{336 \sqrt {2} c^{13/4} (b+2 c x)} \]

[Out]

-5/168*(-4*a*c+b^2)*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(1/4)/c^3+1/14*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(
5/4)/c^2+2/9*e*(c*x^2+b*x+a)^(9/4)/c+5/672*(-4*a*c+b^2)^(9/4)*(-b*e+2*c*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)))*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^(13/4)/(2*c*x+b)*2^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 626, 637, 226} \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/4} \, dx=\frac {5 \left (b^2-4 a c\right )^{9/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) (2 c d-b e) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{336 \sqrt {2} c^{13/4} (b+2 c x)}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{168 c^3}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c} \]

[In]

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

[Out]

(-5*(b^2 - 4*a*c)*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(168*c^3) + ((2*c*d - b*e)*(b + 2*c*x)*(a
+ b*x + c*x^2)^(5/4))/(14*c^2) + (2*e*(a + b*x + c*x^2)^(9/4))/(9*c) + (5*(b^2 - 4*a*c)^(9/4)*(2*c*d - b*e)*S
qrt[(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])/(336*Sqrt[2]*c^(13/4)*(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 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 654

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] + Dist[(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[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac {(2 c d-b e) \int \left (a+b x+c x^2\right )^{5/4} \, dx}{2 c} \\ & = \frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}-\frac {\left (5 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \sqrt [4]{a+b x+c x^2} \, dx}{56 c^2} \\ & = -\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{672 c^3} \\ & = -\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e) \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 )}{168 c^3 (b+2 c x)} \\ & = -\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac {5 \left (b^2-4 a c\right )^{9/4} (2 c d-b e) \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 )}{336 \sqrt {2} c^{13/4} (b+2 c x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.61 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/4} \, dx=\frac {2 e (a+x (b+c x))^{9/4}}{9 c}+\frac {(2 c d-b e) \left (24 c^2 (b+2 c x) (a+x (b+c x))^2-5 \left (b^2-4 a c\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )\right )\right )}{336 c^4 (a+x (b+c x))^{3/4}} \]

[In]

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

[Out]

(2*e*(a + x*(b + c*x))^(9/4))/(9*c) + ((2*c*d - b*e)*(24*c^2*(b + 2*c*x)*(a + x*(b + c*x))^2 - 5*(b^2 - 4*a*c)
*(2*c*(b + 2*c*x)*(a + x*(b + c*x)) - Sqrt[2]*(b^2 - 4*a*c)^(3/2)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)
*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2])))/(336*c^4*(a + x*(b + c*x))^(3/4))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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