Integrand size = 30, antiderivative size = 402 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^9} \, dx=-\frac {c \sqrt {a+b x^4}}{8 x^8}+\frac {2 f \sqrt {a+b x^4}}{15 x^5}-\frac {b c \sqrt {a+b x^4}}{16 a x^4}-\frac {2 b d \sqrt {a+b x^4}}{21 a x^3}-\frac {2 b f \sqrt {a+b x^4}}{5 \sqrt {a} x \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (3 d+7 f x^2\right ) \sqrt {a+b x^4}}{21 x^7}-\frac {e \left (a+b x^4\right )^{3/2}}{6 a x^6}+\frac {b^2 c \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 a^{3/2}}-\frac {2 b^{5/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{3/4} \sqrt {a+b x^4}}-\frac {b^{5/4} \left (5 \sqrt {b} d-21 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 a^{5/4} \sqrt {a+b x^4}} \] Output:
-1/8*c*(b*x^4+a)^(1/2)/x^8+2/15*f*(b*x^4+a)^(1/2)/x^5-1/16*b*c*(b*x^4+a)^( 1/2)/a/x^4-2/21*b*d*(b*x^4+a)^(1/2)/a/x^3-2/5*b*f*(b*x^4+a)^(1/2)/a^(1/2)/ x/(a^(1/2)+b^(1/2)*x^2)-1/21*(7*f*x^2+3*d)*(b*x^4+a)^(1/2)/x^7-1/6*e*(b*x^ 4+a)^(3/2)/a/x^6+1/16*b^2*c*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(3/2)-2/5*b ^(5/4)*f*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*E llipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(3/4)/(b*x^4+a)^( 1/2)-1/105*b^(5/4)*(5*b^(1/2)*d-21*a^(1/2)*f)*(a^(1/2)+b^(1/2)*x^2)*((b*x^ 4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^( 1/4)),1/2*2^(1/2))/a^(5/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.15 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.36 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^9} \, dx=-\frac {\sqrt {a+b x^4} \left (30 a^3 d \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {1}{2},-\frac {3}{4},-\frac {b x^4}{a}\right )+7 x \left (6 a^3 f x \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{2},-\frac {1}{4},-\frac {b x^4}{a}\right )+5 \left (a+b x^4\right ) \sqrt {1+\frac {b x^4}{a}} \left (a^2 e+b^2 c x^6 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},3,\frac {5}{2},1+\frac {b x^4}{a}\right )\right )\right )\right )}{210 a^3 x^7 \sqrt {1+\frac {b x^4}{a}}} \] Input:
Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^9,x]
Output:
-1/210*(Sqrt[a + b*x^4]*(30*a^3*d*Hypergeometric2F1[-7/4, -1/2, -3/4, -((b *x^4)/a)] + 7*x*(6*a^3*f*x*Hypergeometric2F1[-5/4, -1/2, -1/4, -((b*x^4)/a )] + 5*(a + b*x^4)*Sqrt[1 + (b*x^4)/a]*(a^2*e + b^2*c*x^6*Hypergeometric2F 1[3/2, 3, 5/2, 1 + (b*x^4)/a]))))/(a^3*x^7*Sqrt[1 + (b*x^4)/a])
Time = 1.19 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2364, 27, 2372, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^4} \left (c+d x+e x^2+f x^3\right )}{x^9} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -2 b \int -\frac {168 f x^3+140 e x^2+120 d x+105 c}{840 x^5 \sqrt {b x^4+a}}dx-\frac {1}{840} \sqrt {a+b x^4} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{420} b \int \frac {168 f x^3+140 e x^2+120 d x+105 c}{x^5 \sqrt {b x^4+a}}dx-\frac {1}{840} \sqrt {a+b x^4} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right )\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{420} b \int \left (\frac {140 e x^2+105 c}{x^5 \sqrt {b x^4+a}}+\frac {168 f x^2+120 d}{x^4 \sqrt {b x^4+a}}\right )dx-\frac {1}{840} \sqrt {a+b x^4} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{420} b \left (-\frac {4 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {b} d-21 \sqrt {a} f\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{a^{5/4} \sqrt {a+b x^4}}-\frac {168 \sqrt [4]{b} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}+\frac {105 b c \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {105 c \sqrt {a+b x^4}}{4 a x^4}-\frac {40 d \sqrt {a+b x^4}}{a x^3}-\frac {70 e \sqrt {a+b x^4}}{a x^2}-\frac {168 f \sqrt {a+b x^4}}{a x}+\frac {168 \sqrt {b} f x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}\right )-\frac {1}{840} \sqrt {a+b x^4} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right )\) |
Input:
Int[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^9,x]
Output:
-1/840*(((105*c)/x^8 + (120*d)/x^7 + (140*e)/x^6 + (168*f)/x^5)*Sqrt[a + b *x^4]) + (b*((-105*c*Sqrt[a + b*x^4])/(4*a*x^4) - (40*d*Sqrt[a + b*x^4])/( a*x^3) - (70*e*Sqrt[a + b*x^4])/(a*x^2) - (168*f*Sqrt[a + b*x^4])/(a*x) + (168*Sqrt[b]*f*x*Sqrt[a + b*x^4])/(a*(Sqrt[a] + Sqrt[b]*x^2)) + (105*b*c*A rcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(4*a^(3/2)) - (168*b^(1/4)*f*(Sqrt[a] + S qrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan [(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*Sqrt[a + b*x^4]) - (4*b^(1/4)*(5*Sqr t[b]*d - 21*Sqrt[a]*f)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(5/4)*S qrt[a + b*x^4])))/420
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a + b*x^n)^p, x] - Simp[b*n*p Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a, b} , x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1 , 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 ] && !PolyQ[Pq, x^(n/2)]
Result contains complex when optimal does not.
