3.42 \(\int \frac{a+b x^3+c x^6}{(d+e x^3)^{9/2}} \, dx\)

Optimal. Leaf size=389 \[ \frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (247 a e^2+26 b d e+16 c d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt{3}\right )}{1215 \sqrt [4]{3} d^4 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}+\frac{2 x \left (247 a e^2+26 b d e+16 c d^2\right )}{1215 d^4 e^2 \sqrt{d+e x^3}}+\frac{2 x \left (247 a e^2+26 b d e+16 c d^2\right )}{2835 d^3 e^2 \left (d+e x^3\right )^{3/2}}-\frac{2 x \left (-19 a e^2-2 b d e+23 c d^2\right )}{315 d^2 e^2 \left (d+e x^3\right )^{5/2}}+\frac{2 x \left (a e^2-b d e+c d^2\right )}{21 d e^2 \left (d+e x^3\right )^{7/2}} \]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*x)/(21*d*e^2*(d + e*x^3)^(7/2)) - (2*(23*c*d^2 - 2*b*d*e - 19*a*e^2)*x)/(315*d^2*e^
2*(d + e*x^3)^(5/2)) + (2*(16*c*d^2 + 26*b*d*e + 247*a*e^2)*x)/(2835*d^3*e^2*(d + e*x^3)^(3/2)) + (2*(16*c*d^2
+ 26*b*d*e + 247*a*e^2)*x)/(1215*d^4*e^2*Sqrt[d + e*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(16*c*d^2 + 26*b*d*e + 247*a
*e^2)*(d^(1/3) + e^(1/3)*x)*Sqrt[(d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*
x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*d^(1/3) + e^(1/3)*x)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)], -7 - 4*Sqrt[3
]])/(1215*3^(1/4)*d^4*e^(7/3)*Sqrt[(d^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)^2]*Sqrt
[d + e*x^3])

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Rubi [A]  time = 0.394819, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1409, 385, 199, 218} \[ \frac{2 x \left (247 a e^2+26 b d e+16 c d^2\right )}{1215 d^4 e^2 \sqrt{d+e x^3}}+\frac{2 x \left (247 a e^2+26 b d e+16 c d^2\right )}{2835 d^3 e^2 \left (d+e x^3\right )^{3/2}}-\frac{2 x \left (-19 a e^2-2 b d e+23 c d^2\right )}{315 d^2 e^2 \left (d+e x^3\right )^{5/2}}+\frac{2 x \left (a e^2-b d e+c d^2\right )}{21 d e^2 \left (d+e x^3\right )^{7/2}}+\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (247 a e^2+26 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{1215 \sqrt [4]{3} d^4 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3 + c*x^6)/(d + e*x^3)^(9/2),x]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*x)/(21*d*e^2*(d + e*x^3)^(7/2)) - (2*(23*c*d^2 - 2*b*d*e - 19*a*e^2)*x)/(315*d^2*e^
2*(d + e*x^3)^(5/2)) + (2*(16*c*d^2 + 26*b*d*e + 247*a*e^2)*x)/(2835*d^3*e^2*(d + e*x^3)^(3/2)) + (2*(16*c*d^2
+ 26*b*d*e + 247*a*e^2)*x)/(1215*d^4*e^2*Sqrt[d + e*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(16*c*d^2 + 26*b*d*e + 247*a
*e^2)*(d^(1/3) + e^(1/3)*x)*Sqrt[(d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*
x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*d^(1/3) + e^(1/3)*x)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)], -7 - 4*Sqrt[3
]])/(1215*3^(1/4)*d^4*e^(7/3)*Sqrt[(d^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)^2]*Sqrt
[d + e*x^3])

Rule 1409

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> -Simp[((c*d^2 - b*
d*e + a*e^2)*x*(d + e*x^n)^(q + 1))/(d*e^2*n*(q + 1)), x] + Dist[1/(n*(q + 1)*d*e^2), Int[(d + e*x^n)^(q + 1)*
Simp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, n}, x] &
& EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 218

