3.5.61 \(\int (a+b x^3)^{3/2} (c+d x+e x^2+f x^3+g x^4) \, dx\) [461]

Optimal. Leaf size=694 \[ \frac {2 a^2 e \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {54 a^2 (19 b d-4 a g) \sqrt {a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 b d-4 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1616615 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

2/692835*(b*x^3+a)^(3/2)*(36465*g*x^5+40755*f*x^4+46189*e*x^3+53295*d*x^2+62985*c*x)+2/15*a^2*e*(b*x^3+a)^(1/2
)/b+54/935*a^2*f*x*(b*x^3+a)^(1/2)/b+54/1729*a^2*g*x^2*(b*x^3+a)^(1/2)/b+2/4849845*a*(176715*g*x^5+233415*f*x^
4+323323*e*x^3+479655*d*x^2+793611*c*x)*(b*x^3+a)^(1/2)+54/1729*a^2*(-4*a*g+19*b*d)*(b*x^3+a)^(1/2)/b^(5/3)/(b
^(1/3)*x+a^(1/3)*(1+3^(1/2)))-27/1729*3^(1/4)*a^(7/3)*(-4*a*g+19*b*d)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x
+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*((a^(2/3)-a^(1/
3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)/b^(5/3)/(b*x^3+a)^(1/2)/(a^(1/3)*(a^(1/3)+b
^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)+18/1616615*3^(3/4)*a^2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/
3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1729*b^(1/3)*(-2*a*f+17*b*c)-935*a^(
1/3)*(-4*a*g+19*b*d)*(1-3^(1/2)))*(1/2*6^(1/2)+1/2*2^(1/2))*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*
x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)/b^(5/3)/(b*x^3+a)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+3^(
1/2)))^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.61, antiderivative size = 694, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {1867, 1902, 1900, 267, 1892, 224, 1891} \begin {gather*} -\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b d-4 a g) E\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 \sqrt {a+b x^3} (19 b d-4 a g)}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 a^2 e \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right )}{1616615 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*e*Sqrt[a + b*x^3])/(15*b) + (54*a^2*f*x*Sqrt[a + b*x^3])/(935*b) + (54*a^2*g*x^2*Sqrt[a + b*x^3])/(1729
*b) + (54*a^2*(19*b*d - 4*a*g)*Sqrt[a + b*x^3])/(1729*b^(5/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) + (2*(a + b
*x^3)^(3/2)*(62985*c*x + 53295*d*x^2 + 46189*e*x^3 + 40755*f*x^4 + 36465*g*x^5))/692835 + (2*a*Sqrt[a + b*x^3]
*(793611*c*x + 479655*d*x^2 + 323323*e*x^3 + 233415*f*x^4 + 176715*g*x^5))/4849845 - (27*3^(1/4)*Sqrt[2 - Sqrt
[3]]*a^(7/3)*(19*b*d - 4*a*g)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqr
t[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^
(1/3)*x)], -7 - 4*Sqrt[3]])/(1729*b^(5/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3
)*x)^2]*Sqrt[a + b*x^3]) + (18*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*(1729*b^(1/3)*(17*b*c - 2*a*f) - 935*(1 - Sqrt[3]
)*a^(1/3)*(19*b*d - 4*a*g))*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[
3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1
/3)*x)], -7 - 4*Sqrt[3]])/(1616615*b^(5/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/
3)*x)^2]*Sqrt[a + b*x^3])

Rule 224

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]*Sq
rt[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 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 1867

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a + b*x^n)^p*Sum[Co
eff[Pq, x, i]*(x^(i + 1)/(n*p + i + 1)), {i, 0, q}], x] + Dist[a*n*p, Int[(a + b*x^n)^(p - 1)*Sum[Coeff[Pq, x,
i]*(x^i/(n*p + i + 1)), {i, 0, q}], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[(n - 1)/2, 0] && GtQ[
p, 0]

Rule 1891

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[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
] - Simp[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] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rule 1892

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a,
3]]}, Dist[(c*r - (1 - Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 - Sqrt[3])*s + r*x)
/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]

