Integrand size = 35, antiderivative size = 620 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx=\frac {2 a f \sqrt {a+b x^3}}{9 b}+\frac {6 a g x \sqrt {a+b x^3}}{55 b}+\frac {6 a e \sqrt {a+b x^3}}{7 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}-\frac {2}{3} \sqrt {a} c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (77 b d-55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-14 a g\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{385 b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
2/9*a*f*(b*x^3+a)^(1/2)/b+6/55*a*g*x*(b*x^3+a)^(1/2)/b+6/7*a*e*(b*x^3+a)^( 1/2)/b^(2/3)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)+2/3465*(b*x^3+a)^(1/2)*(315*g *x^5+385*f*x^4+495*e*x^3+693*d*x^2+1155*c*x)/x-2/3*a^(1/2)*c*arctanh((b*x^ 3+a)^(1/2)/a^(1/2))-3/7*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a^(4/3)*e*(a^(1/ 3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3 )+b^(1/3)*x)^2)^(1/2)*EllipticE(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2 ))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)/b^(2/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/ ((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)+2/385*3^(3/4)*(1/ 2*6^(1/2)+1/2*2^(1/2))*a*(77*b*d-55*(1-3^(1/2))*a^(1/3)*b^(2/3)*e-14*a*g)* (a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))* a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+ 3^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)/b^(4/3)/(a^(1/3)*(a^(1/3)+b^(1/ 3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.16 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx=\frac {4 \sqrt {1+\frac {b x^3}{a}} \left (\sqrt {a+b x^3} \left (33 b c+11 a f+9 a g x+11 b f x^3+9 b g x^4\right )-33 \sqrt {a} b c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )+18 (11 b d-2 a g) x \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+99 b e x^2 \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{198 b \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x,x]
Output:
(4*Sqrt[1 + (b*x^3)/a]*(Sqrt[a + b*x^3]*(33*b*c + 11*a*f + 9*a*g*x + 11*b* f*x^3 + 9*b*g*x^4) - 33*Sqrt[a]*b*c*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]) + 18 *(11*b*d - 2*a*g)*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b *x^3)/a)] + 99*b*e*x^2*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 2/3, 5/3, - ((b*x^3)/a)])/(198*b*Sqrt[1 + (b*x^3)/a])
Time = 1.73 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2365, 27, 2371, 798, 73, 221, 2427, 27, 2425, 793, 2417, 759, 2416}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2365 |
| \(\displaystyle \frac {3}{2} a \int \frac {2 \left (315 g x^4+385 f x^3+495 e x^2+693 d x+1155 c\right )}{3465 x \sqrt {b x^3+a}}dx+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {315 g x^4+385 f x^3+495 e x^2+693 d x+1155 c}{x \sqrt {b x^3+a}}dx}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 2371 |
| \(\displaystyle \frac {a \left (1155 c \int \frac {1}{x \sqrt {b x^3+a}}dx+\int \frac {315 g x^3+385 f x^2+495 e x+693 d}{\sqrt {b x^3+a}}dx\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {a \left (385 c \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3+\int \frac {315 g x^3+385 f x^2+495 e x+693 d}{\sqrt {b x^3+a}}dx\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a \left (\frac {770 c \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{b}+\int \frac {315 g x^3+385 f x^2+495 e x+693 d}{\sqrt {b x^3+a}}dx\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (\int \frac {315 g x^3+385 f x^2+495 e x+693 d}{\sqrt {b x^3+a}}dx-\frac {770 c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {a \left (\frac {2 \int \frac {5 \left (385 b f x^2+495 b e x+63 (11 b d-2 a g)\right )}{2 \sqrt {b x^3+a}}dx}{5 b}-\frac {770 c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {126 g x \sqrt {a+b x^3}}{b}\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\int \frac {385 b f x^2+495 b e x+63 (11 b d-2 a g)}{\sqrt {b x^3+a}}dx}{b}-\frac {770 c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {126 g x \sqrt {a+b x^3}}{b}\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 2425 |
| \(\displaystyle \frac {a \left (\frac {\int \frac {63 (11 b d-2 a g)+495 b e x}{\sqrt {b x^3+a}}dx+385 b f \int \frac {x^2}{\sqrt {b x^3+a}}dx}{b}-\frac {770 c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {126 g x \sqrt {a+b x^3}}{b}\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 793 |
