3.1.36 \(\int (d+e x^3)^{3/2} (a+b x^3+c x^6) \, dx\) [36]

Optimal. Leaf size=356 \[ \frac {18 d \left (16 c d^2-46 b d e+391 a e^2\right ) x \sqrt {d+e x^3}}{21505 e^2}+\frac {2 \left (16 c d^2-46 b d e+391 a e^2\right ) x \left (d+e x^3\right )^{3/2}}{4301 e^2}-\frac {2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} d^2 \left (16 c d^2-46 b d e+391 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right )|-7-4 \sqrt {3}\right )}{21505 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \]

[Out]

2/4301*(391*a*e^2-46*b*d*e+16*c*d^2)*x*(e*x^3+d)^(3/2)/e^2-2/391*(-23*b*e+8*c*d)*x*(e*x^3+d)^(5/2)/e^2+2/23*c*
x^4*(e*x^3+d)^(5/2)/e+18/21505*d*(391*a*e^2-46*b*d*e+16*c*d^2)*x*(e*x^3+d)^(1/2)/e^2+18/21505*3^(3/4)*d^2*(391
*a*e^2-46*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)/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 = 0.21, antiderivative size = 356, 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 = {1425, 396, 201, 224} \begin {gather*} \frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} d^2 \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 (391 a e^2-46 b d e+16 c d^2\right ) F\left (\text {ArcSin}\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 )}{21505 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 (d+e x^3\right )^{3/2} \left (391 a e^2-46 b d e+16 c d^2\right )}{4301 e^2}+\frac {18 d x \sqrt {d+e x^3} \left (391 a e^2-46 b d e+16 c d^2\right )}{21505 e^2}-\frac {2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(18*d*(16*c*d^2 - 46*b*d*e + 391*a*e^2)*x*Sqrt[d + e*x^3])/(21505*e^2) + (2*(16*c*d^2 - 46*b*d*e + 391*a*e^2)*
x*(d + e*x^3)^(3/2))/(4301*e^2) - (2*(8*c*d - 23*b*e)*x*(d + e*x^3)^(5/2))/(391*e^2) + (2*c*x^4*(d + e*x^3)^(5
/2))/(23*e) + (18*3^(3/4)*Sqrt[2 + Sqrt[3]]*d^2*(16*c*d^2 - 46*b*d*e + 391*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]])/(21505*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 201

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

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 396

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

Rule 1425

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

Rubi steps

\begin {align*} \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx &=\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac {2 \int \left (d+e x^3\right )^{3/2} \left (\frac {23 a e}{2}-\left (4 c d-\frac {23 b e}{2}\right ) x^3\right ) \, dx}{23 e}\\ &=-\frac {2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}-\frac {1}{391} \left (-391 a-\frac {2 d (8 c d-23 b e)}{e^2}\right ) \int \left (d+e x^3\right )^{3/2} \, dx\\ &=\frac {2 \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{4301}-\frac {2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac {\left (9 d \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right )\right ) \int \sqrt {d+e x^3} \, dx}{4301}\\ &=\frac {18 d \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right ) x \sqrt {d+e x^3}}{21505}+\frac {2 \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{4301}-\frac {2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac {\left (27 d^2 \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right )\right ) \int \frac {1}{\sqrt {d+e x^3}} \, dx}{21505}\\ &=\frac {18 d \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right ) x \sqrt {d+e x^3}}{21505}+\frac {2 \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{4301}-\frac {2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} d^2 \left (16 c d^2-46 b d e+391 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 )}{21505 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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 6.96, size = 101, normalized size = 0.28 \begin {gather*} \frac {x \sqrt {d+e x^3} \left (-2 \left (d+e x^3\right )^2 \left (8 c d-23 b e-17 c e x^3\right )+\frac {\left (16 c d^3+23 d e (-2 b d+17 a e)\right ) \, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {e x^3}{d}\right )}{\sqrt {1+\frac {e x^3}{d}}}\right )}{391 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[d + e*x^3]*(-2*(d + e*x^3)^2*(8*c*d - 23*b*e - 17*c*e*x^3) + ((16*c*d^3 + 23*d*e*(-2*b*d + 17*a*e))*Hy
pergeometric2F1[-3/2, 1/3, 4/3, -((e*x^3)/d)])/Sqrt[1 + (e*x^3)/d]))/(391*e^2)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1009 vs. \(2 (285 ) = 570\).
time = 0.22, size = 1010, normalized size = 2.84