Rule 1900

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[Coeff[Pq, x, n - 1], Int[x^(n - 1)*(a + b*x^n)^p, x
], x] + Int[ExpandToSum[Pq - Coeff[Pq, x, n - 1]*x^(n - 1), x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, p}, x] && Pol
yQ[Pq, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1

Rule 1902

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, D
ist[1/(b*(q + n*p + 1)), Int[ExpandToSum[b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a +
b*x^n)^p, x], x] + Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), x]] /; NeQ[q + n*p + 1, 0]
&& q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IG
tQ[n, 0]

Rubi steps

\begin {align*} \int \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx &=\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {1}{2} (9 a) \int \sqrt {a+b x^3} \left (\frac {2 c}{11}+\frac {2 d x}{13}+\frac {2 e x^2}{15}+\frac {2 f x^3}{17}+\frac {2 g x^4}{19}\right ) \, dx\\ &=\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {1}{4} \left (27 a^2\right ) \int \frac {\frac {4 c}{55}+\frac {4 d x}{91}+\frac {4 e x^2}{135}+\frac {4 f x^3}{187}+\frac {4 g x^4}{247}}{\sqrt {a+b x^3}} \, dx\\ &=\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {\left (27 a^2\right ) \int \frac {\frac {14 b c}{55}+\frac {2}{247} (19 b d-4 a g) x+\frac {14}{135} b e x^2+\frac {14}{187} b f x^3}{\sqrt {a+b x^3}} \, dx}{14 b}\\ &=\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {\left (27 a^2\right ) \int \frac {\frac {7}{187} b (17 b c-2 a f)+\frac {5}{247} b (19 b d-4 a g) x+\frac {7}{27} b^2 e x^2}{\sqrt {a+b x^3}} \, dx}{35 b^2}\\ &=\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {\left (27 a^2\right ) \int \frac {\frac {7}{187} b (17 b c-2 a f)+\frac {5}{247} b (19 b d-4 a g) x}{\sqrt {a+b x^3}} \, dx}{35 b^2}+\frac {1}{5} \left (a^2 e\right ) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx\\ &=\frac {2 a^2 e \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {\left (27 a^2 (19 b d-4 a g)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{1729 b^{4/3}}+\frac {\left (27 a^2 \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{1616615 b^{4/3}}\\ &=\frac {2 a^2 e \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {54 a^2 (19 b d-4 a g) \sqrt {a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 b d-4 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1616615 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 9.32, size = 139, normalized size = 0.20 \begin {gather*} \frac {\sqrt {a+b x^3} \left (4 \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} (323 e+15 x (19 f+17 g x))-570 a (-17 b c+2 a f) x \, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )-255 a (-19 b d+4 a g) x^2 \, _2F_1\left (-\frac {3}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )\right )}{9690 b \sqrt {1+\frac {b x^3}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^3]*(4*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*(323*e + 15*x*(19*f + 17*g*x)) - 570*a*(-17*b*c + 2*a*f)
*x*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] - 255*a*(-19*b*d + 4*a*g)*x^2*Hypergeometric2F1[-3/2, 2/3,
5/3, -((b*x^3)/a)]))/(9690*b*Sqrt[1 + (b*x^3)/a])

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1628 vs. \(2 (542 ) = 1084\).
time = 0.38, size = 1629, normalized size = 2.35

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1024\)
risch	\(\text {Expression too large to display}\)	\(1138\)
default	\(\text {Expression too large to display}\)	\(1629\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c),x,method=_RETURNVERBOSE)

[Out]