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

Rubi steps

\begin{align*} \int \frac{a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx &=\frac{2 \left (c d^2-b d e+a e^2\right ) x}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac{2 \int \frac{\frac{1}{2} \left (2 c d^2-e (2 b d+19 a e)\right )-\frac{21}{2} c d e x^3}{\left (d+e x^3\right )^{7/2}} \, dx}{21 d e^2}\\ &=\frac{2 \left (c d^2-b d e+a e^2\right ) x}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac{2 \left (23 c d^2-2 b d e-19 a e^2\right ) x}{315 d^2 e^2 \left (d+e x^3\right )^{5/2}}+\frac{\left (16 c d^2+26 b d e+247 a e^2\right ) \int \frac{1}{\left (d+e x^3\right )^{5/2}} \, dx}{315 d^2 e^2}\\ &=\frac{2 \left (c d^2-b d e+a e^2\right ) x}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac{2 \left (23 c d^2-2 b d e-19 a e^2\right ) x}{315 d^2 e^2 \left (d+e x^3\right )^{5/2}}+\frac{2 \left (16 c d^2+26 b d e+247 a e^2\right ) x}{2835 d^3 e^2 \left (d+e x^3\right )^{3/2}}+\frac{\left (16 c d^2+26 b d e+247 a e^2\right ) \int \frac{1}{\left (d+e x^3\right )^{3/2}} \, dx}{405 d^3 e^2}\\ &=\frac{2 \left (c d^2-b d e+a e^2\right ) x}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac{2 \left (23 c d^2-2 b d e-19 a e^2\right ) x}{315 d^2 e^2 \left (d+e x^3\right )^{5/2}}+\frac{2 \left (16 c d^2+26 b d e+247 a e^2\right ) x}{2835 d^3 e^2 \left (d+e x^3\right )^{3/2}}+\frac{2 \left (16 c d^2+26 b d e+247 a e^2\right ) x}{1215 d^4 e^2 \sqrt{d+e x^3}}+\frac{\left (16 c d^2+26 b d e+247 a e^2\right ) \int \frac{1}{\sqrt{d+e x^3}} \, dx}{1215 d^4 e^2}\\ &=\frac{2 \left (c d^2-b d e+a e^2\right ) x}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac{2 \left (23 c d^2-2 b d e-19 a e^2\right ) x}{315 d^2 e^2 \left (d+e x^3\right )^{5/2}}+\frac{2 \left (16 c d^2+26 b d e+247 a e^2\right ) x}{2835 d^3 e^2 \left (d+e x^3\right )^{3/2}}+\frac{2 \left (16 c d^2+26 b d e+247 a e^2\right ) x}{1215 d^4 e^2 \sqrt{d+e x^3}}+\frac{2 \sqrt{2+\sqrt{3}} \left (16 c d^2+26 b d e+247 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right )|-7-4 \sqrt{3}\right )}{1215 \sqrt [4]{3} d^4 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}\\ \end{align*}

Mathematica [C]  time = 0.243114, size = 200, normalized size = 0.51 \[ \frac{2 x \left (e \left (a e \left (7182 d^2 e x^3+3388 d^3+5928 d e^2 x^6+1729 e^3 x^9\right )+b d \left (756 d^2 e x^3-91 d^3+624 d e^2 x^6+182 e^3 x^9\right )\right )+c d^2 \left (-189 d^2 e x^3-56 d^3+384 d e^2 x^6+112 e^3 x^9\right )\right )+7 x \sqrt{\frac{e x^3}{d}+1} \left (d+e x^3\right )^3 \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{e x^3}{d}\right ) \left (13 e (19 a e+2 b d)+16 c d^2\right )}{8505 d^4 e^2 \left (d+e x^3\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(9/2),x]

[Out]

