3.2528 \(\int \frac{(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=637 \[ \frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{40 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (40 a e+147 b d)+77 b^2 e^2+66 c e x (2 c d-b e)+360 c^2 d^2\right )}{210 c^3}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right )}{20 c^{7/2} \sqrt{b^2-4 a c} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c} \]

[Out]

(2*e*(d + e*x)^2*(a + b*x + c*x^2)^(3/4))/(7*c) + (e*(360*c^2*d^2 + 77*b^2*e^2 - 2*c*e*(147*b*d + 40*a*e) + 66
*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/4))/(210*c^3) + ((2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*
b*d + 6*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(20*c^(7/2)*Sqrt[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)^(3/4)*(2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*b*d + 6
*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])/(20*Sqrt[2]*c^(15/4)*(b + 2*c*x)) + ((b^2 - 4*a*c)^(3/4)*(2*c*d - b*e)*(20*c^2*d
^2 + 11*b^2*e^2 - 4*c*e*(5*b*d + 6*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])/(40*Sqrt[2]*c^(15/4)*(b + 2*c*x))

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Rubi [A]  time = 0.727933, antiderivative size = 637, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {742, 779, 623, 305, 220, 1196} \[ \frac{e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (40 a e+147 b d)+77 b^2 e^2+66 c e x (2 c d-b e)+360 c^2 d^2\right )}{210 c^3}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right )}{20 c^{7/2} \sqrt{b^2-4 a c} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{40 \sqrt{2} c^{15/4} (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(d + e*x)^2*(a + b*x + c*x^2)^(3/4))/(7*c) + (e*(360*c^2*d^2 + 77*b^2*e^2 - 2*c*e*(147*b*d + 40*a*e) + 66
*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/4))/(210*c^3) + ((2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*
b*d + 6*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(20*c^(7/2)*Sqrt[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)^(3/4)*(2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*b*d + 6
*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])/(20*Sqrt[2]*c^(15/4)*(b + 2*c*x)) + ((b^2 - 4*a*c)^(3/4)*(2*c*d - b*e)*(20*c^2*d
^2 + 11*b^2*e^2 - 4*c*e*(5*b*d + 6*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])/(40*Sqrt[2]*c^(15/4)*(b + 2*c*x))

Rule 742

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 779

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 623

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 305

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 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

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)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin{align*} \int \frac{(d+e x)^3}{\sqrt [4]{a+b x+c x^2}} \, dx &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}+\frac{2 \int \frac{(d+e x) \left (\frac{1}{4} \left (14 c d^2-3 b d e-8 a e^2\right )+\frac{11}{4} e (2 c d-b e) x\right )}{\sqrt [4]{a+b x+c x^2}} \, dx}{7 c}\\ &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}+\frac{e \left (360 c^2 d^2+77 b^2 e^2-2 c e (147 b d+40 a e)+66 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4}}{210 c^3}+\frac{\left ((2 c d-b e) \left (20 c^2 d^2+11 b^2 e^2-4 c e (5 b d+6 a e)\right )\right ) \int \frac{1}{\sqrt [4]{a+b x+c x^2}} \, dx}{40 c^3}\\ &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}+\frac{e \left (360 c^2 d^2+77 b^2 e^2-2 c e (147 b d+40 a e)+66 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4}}{210 c^3}+\frac{\left ((2 c d-b e) \left (20 c^2 d^2+11 b^2 e^2-4 c e (5 b d+6 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{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 )}{10 c^3 (b+2 c x)}\\ &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}+\frac{e \left (360 c^2 d^2+77 b^2 e^2-2 c e (147 b d+40 a e)+66 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4}}{210 c^3}+\frac{\left (\sqrt{b^2-4 a c} (2 c d-b e) \left (20 c^2 d^2+11 b^2 e^2-4 c e (5 b d+6 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{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 )}{20 c^{7/2} (b+2 c x)}-\frac{\left (\sqrt{b^2-4 a c} (2 c d-b e) \left (20 c^2 d^2+11 b^2 e^2-4 c e (5 b d+6 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{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 )}{20 c^{7/2} (b+2 c x)}\\ &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{3/4}}{7 c}+\frac{e \left (360 c^2 d^2+77 b^2 e^2-2 c e (147 b d+40 a e)+66 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4}}{210 c^3}+\frac{(2 c d-b e) \left (20 c^2 d^2+11 b^2 e^2-4 c e (5 b d+6 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{20 c^{7/2} \sqrt{b^2-4 a c} \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 )^{3/4} (2 c d-b e) \left (20 c^2 d^2+11 b^2 e^2-4 c e (5 b d+6 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 )}{20 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \left (20 c^2 d^2+11 b^2 e^2-4 c e (5 b d+6 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 )}{40 \sqrt{2} c^{15/4} (b+2 c x)}\\ \end{align*}

Mathematica [C]  time = 0.306702, size = 211, normalized size = 0.33 \[ \frac{\frac{21 (b+2 c x) \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} (2 c d-b e) \left (-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{8 \sqrt{2} c^3}+\frac{e (a+x (b+c x)) \left (-2 c e (40 a e+147 b d+33 b e x)+77 b^2 e^2+12 c^2 d (30 d+11 e x)\right )}{2 c^2}+30 e (d+e x)^2 (a+x (b+c x))}{105 c \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(30*e*(d + e*x)^2*(a + x*(b + c*x)) + (e*(a + x*(b + c*x))*(77*b^2*e^2 + 12*c^2*d*(30*d + 11*e*x) - 2*c*e*(147
*b*d + 40*a*e + 33*b*e*x)))/(2*c^2) + (21*(2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*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)])/(8*Sqrt[2]*c^3))/(105*c*(a + x*(b + c*x))^(1/4))

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Maple [F]  time = 0.994, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{3}{\frac{1}{\sqrt [4]{c{x}^{2}+bx+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+b*x+a)^(1/4),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{3}}{\sqrt [4]{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+b*x+a)^(1/4),x, algorithm="giac")

[Out]

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