Time = 3.85 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (672 b f \,x^{7}+280 b e \,x^{6}+160 x^{5} b d +105 b c \,x^{4}+336 a f \,x^{3}+280 a e \,x^{2}+240 a d x +210 a c \right )}{1680 x^{8} a}-\frac {b^{2} \left (\frac {80 d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {105 c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2 \sqrt {a}}-\frac {336 i f \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )}{840 a}\) | \(277\) |
| elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {d \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {e \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {f \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {b c \sqrt {b \,x^{4}+a}}{16 a \,x^{4}}-\frac {2 b d \sqrt {b \,x^{4}+a}}{21 a \,x^{3}}-\frac {e b \sqrt {b \,x^{4}+a}}{6 a \,x^{2}}-\frac {2 b f \sqrt {b \,x^{4}+a}}{5 a x}-\frac {2 d \,b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {2 i b^{\frac {3}{2}} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{2} c \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{16 a^{\frac {3}{2}}}\) | \(335\) |
| default | \(c \left (-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}}}{8 a \,x^{8}}+\frac {b \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{16 a^{2} x^{4}}+\frac {b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}-\frac {b^{2} \sqrt {b \,x^{4}+a}}{16 a^{2}}\right )+d \left (-\frac {\sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {2 b \sqrt {b \,x^{4}+a}}{21 a \,x^{3}}-\frac {2 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {e \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}+f \left (-\frac {\sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {2 b \sqrt {b \,x^{4}+a}}{5 a x}+\frac {2 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(348\) |
Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^9,x,method=_RETURNVERBOSE)
Output:
-1/1680*(b*x^4+a)^(1/2)*(672*b*f*x^7+280*b*e*x^6+160*b*d*x^5+105*b*c*x^4+3 36*a*f*x^3+280*a*e*x^2+240*a*d*x+210*a*c)/x^8/a-1/840/a*b^2*(80*d/(I/a^(1/ 2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2 )^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-105/2*c/a ^(1/2)*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2)-336*I*f*a^(1/2)/(I/a^(1/2)* b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^( 1/2)/(b*x^4+a)^(1/2)/b^(1/2)*(EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-Ell ipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I)))
Time = 0.12 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.49 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^9} \, dx=-\frac {1344 \, a^{\frac {3}{2}} b f x^{8} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 105 \, \sqrt {a} b^{2} c x^{8} \log \left (-\frac {b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 64 \, {\left (5 \, a b d + 21 \, a b f\right )} \sqrt {a} x^{8} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (672 \, a b f x^{7} + 280 \, a b e x^{6} + 160 \, a b d x^{5} + 105 \, a b c x^{4} + 336 \, a^{2} f x^{3} + 280 \, a^{2} e x^{2} + 240 \, a^{2} d x + 210 \, a^{2} c\right )} \sqrt {b x^{4} + a}}{3360 \, a^{2} x^{8}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^9,x, algorithm="fricas")
Output:
-1/3360*(1344*a^(3/2)*b*f*x^8*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4 )), -1) - 105*sqrt(a)*b^2*c*x^8*log(-(b*x^4 + 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x^4) - 64*(5*a*b*d + 21*a*b*f)*sqrt(a)*x^8*(-b/a)^(3/4)*elliptic_f(ar csin(x*(-b/a)^(1/4)), -1) + 2*(672*a*b*f*x^7 + 280*a*b*e*x^6 + 160*a*b*d*x ^5 + 105*a*b*c*x^4 + 336*a^2*f*x^3 + 280*a^2*e*x^2 + 240*a^2*d*x + 210*a^2 *c)*sqrt(b*x^4 + a))/(a^2*x^8)
Result contains complex when optimal does not.
