Integrand size = 30, antiderivative size = 377 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^8} \, dx=\frac {2 e \sqrt {a+b x^4}}{15 x^5}-\frac {f \sqrt {a+b x^4}}{4 x^4}-\frac {2 b c \sqrt {a+b x^4}}{21 a x^3}-\frac {2 b e \sqrt {a+b x^4}}{5 \sqrt {a} x \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (3 c+7 e x^2\right ) \sqrt {a+b x^4}}{21 x^7}-\frac {d \left (a+b x^4\right )^{3/2}}{6 a x^6}-\frac {b f \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {2 b^{5/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{3/4} \sqrt {a+b x^4}}-\frac {b^{5/4} \left (5 \sqrt {b} c-21 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 a^{5/4} \sqrt {a+b x^4}} \] Output:
2/15*e*(b*x^4+a)^(1/2)/x^5-1/4*f*(b*x^4+a)^(1/2)/x^4-2/21*b*c*(b*x^4+a)^(1 /2)/a/x^3-2/5*b*e*(b*x^4+a)^(1/2)/a^(1/2)/x/(a^(1/2)+b^(1/2)*x^2)-1/21*(7* e*x^2+3*c)*(b*x^4+a)^(1/2)/x^7-1/6*d*(b*x^4+a)^(3/2)/a/x^6-1/4*b*f*arctanh ((b*x^4+a)^(1/2)/a^(1/2))/a^(1/2)-2/5*b^(5/4)*e*(a^(1/2)+b^(1/2)*x^2)*((b* x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*EllipticE(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)*c-21* a^(1/2)*e)*(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 = 10.17 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.38 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^8} \, dx=-\frac {\sqrt {a+b x^4} \left (35 x \left (\sqrt {1+\frac {b x^4}{a}} \left (2 a d+3 a f x^2+2 b d x^4\right )+3 b f x^6 \text {arctanh}\left (\sqrt {1+\frac {b x^4}{a}}\right )\right )+60 a c \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {1}{2},-\frac {3}{4},-\frac {b x^4}{a}\right )+84 a e x^2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{2},-\frac {1}{4},-\frac {b x^4}{a}\right )\right )}{420 a x^7 \sqrt {1+\frac {b x^4}{a}}} \] Input:
Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^8,x]
Output:
-1/420*(Sqrt[a + b*x^4]*(35*x*(Sqrt[1 + (b*x^4)/a]*(2*a*d + 3*a*f*x^2 + 2* b*d*x^4) + 3*b*f*x^6*ArcTanh[Sqrt[1 + (b*x^4)/a]]) + 60*a*c*Hypergeometric 2F1[-7/4, -1/2, -3/4, -((b*x^4)/a)] + 84*a*e*x^2*Hypergeometric2F1[-5/4, - 1/2, -1/4, -((b*x^4)/a)]))/(a*x^7*Sqrt[1 + (b*x^4)/a])
Time = 1.13 (sec) , antiderivative size = 365, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^4} \left (c+d x+e x^2+f x^3\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -2 b \int -\frac {105 f x^3+84 e x^2+70 d x+60 c}{420 x^4 \sqrt {b x^4+a}}dx-\frac {1}{420} \sqrt {a+b x^4} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} b \int \frac {105 f x^3+84 e x^2+70 d x+60 c}{x^4 \sqrt {b x^4+a}}dx-\frac {1}{420} \sqrt {a+b x^4} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{210} b \int \left (\frac {84 e x^2+60 c}{x^4 \sqrt {b x^4+a}}+\frac {105 f x^2+70 d}{x^3 \sqrt {b x^4+a}}\right )dx-\frac {1}{420} \sqrt {a+b x^4} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{210} b \left (-\frac {2 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {b} c-21 \sqrt {a} e\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{a^{5/4} \sqrt {a+b x^4}}-\frac {84 \sqrt [4]{b} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {105 f \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {20 c \sqrt {a+b x^4}}{a x^3}-\frac {35 d \sqrt {a+b x^4}}{a x^2}-\frac {84 e \sqrt {a+b x^4}}{a x}+\frac {84 \sqrt {b} e x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}\right )-\frac {1}{420} \sqrt {a+b x^4} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )\) |
Input:
Int[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^8,x]
Output:
-1/420*(((60*c)/x^7 + (70*d)/x^6 + (84*e)/x^5 + (105*f)/x^4)*Sqrt[a + b*x^ 4]) + (b*((-20*c*Sqrt[a + b*x^4])/(a*x^3) - (35*d*Sqrt[a + b*x^4])/(a*x^2) - (84*e*Sqrt[a + b*x^4])/(a*x) + (84*Sqrt[b]*e*x*Sqrt[a + b*x^4])/(a*(Sqr t[a] + Sqrt[b]*x^2)) - (105*f*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(2*Sqrt[a] ) - (84*b^(1/4)*e*(Sqrt[a] + Sqrt[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]) - (2*b^(1/4)*(5*Sqrt[b]*c - 21*Sqrt[a]*e)*(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)*Sqrt[a + b*x^4])))/210
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 = 3.39 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (168 b e \,x^{6}+70 x^{5} b d +40 b c \,x^{4}+105 a f \,x^{3}+84 a e \,x^{2}+70 a d x +60 a c \right )}{420 x^{7} a}+\frac {b \left (-\frac {20 c b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {105 \sqrt {a}\, f \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}+\frac {84 i \sqrt {b}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )}{210 a}\) | \(269\) |
| elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {d \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {e \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {f \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {2 b c \sqrt {b \,x^{4}+a}}{21 a \,x^{3}}-\frac {d b \sqrt {b \,x^{4}+a}}{6 a \,x^{2}}-\frac {2 e b \sqrt {b \,x^{4}+a}}{5 a x}-\frac {2 b^{2} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {2 i e \,b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {f b \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4 \sqrt {a}}\) | \(314\) |