| \(\displaystyle \frac {a \left (\frac {\int \frac {63 (11 b d-2 a g)+495 b e x}{\sqrt {b x^3+a}}dx+\frac {770}{3} f \sqrt {a+b x^3}}{b}-\frac {770 c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {126 g x \sqrt {a+b x^3}}{b}\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {a \left (\frac {9 \left (-55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-14 a g+77 b d\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+495 b^{2/3} e \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {770}{3} f \sqrt {a+b x^3}}{b}-\frac {770 c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {126 g x \sqrt {a+b x^3}}{b}\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {a \left (\frac {495 b^{2/3} e \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (-55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-14 a g+77 b d\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {770}{3} f \sqrt {a+b x^3}}{b}-\frac {770 c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {126 g x \sqrt {a+b x^3}}{b}\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {a \left (\frac {\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (-55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-14 a g+77 b d\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+495 b^{2/3} e \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )+\frac {770}{3} f \sqrt {a+b x^3}}{b}-\frac {770 c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {126 g x \sqrt {a+b x^3}}{b}\right )}{1155}+\frac {2 \sqrt {a+b x^3} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x}\) |
Input:
Int[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x,x]
Output:
(2*Sqrt[a + b*x^3]*(1155*c*x + 693*d*x^2 + 495*e*x^3 + 385*f*x^4 + 315*g*x ^5))/(3465*x) + (a*((126*g*x*Sqrt[a + b*x^3])/b - (770*c*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/Sqrt[a] + ((770*f*Sqrt[a + b*x^3])/3 + 495*b^(2/3)*e*((2* Sqrt[a + b*x^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^(1/4)* Sqrt[2 - Sqrt[3]]*a^(1/3)*(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]*EllipticE[Ar cSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)* x)], -7 - 4*Sqrt[3]])/(b^(1/3)*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])) + (6*3^(3/4)*Sqrt[2 + S qrt[3]]*(77*b*d - 55*(1 - Sqrt[3])*a^(1/3)*b^(2/3)*e - 14*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]])/(b^(1/3)*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]))/b))/1155
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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]*Sqrt[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]
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> M odule[{q = Expon[Pq, x], i}, Simp[(c*x)^m*(a + b*x^n)^p*Sum[Coeff[Pq, x, i] *(x^(i + 1)/(m + n*p + i + 1)), {i, 0, q}], x] + Simp[a*n*p Int[(c*x)^m*( a + b*x^n)^(p - 1)*Sum[Coeff[Pq, x, i]*(x^i/(m + n*p + i + 1)), {i, 0, q}], x], x]] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]
Int[(Pq_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Simp[Coeff[Pq, x, 0] Int[1/(x*Sqrt[a + b*x^n]), x], x] + Int[ExpandToSum[(Pq - Coeff[Pq, x, 0])/x, x]/Sqrt[a + b*x^n], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IG tQ[n, 0] && NeQ[Coeff[Pq, x, 0], 0]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[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] - S imp[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] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 - Sqrt[3])*d*s)/r Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r Int[((1 - Sqrt[3])*s + r*x)/Sq rt[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]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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] && PolyQ[P q, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x ]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), x] + Simp[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]] /; 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] && IGtQ[n, 0]
Time = 0.49 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.37
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(848\) |
| default | \(\text {Expression too large to display}\) | \(1118\) |
Input:
int((b*x^3+a)^(1/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x,x,method=_RETURNVERBOSE)
Output:
2/11*g*x^4*(b*x^3+a)^(1/2)+2/9*f*x^3*(b*x^3+a)^(1/2)+2/7*e*x^2*(b*x^3+a)^( 1/2)+2/5*(3/11*a*g+b*d)/b*x*(b*x^3+a)^(1/2)+2/3*(1/3*a*f+c*b)/b*(b*x^3+a)^ (1/2)-2/3*I*(a*d-2/5*(3/11*a*g+b*d)/b*a)*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))-2/7*I*a*e*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/...