method	result	size
risch	\(\frac {2 x \left (935 e^{3} c \,x^{9}+1265 e^{3} b \,x^{6}+1430 d \,e^{2} c \,x^{6}+1955 a \,e^{3} x^{3}+2300 b d \,e^{2} x^{3}+135 c \,d^{2} e \,x^{3}+5474 d \,e^{2} a +621 d^{2} e b -216 c \,d^{3}\right ) \sqrt {e \,x^{3}+d}}{21505 e^{2}}-\frac {18 i d^{2} \left (391 a \,e^{2}-46 d e b +16 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{21505 e^{3} \sqrt {e \,x^{3}+d}}\)	\(398\)
elliptic	\(\frac {2 c e \,x^{10} \sqrt {e \,x^{3}+d}}{23}+\frac {2 \left (e^{2} b +\frac {26}{23} c d e \right ) x^{7} \sqrt {e \,x^{3}+d}}{17 e}+\frac {2 \left (a \,e^{2}+2 d e b +c \,d^{2}-\frac {14 d \left (e^{2} b +\frac {26}{23} c d e \right )}{17 e}\right ) x^{4} \sqrt {e \,x^{3}+d}}{11 e}+\frac {2 \left (2 a d e +d^{2} b -\frac {8 d \left (a \,e^{2}+2 d e b +c \,d^{2}-\frac {14 d \left (e^{2} b +\frac {26}{23} c d e \right )}{17 e}\right )}{11 e}\right ) x \sqrt {e \,x^{3}+d}}{5 e}-\frac {2 i \left (a \,d^{2}-\frac {2 d \left (2 a d e +d^{2} b -\frac {8 d \left (a \,e^{2}+2 d e b +c \,d^{2}-\frac {14 d \left (e^{2} b +\frac {26}{23} c d e \right )}{17 e}\right )}{11 e}\right )}{5 e}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{3 e \sqrt {e \,x^{3}+d}}\)	\(505\)
default	\(\text {Expression too large to display}\)	\(1010\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(2/23*e*x^10*(e*x^3+d)^(1/2)+52/391*d*x^7*(e*x^3+d)^(1/2)+54/4301*d^2/e*x^4*(e*x^3+d)^(1/2)-432/21505*d^3/e^
2*x*(e*x^3+d)^(1/2)-288/21505*I*d^4/e^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)
))+b*(2/17*e*x^7*(e*x^3+d)^(1/2)+40/187*d*x^4*(e*x^3+d)^(1/2)+54/935*d^2/e*x*(e*x^3+d)^(1/2)+36/935*I*d^3/e^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)))+a*(2/11*e*x^4*(e*x^3+d)^(1/2)+28/55*d
*x*(e*x^3+d)^(1/2)-18/55*I*d^2*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 = 0.00, 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((e*x^3+d)^(3/2)*(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.09, size = 130, normalized size = 0.37 \begin {gather*} \frac {2}{21505} \, {\left (27 \, {\left (16 \, c d^{4} - 46 \, b d^{3} e + 391 \, a d^{2} e^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) - {\left (216 \, c d^{3} x e - 5 \, {\left (187 \, c x^{10} + 253 \, b x^{7} + 391 \, a x^{4}\right )} e^{4} - 2 \, {\left (715 \, c d x^{7} + 1150 \, b d x^{4} + 2737 \, a d x\right )} e^{3} - 27 \, {\left (5 \, c d^{2} x^{4} + 23 \, b d^{2} x\right )} e^{2}\right )} \sqrt {x^{3} e + d}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21505*(27*(16*c*d^4 - 46*b*d^3*e + 391*a*d^2*e^2)*e^(1/2)*weierstrassPInverse(0, -4*d*e^(-1), x) - (216*c*d^
3*x*e - 5*(187*c*x^10 + 253*b*x^7 + 391*a*x^4)*e^4 - 2*(715*c*d*x^7 + 1150*b*d*x^4 + 2737*a*d*x)*e^3 - 27*(5*c
*d^2*x^4 + 23*b*d^2*x)*e^2)*sqrt(x^3*e + d))*e^(-3)

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Sympy [A]
time = 2.87, size = 257, normalized size = 0.72 \begin {gather*} \frac {a d^{\frac {3}{2}} 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 {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a \sqrt {d} e 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 {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {b d^{\frac {3}{2}} 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 {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {b \sqrt {d} e 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 {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {c d^{\frac {3}{2}} 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 {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {c \sqrt {d} e x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**(3/2)*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(4/3)) + a*sqrt(d)*e*x**4
*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(7/3)) + b*d**(3/2)*x**4*gamma(4/3)*h
yper((-1/2, 4/3), (7/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(7/3)) + b*sqrt(d)*e*x**7*gamma(7/3)*hyper((-1/2,
7/3), (10/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(10/3)) + c*d**(3/2)*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3
,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(10/3)) + c*sqrt(d)*e*x**10*gamma(10/3)*hyper((-1/2, 10/3), (13/3,), e*x
**3*exp_polar(I*pi)/d)/(3*gamma(13/3))

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (e\,x^3+d\right )}^{3/2}\,\left (c\,x^6+b\,x^3+a\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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