| default | \(c \left (-\frac {\sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {2 b \sqrt {b \,x^{4}+a}}{21 a \,x^{3}}-\frac {2 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {d \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}+e \left (-\frac {\sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {2 b \sqrt {b \,x^{4}+a}}{5 a x}+\frac {2 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+f \left (-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4 \sqrt {a}}+\frac {b \sqrt {b \,x^{4}+a}}{4 a}\right )\) | \(326\) |
Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^8,x,method=_RETURNVERBOSE)
Output:
-1/420*(b*x^4+a)^(1/2)*(168*b*e*x^6+70*b*d*x^5+40*b*c*x^4+105*a*f*x^3+84*a *e*x^2+70*a*d*x+60*a*c)/x^7/a+1/210*b/a*(-20*c*b/(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*a^(1/2)*f*ln((2*a+2* a^(1/2)*(b*x^4+a)^(1/2))/x^2)+84*I*b^(1/2)*e*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)*(EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2 )*b^(1/2))^(1/2),I)))
Time = 0.13 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.46 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^8} \, dx=-\frac {336 \, \sqrt {a} b e x^{7} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 105 \, \sqrt {a} b f x^{7} \log \left (-\frac {b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 16 \, {\left (5 \, b c + 21 \, b e\right )} \sqrt {a} x^{7} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (168 \, b e x^{6} + 70 \, b d x^{5} + 40 \, b c x^{4} + 105 \, a f x^{3} + 84 \, a e x^{2} + 70 \, a d x + 60 \, a c\right )} \sqrt {b x^{4} + a}}{840 \, a x^{7}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^8,x, algorithm="fricas")
Output:
-1/840*(336*sqrt(a)*b*e*x^7*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)) , -1) - 105*sqrt(a)*b*f*x^7*log(-(b*x^4 - 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a) /x^4) - 16*(5*b*c + 21*b*e)*sqrt(a)*x^7*(-b/a)^(3/4)*elliptic_f(arcsin(x*( -b/a)^(1/4)), -1) + 2*(168*b*e*x^6 + 70*b*d*x^5 + 40*b*c*x^4 + 105*a*f*x^3 + 84*a*e*x^2 + 70*a*d*x + 60*a*c)*sqrt(b*x^4 + a))/(a*x^7)
Result contains complex when optimal does not.
Time = 2.67 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.51 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^8} \, dx=\frac {\sqrt {a} c \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {\sqrt {a} e \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} - \frac {\sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {\sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} - \frac {b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{4}} + 1}}{6 a} - \frac {b f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 \sqrt {a}} \] Input:
integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(1/2)/x**8,x)
Output:
sqrt(a)*c*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)*e*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4, ), b*x**4*exp_polar(I*pi)/a)/(4*x**5*gamma(-1/4)) - sqrt(b)*d*sqrt(a/(b*x* *4) + 1)/(6*x**4) - sqrt(b)*f*sqrt(a/(b*x**4) + 1)/(4*x**2) - b**(3/2)*d*s qrt(a/(b*x**4) + 1)/(6*a) - b*f*asinh(sqrt(a)/(sqrt(b)*x**2))/(4*sqrt(a))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^8} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{8}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^8,x, algorithm="maxima")
Output:
integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^8, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^8} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{8}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^8,x, algorithm="giac")
Output:
integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^8, x)
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^8} \, dx=\int \frac {\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^8} \,d x \] Input:
int(((a + b*x^4)^(1/2)*(c + d*x + e*x^2 + f*x^3))/x^8,x)
Output:
int(((a + b*x^4)^(1/2)*(c + d*x + e*x^2 + f*x^3))/x^8, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^8} \, dx=\frac {-24 \sqrt {b \,x^{4}+a}\, a c -20 \sqrt {b \,x^{4}+a}\, a d x -40 \sqrt {b \,x^{4}+a}\, a e \,x^{2}-30 \sqrt {b \,x^{4}+a}\, a f \,x^{3}-20 \sqrt {b \,x^{4}+a}\, b d \,x^{5}+15 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {a}\right ) b f \,x^{7}-15 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {a}\right ) b f \,x^{7}-48 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{12}+a \,x^{8}}d x \right ) a^{2} c \,x^{7}-80 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{10}+a \,x^{6}}d x \right ) a^{2} e \,x^{7}}{120 a \,x^{7}} \] Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^8,x)
Output:
( - 24*sqrt(a + b*x**4)*a*c - 20*sqrt(a + b*x**4)*a*d*x - 40*sqrt(a + b*x* *4)*a*e*x**2 - 30*sqrt(a + b*x**4)*a*f*x**3 - 20*sqrt(a + b*x**4)*b*d*x**5 + 15*sqrt(a)*log(sqrt(a + b*x**4) - sqrt(a))*b*f*x**7 - 15*sqrt(a)*log(sq rt(a + b*x**4) + sqrt(a))*b*f*x**7 - 48*int(sqrt(a + b*x**4)/(a*x**8 + b*x **12),x)*a**2*c*x**7 - 80*int(sqrt(a + b*x**4)/(a*x**6 + b*x**10),x)*a**2* e*x**7)/(120*a*x**7)