g*(2/19*b*x^8*(b*x^3+a)^(1/2)+44/247*a*x^5*(b*x^3+a)^(1/2)+54/1729*a^2*x^2*(b*x^3+a)^(1/2)/b+72/1729*I/b^2*a^3
*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(
1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^
2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/
3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(
1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b
^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2
)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-
a*b^2)^(1/3)))^(1/2))))+f*(2/17*b*x^7*(b*x^3+a)^(1/2)+40/187*a*x^4*(b*x^3+a)^(1/2)+54/935*a^2*x*(b*x^3+a)^(1/2
)/b+36/935*I/b^2*a^3*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)
*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)
*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*E
llipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2
),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)))+2/15*e*(b*x^3+a)
^(5/2)/b+d*(2/13*b*x^5*(b*x^3+a)^(1/2)+32/91*a*x^2*(b*x^3+a)^(1/2)-18/91*I*a^2*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+
1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(
-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b
^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1
/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3)
)^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2
)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1
/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))))+c*(2/1
1*b*x^4*(b*x^3+a)^(1/2)+28/55*a*x*(b*x^3+a)^(1/2)-18/55*I*a^2*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1
/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(
1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2
)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a
*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/
b*(-a*b^2)^(1/3)))^(1/2)))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.09, size = 201, normalized size = 0.29 \begin {gather*} \frac {2 \, {\left (140049 \, {\left (17 \, a^{2} b c - 2 \, a^{3} f\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 75735 \, {\left (19 \, a^{2} b d - 4 \, a^{3} g\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (255255 \, b^{3} g x^{8} + 285285 \, b^{3} f x^{7} + 323323 \, b^{3} e x^{6} + 646646 \, a b^{2} e x^{3} + 19635 \, {\left (19 \, b^{3} d + 22 \, a b^{2} g\right )} x^{5} + 25935 \, {\left (17 \, b^{3} c + 20 \, a b^{2} f\right )} x^{4} + 323323 \, a^{2} b e + 2805 \, {\left (304 \, a b^{2} d + 27 \, a^{2} b g\right )} x^{2} + 5187 \, {\left (238 \, a b^{2} c + 27 \, a^{2} b f\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{4849845 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*(140049*(17*a^2*b*c - 2*a^3*f)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) - 75735*(19*a^2*b*d - 4*a^3
*g)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (255255*b^3*g*x^8 + 285285*b^3*f*x
^7 + 323323*b^3*e*x^6 + 646646*a*b^2*e*x^3 + 19635*(19*b^3*d + 22*a*b^2*g)*x^5 + 25935*(17*b^3*c + 20*a*b^2*f)
*x^4 + 323323*a^2*b*e + 2805*(304*a*b^2*d + 27*a^2*b*g)*x^2 + 5187*(238*a*b^2*c + 27*a^2*b*f)*x)*sqrt(b*x^3 +
a))/b^2

________________________________________________________________________________________

Sympy [A]
time = 4.15, size = 444, normalized size = 0.64 \begin {gather*} \frac {a^{\frac {3}{2}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {3}{2}} d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {a^{\frac {3}{2}} f x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {a^{\frac {3}{2}} g x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} b c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt {a} b d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} b f x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt {a} b g x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} + a e \left (\begin {cases} \frac {\sqrt {a} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{9 b} & \text {otherwise} \end {cases}\right ) + b e \left (\begin {cases} - \frac {4 a^{2} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 a x^{3} \sqrt {a + b x^{3}}}{45 b} + \frac {2 x^{6} \sqrt {a + b x^{3}}}{15} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(3/2)*c*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) + a**(3/2)*d*x**2*
gamma(2/3)*hyper((-1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(5/3)) + a**(3/2)*f*x**4*gamma(4/3)*hy
per((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + a**(3/2)*g*x**5*gamma(5/3)*hyper((-1/2, 5/
3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(8/3)) + sqrt(a)*b*c*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,),
b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + sqrt(a)*b*d*x**5*gamma(5/3)*hyper((-1/2, 5/3), (8/3,), b*x**3*exp_p
olar(I*pi)/a)/(3*gamma(8/3)) + sqrt(a)*b*f*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/
a)/(3*gamma(10/3)) + sqrt(a)*b*g*x**8*gamma(8/3)*hyper((-1/2, 8/3), (11/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamm
a(11/3)) + a*e*Piecewise((sqrt(a)*x**3/3, Eq(b, 0)), (2*(a + b*x**3)**(3/2)/(9*b), True)) + b*e*Piecewise((-4*
a**2*sqrt(a + b*x**3)/(45*b**2) + 2*a*x**3*sqrt(a + b*x**3)/(45*b) + 2*x**6*sqrt(a + b*x**3)/15, Ne(b, 0)), (s
qrt(a)*x**6/6, True))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________