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

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

[Out]

2/21*(a*e^2-b*d*e+c*d^2)*x/d/e^2/(e*x^3+d)^(7/2)-2/315*(-19*a*e^2-2*b*d*e+23*c*d^2)*x/d^2/e^2/(e*x^3+d)^(5/2)+
2/2835*(247*a*e^2+26*b*d*e+16*c*d^2)*x/d^3/e^2/(e*x^3+d)^(3/2)+2/1215*(247*a*e^2+26*b*d*e+16*c*d^2)*x/d^4/e^2/
(e*x^3+d)^(1/2)+2/3645*(247*a*e^2+26*b*d*e+16*c*d^2)*(d^(1/3)+e^(1/3)*x)*EllipticF((e^(1/3)*x+d^(1/3)*(1-3^(1/
2)))/(e^(1/3)*x+d^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)+1/2*2^(1/2))*((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2
/3)*x^2)/(e^(1/3)*x+d^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(3/4)/d^4/e^(7/3)/(e*x^3+d)^(1/2)/(d^(1/3)*(d^(1/3)+e^(1/3
)*x)/(e^(1/3)*x+d^(1/3)*(1+3^(1/2)))^2)^(1/2)

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Rubi [A]  time = 0.39, antiderivative size = 389, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [C]  time = 0.24, size = 200, normalized size = 0.51 \[ \frac {7 x \sqrt {\frac {e x^3}{d}+1} \left (d+e x^3\right )^3 \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-\frac {e x^3}{d}\right ) \left (13 e (19 a e+2 b d)+16 c d^2\right )+2 x \left (e \left (a e \left (3388 d^3+7182 d^2 e x^3+5928 d e^2 x^6+1729 e^3 x^9\right )+b d \left (-91 d^3+756 d^2 e x^3+624 d e^2 x^6+182 e^3 x^9\right )\right )+c d^2 \left (-56 d^3-189 d^2 e x^3+384 d e^2 x^6+112 e^3 x^9\right )\right )}{8505 d^4 e^2 \left (d+e x^3\right )^{7/2}} \]

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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fricas [F]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{6} + b x^{3} + a\right )} \sqrt {e x^{3} + d}}{e^{5} x^{15} + 5 \, d e^{4} x^{12} + 10 \, d^{2} e^{3} x^{9} + 10 \, d^{3} e^{2} x^{6} + 5 \, d^{4} e x^{3} + d^{5}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac {9}{2}}}\,{d x} \]

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)

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maple [B]  time = 0.06, size = 1182, normalized size = 3.04 \[ \text {result too large to display} \]

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*(-
d*e^2)^(1/3)/e-1/2*I*3^(1/2)*(-d*e^2)^(1/3)/e)*3^(1/2)/(-d*e^2)^(1/3)*e)^(1/2)*((x-(-d*e^2)^(1/3)/e)/(-3/2*(-d
*e^2)^(1/3)/e+1/2*I*3^(1/2)*(-d*e^2)^(1/3)/e))^(1/2)*(-I*(x+1/2*(-d*e^2)^(1/3)/e+1/2*I*3^(1/2)*(-d*e^2)^(1/3)/
e)*3^(1/2)/(-d*e^2)^(1/3)*e)^(1/2)/(e*x^3+d)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2*(-d*e^2)^(1/3)/e-1/2*I*3^(1
/2)*(-d*e^2)^(1/3)/e)*3^(1/2)/(-d*e^2)^(1/3)*e)^(1/2),(I*3^(1/2)*(-d*e^2)^(1/3)/(-3/2*(-d*e^2)^(1/3)/e+1/2*I*3
^(1/2)*(-d*e^2)^(1/3)/e)/e)^(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*(-d*e^2)^(1/3)/e-1/2*I*3^(1/2)*(-d*e^2)^(1/3)/e)*3^(1/2)/(-d*e^2)^(1/3)*e)^(
1/2)*((x-(-d*e^2)^(1/3)/e)/(-3/2*(-d*e^2)^(1/3)/e+1/2*I*3^(1/2)*(-d*e^2)^(1/3)/e))^(1/2)*(-I*(x+1/2*(-d*e^2)^(
1/3)/e+1/2*I*3^(1/2)*(-d*e^2)^(1/3)/e)*3^(1/2)/(-d*e^2)^(1/3)*e)^(1/2)/(e*x^3+d)^(1/2)*EllipticF(1/3*3^(1/2)*(
I*(x+1/2*(-d*e^2)^(1/3)/e-1/2*I*3^(1/2)*(-d*e^2)^(1/3)/e)*3^(1/2)/(-d*e^2)^(1/3)*e)^(1/2),(I*3^(1/2)*(-d*e^2)^
(1/3)/(-3/2*(-d*e^2)^(1/3)/e+1/2*I*3^(1/2)*(-d*e^2)^(1/3)/e)/e)^(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)*(-d*e^2)^(1/3)/e*(I*(x+1/2*(-d*e^2)^(1/3)/e-1/2*I*3^(1/2)*(-d*e^
2)^(1/3)/e)*3^(1/2)/(-d*e^2)^(1/3)*e)^(1/2)*((x-(-d*e^2)^(1/3)/e)/(-3/2*(-d*e^2)^(1/3)/e+1/2*I*3^(1/2)*(-d*e^2
)^(1/3)/e))^(1/2)*(-I*(x+1/2*(-d*e^2)^(1/3)/e+1/2*I*3^(1/2)*(-d*e^2)^(1/3)/e)*3^(1/2)/(-d*e^2)^(1/3)*e)^(1/2)/
(e*x^3+d)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2*(-d*e^2)^(1/3)/e-1/2*I*3^(1/2)*(-d*e^2)^(1/3)/e)*3^(1/2)/(-d*e
^2)^(1/3)*e)^(1/2),(I*3^(1/2)*(-d*e^2)^(1/3)/(-3/2*(-d*e^2)^(1/3)/e+1/2*I*3^(1/2)*(-d*e^2)^(1/3)/e)/e)^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac {9}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {c\,x^6+b\,x^3+a}{{\left (e\,x^3+d\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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