Time = 3.73 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.61 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^9} \, dx=\frac {\sqrt {a} d \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {\sqrt {a} f \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} - \frac {a c}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {3 \sqrt {b} c}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {\sqrt {b} e \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {b^{\frac {3}{2}} c}{16 a x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{6 a} + \frac {b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 a^{\frac {3}{2}}} \] Input:
integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(1/2)/x**9,x)
Output:
sqrt(a)*d*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), b*x**4*exp_polar(I*pi)/ a)/(4*x**7*gamma(-3/4)) + sqrt(a)*f*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4, ), b*x**4*exp_polar(I*pi)/a)/(4*x**5*gamma(-1/4)) - a*c/(8*sqrt(b)*x**10*s qrt(a/(b*x**4) + 1)) - 3*sqrt(b)*c/(16*x**6*sqrt(a/(b*x**4) + 1)) - sqrt(b )*e*sqrt(a/(b*x**4) + 1)/(6*x**4) - b**(3/2)*c/(16*a*x**2*sqrt(a/(b*x**4) + 1)) - b**(3/2)*e*sqrt(a/(b*x**4) + 1)/(6*a) + b**2*c*asinh(sqrt(a)/(sqrt (b)*x**2))/(16*a**(3/2))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^9} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{9}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^9,x, algorithm="maxima")
Output:
-1/32*(b^2*log((sqrt(b*x^4 + a) - sqrt(a))/(sqrt(b*x^4 + a) + sqrt(a)))/a^ (3/2) + 2*((b*x^4 + a)^(3/2)*b^2 + sqrt(b*x^4 + a)*a*b^2)/((b*x^4 + a)^2*a - 2*(b*x^4 + a)*a^2 + a^3))*c + integrate(sqrt(b*x^4 + a)*(f*x^2 + e*x + d)/x^8, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^9} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{9}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^9,x, algorithm="giac")
Output:
integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^9, x)
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^9} \, dx=\int \frac {\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^9} \,d x \] Input:
int(((a + b*x^4)^(1/2)*(c + d*x + e*x^2 + f*x^3))/x^9,x)
Output:
int(((a + b*x^4)^(1/2)*(c + d*x + e*x^2 + f*x^3))/x^9, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^9} \, dx=\frac {-60 \sqrt {b \,x^{4}+a}\, a^{2} c -96 \sqrt {b \,x^{4}+a}\, a^{2} d x -80 \sqrt {b \,x^{4}+a}\, a^{2} e \,x^{2}-160 \sqrt {b \,x^{4}+a}\, a^{2} f \,x^{3}-30 \sqrt {b \,x^{4}+a}\, a b c \,x^{4}-80 \sqrt {b \,x^{4}+a}\, a b e \,x^{6}-15 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {a}\right ) b^{2} c \,x^{8}+15 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {a}\right ) b^{2} c \,x^{8}-192 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{12}+a \,x^{8}}d x \right ) a^{3} d \,x^{8}-320 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{10}+a \,x^{6}}d x \right ) a^{3} f \,x^{8}}{480 a^{2} x^{8}} \] Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^9,x)
Output:
( - 60*sqrt(a + b*x**4)*a**2*c - 96*sqrt(a + b*x**4)*a**2*d*x - 80*sqrt(a + b*x**4)*a**2*e*x**2 - 160*sqrt(a + b*x**4)*a**2*f*x**3 - 30*sqrt(a + b*x **4)*a*b*c*x**4 - 80*sqrt(a + b*x**4)*a*b*e*x**6 - 15*sqrt(a)*log(sqrt(a + b*x**4) - sqrt(a))*b**2*c*x**8 + 15*sqrt(a)*log(sqrt(a + b*x**4) + sqrt(a ))*b**2*c*x**8 - 192*int(sqrt(a + b*x**4)/(a*x**8 + b*x**12),x)*a**3*d*x** 8 - 320*int(sqrt(a + b*x**4)/(a*x**6 + b*x**10),x)*a**3*f*x**8)/(480*a**2* x**8)