(2*x*(c*d^2*(-56*d^3 - 189*d^2*e*x^3 + 384*d*e^2*x^6 + 112*e^3*x^9) + e*(b*d*(-91*d^3 + 756*d^2*e*x^3 + 624*d*
e^2*x^6 + 182*e^3*x^9) + a*e*(3388*d^3 + 7182*d^2*e*x^3 + 5928*d*e^2*x^6 + 1729*e^3*x^9))) + 7*(16*c*d^2 + 13*
e*(2*b*d + 19*a*e))*x*(d + e*x^3)^3*Sqrt[1 + (e*x^3)/d]*Hypergeometric2F1[1/3, 1/2, 4/3, -((e*x^3)/d)])/(8505*
d^4*e^2*(d + e*x^3)^(7/2))

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Maple [B]  time = 0.048, size = 1182, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^6+b*x^3+a)/(e*x^3+d)^(9/2),x)

[Out]

c*(2/21*d*x/e^6*(e*x^3+d)^(1/2)/(x^3+d/e)^4-46/315*x/e^5*(e*x^3+d)^(1/2)/(x^3+d/e)^3+32/2835/d*x/e^4*(e*x^3+d)
^(1/2)/(x^3+d/e)^2+32/1215/e^2/d^2*x/((x^3+d/e)*e)^(1/2)-32/3645*I/e^3/d^2*3^(1/2)*(-d*e^2)^(1/3)*(I*(x+1/2/e*
(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)*((x-1/e*(-d*e^2)^(1/3))/(-3/2/e
*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1/2)*(-I*(x+1/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1
/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)/(e*x^3+d)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^
(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2),(I*3^(1/2)/e*(-d*e^2)^(1/3)/(-3/2/e*(-d*e^2)^(1/3)+1/2
*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1/2)))+b*(-2/21*x/e^5*(e*x^3+d)^(1/2)/(x^3+d/e)^4+4/315/d*x/e^4*(e*x^3+d)^(1/2)
/(x^3+d/e)^3+52/2835/d^2*x/e^3*(e*x^3+d)^(1/2)/(x^3+d/e)^2+52/1215/e/d^3*x/((x^3+d/e)*e)^(1/2)-52/3645*I/e^2/d
^3*3^(1/2)*(-d*e^2)^(1/3)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))
^(1/2)*((x-1/e*(-d*e^2)^(1/3))/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1/2)*(-I*(x+1/2/e*(-d*
e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)/(e*x^3+d)^(1/2)*EllipticF(1/3*3^(1/
2)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2),(I*3^(1/2)/e*(-d
*e^2)^(1/3)/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1/2)))+a*(2/21/d*x/e^4*(e*x^3+d)^(1/2)/(x
^3+d/e)^4+38/315/d^2*x/e^3*(e*x^3+d)^(1/2)/(x^3+d/e)^3+494/2835/d^3*x/e^2*(e*x^3+d)^(1/2)/(x^3+d/e)^2+494/1215
/d^4*x/((x^3+d/e)*e)^(1/2)-494/3645*I/d^4*3^(1/2)/e*(-d*e^2)^(1/3)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e*
(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)*((x-1/e*(-d*e^2)^(1/3))/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e
*(-d*e^2)^(1/3)))^(1/2)*(-I*(x+1/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^
(1/2)/(e*x^3+d)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)
*e/(-d*e^2)^(1/3))^(1/2),(I*3^(1/2)/e*(-d*e^2)^(1/3)/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(
1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{6} + b x^{3} + a\right )} \sqrt{e x^{3} + d}}{e^{5} x^{15} + 5 \, d e^{4} x^{12} + 10 \, d^{2} e^{3} x^{9} + 10 \, d^{3} e^{2} x^{6} + 5 \, d^{4} e x^{3} + d^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(9/2),x, algorithm="fricas")

[Out]

integral((c*x^6 + b*x^3 + a)*sqrt(e*x^3 + d)/(e^5*x^15 + 5*d*e^4*x^12 + 10*d^2*e^3*x^9 + 10*d^3*e^2*x^6 + 5*d^
4*e*x^3 + d^5), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(9/2), x)