Time = 0.23 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx=\left [\frac {1155 \, \sqrt {a} b^{2} c \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 5940 \, a b^{\frac {3}{2}} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 756 \, {\left (11 \, a b d - 2 \, a^{2} g\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 4 \, {\left (315 \, b^{2} g x^{4} + 385 \, b^{2} f x^{3} + 495 \, b^{2} e x^{2} + 1155 \, b^{2} c + 385 \, a b f + 63 \, {\left (11 \, b^{2} d + 3 \, a b g\right )} x\right )} \sqrt {b x^{3} + a}}{6930 \, b^{2}}, \frac {1155 \, \sqrt {-a} b^{2} c \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) - 2970 \, a b^{\frac {3}{2}} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 378 \, {\left (11 \, a b d - 2 \, a^{2} g\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 2 \, {\left (315 \, b^{2} g x^{4} + 385 \, b^{2} f x^{3} + 495 \, b^{2} e x^{2} + 1155 \, b^{2} c + 385 \, a b f + 63 \, {\left (11 \, b^{2} d + 3 \, a b g\right )} x\right )} \sqrt {b x^{3} + a}}{3465 \, b^{2}}\right ] \] Input:
integrate((b*x^3+a)^(1/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x,x, algorithm="fricas ")
Output:
[1/6930*(1155*sqrt(a)*b^2*c*log(-(b^2*x^6 + 8*a*b*x^3 - 4*(b*x^3 + 2*a)*sq rt(b*x^3 + a)*sqrt(a) + 8*a^2)/x^6) - 5940*a*b^(3/2)*e*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + 756*(11*a*b*d - 2*a^2*g)*sqrt (b)*weierstrassPInverse(0, -4*a/b, x) + 4*(315*b^2*g*x^4 + 385*b^2*f*x^3 + 495*b^2*e*x^2 + 1155*b^2*c + 385*a*b*f + 63*(11*b^2*d + 3*a*b*g)*x)*sqrt( b*x^3 + a))/b^2, 1/3465*(1155*sqrt(-a)*b^2*c*arctan(1/2*(b*x^3 + 2*a)*sqrt (b*x^3 + a)*sqrt(-a)/(a*b*x^3 + a^2)) - 2970*a*b^(3/2)*e*weierstrassZeta(0 , -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + 378*(11*a*b*d - 2*a^2*g)*sq rt(b)*weierstrassPInverse(0, -4*a/b, x) + 2*(315*b^2*g*x^4 + 385*b^2*f*x^3 + 495*b^2*e*x^2 + 1155*b^2*c + 385*a*b*f + 63*(11*b^2*d + 3*a*b*g)*x)*sqr t(b*x^3 + a))/b^2]
Time = 4.13 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx=- \frac {2 \sqrt {a} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3} + \frac {\sqrt {a} d x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {\sqrt {a} e x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {\sqrt {a} g x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {2 a c}{3 \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 \sqrt {b} c x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} + f \left (\begin {cases} \frac {\sqrt {a} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{9 b} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((b*x**3+a)**(1/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x,x)
Output:
-2*sqrt(a)*c*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/3 + sqrt(a)*d*x*gamma(1/3)* hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) + sqrt (a)*e*x**2*gamma(2/3)*hyper((-1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a) /(3*gamma(5/3)) + sqrt(a)*g*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), b*x **3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + 2*a*c/(3*sqrt(b)*x**(3/2)*sqrt(a/( b*x**3) + 1)) + 2*sqrt(b)*c*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) + f*Piecewis e((sqrt(a)*x**3/3, Eq(b, 0)), (2*(a + b*x**3)**(3/2)/(9*b), True))
\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x,x, algorithm="maxima ")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*sqrt(b*x^3 + a)/x, x)
\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x,x, algorithm="giac")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*sqrt(b*x^3 + a)/x, x)
Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx=\int \frac {\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x} \,d x \] Input:
int(((a + b*x^3)^(1/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x,x)
Output:
int(((a + b*x^3)^(1/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x, x)
\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x} \, dx=\frac {770 \sqrt {b \,x^{3}+a}\, a f +378 \sqrt {b \,x^{3}+a}\, a g x +2310 \sqrt {b \,x^{3}+a}\, b c +1386 \sqrt {b \,x^{3}+a}\, b d x +990 \sqrt {b \,x^{3}+a}\, b e \,x^{2}+770 \sqrt {b \,x^{3}+a}\, b f \,x^{3}+630 \sqrt {b \,x^{3}+a}\, b g \,x^{4}+1155 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {a}\right ) b c -1155 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {a}\right ) b c -378 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a^{2} g +2079 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a b d +1485 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) a b e}{3465 b} \] Input:
int((b*x^3+a)^(1/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x,x)
Output:
(770*sqrt(a + b*x**3)*a*f + 378*sqrt(a + b*x**3)*a*g*x + 2310*sqrt(a + b*x **3)*b*c + 1386*sqrt(a + b*x**3)*b*d*x + 990*sqrt(a + b*x**3)*b*e*x**2 + 7 70*sqrt(a + b*x**3)*b*f*x**3 + 630*sqrt(a + b*x**3)*b*g*x**4 + 1155*sqrt(a )*log(sqrt(a + b*x**3) - sqrt(a))*b*c - 1155*sqrt(a)*log(sqrt(a + b*x**3) + sqrt(a))*b*c - 378*int(sqrt(a + b*x**3)/(a + b*x**3),x)*a**2*g + 2079*in t(sqrt(a + b*x**3)/(a + b*x**3),x)*a*b*d + 1485*int((sqrt(a + b*x**3)*x)/( a + b*x**3),x)*a*b*e)/(3465*b)