Integrand size = 30, antiderivative size = 316 \[ \int \frac {c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=\frac {c+e x^2}{2 a x \sqrt {a+b x^4}}+\frac {d+f x^2}{2 a \sqrt {a+b x^4}}-\frac {3 c \sqrt {a+b x^4}}{2 a^{3/2} x \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {3 \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a+b x^4}}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt {a+b x^4}} \] Output:
1/2*(e*x^2+c)/a/x/(b*x^4+a)^(1/2)+1/2*(f*x^2+d)/a/(b*x^4+a)^(1/2)-3/2*c*(b *x^4+a)^(1/2)/a^(3/2)/x/(a^(1/2)+b^(1/2)*x^2)-1/2*d*arctanh((b*x^4+a)^(1/2 )/a^(1/2))/a^(3/2)-3/2*b^(1/4)*c*(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^(7/4)/(b*x^4+a)^(1/2)+1/4*(3*b^(1/2)*c+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^(7/4)/b^(1/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.39 \[ \int \frac {c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=\frac {d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b x^4}{a}\right )-2 c \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{4},-\frac {b x^4}{a}\right )+x^2 \left (e+f x+e \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )\right )}{2 a x \sqrt {a+b x^4}} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3)/(x^2*(a + b*x^4)^(3/2)),x]
Output:
(d*x*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^4)/a] - 2*c*Sqrt[1 + (b*x^4) /a]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((b*x^4)/a)] + x^2*(e + f*x + e*Sqr t[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)]))/(2*a*x*S qrt[a + b*x^4])
Time = 1.24 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2368, 25, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {\int -\frac {\frac {2 b^2 d x^5}{a}+\frac {b^2 c x^4}{a}+b e x^2+2 b d x+2 b c}{x^2 \sqrt {b x^4+a}}dx}{2 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {2 b^2 d x^5}{a}+\frac {b^2 c x^4}{a}+b e x^2+2 b d x+2 b c}{x^2 \sqrt {b x^4+a}}dx}{2 a b}+\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {\int \left (\frac {\frac {b^2 c x^4}{a}+b e x^2+2 b c}{x^2 \sqrt {b x^4+a}}+\frac {\frac {2 b^2 d x^4}{a}+2 b d}{x \sqrt {b x^4+a}}\right )dx}{2 a b}+\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} e+3 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a+b x^4}}-\frac {3 b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {3 b^{3/2} c x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b c \sqrt {a+b x^4}}{a x}+\frac {b d \sqrt {a+b x^4}}{a}}{2 a b}+\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3)/(x^2*(a + b*x^4)^(3/2)),x]
Output:
(x*(a*e + a*f*x - b*c*x^2 - b*d*x^3))/(2*a^2*Sqrt[a + b*x^4]) + ((b*d*Sqrt [a + b*x^4])/a - (2*b*c*Sqrt[a + b*x^4])/(a*x) + (3*b^(3/2)*c*x*Sqrt[a + b *x^4])/(a*(Sqrt[a] + Sqrt[b]*x^2)) - (b*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]] )/Sqrt[a] - (3*b^(5/4)*c*(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]) + (b^(3/4)*(3*Sqrt[b]*c + 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])/(2*a^(3/4)*Sqrt[a + b*x^4]))/(2*a*b)
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m *Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 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 = 2.24 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.85
| method | result | size |
| elliptic | \(-\frac {2 b \left (\frac {c \,x^{3}}{4 a^{2}}-\frac {x^{2} f}{4 a b}-\frac {x e}{4 a b}-\frac {d}{4 b a}\right )}{\sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {c \sqrt {b \,x^{4}+a}}{a^{2} x}+\frac {e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {3 i \sqrt {b}\, c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {d \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{2 a^{\frac {3}{2}}}\) | \(268\) |
| default | \(e \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+c \left (-\frac {b \,x^{3}}{2 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {\sqrt {b \,x^{4}+a}}{a^{2} x}+\frac {3 i \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {1}{2 a \sqrt {b \,x^{4}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\right )+\frac {f \,x^{2}}{2 \sqrt {b \,x^{4}+a}\, a}\) | \(298\) |
| risch | \(-\frac {c \sqrt {b \,x^{4}+a}}{a^{2} x}+\frac {a^{2} e \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+a^{2} d \left (\frac {1}{2 a \sqrt {b \,x^{4}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\right )+b^{2} c \left (-\frac {x^{3}}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 i \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+\frac {f a \,x^{2}}{2 \sqrt {b \,x^{4}+a}}}{a^{2}}\) | \(310\) |
Input:
int((f*x^3+e*x^2+d*x+c)/x^2/(b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*b*(1/4*c/a^2*x^3-1/4/a/b*x^2*f-1/4/a/b*x*e-1/4*d/b/a)/((x^4+a/b)*b)^(1/ 2)-1/a^2*c*(b*x^4+a)^(1/2)/x+1/2/a*e/(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)*Ellip ticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)+3/2*I*b^(1/2)/a^(3/2)*c/(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))-1/2*d/a^(3/2)*arctanh(a^(1/2)/(b*x^4+a)^(1/2))
Time = 0.12 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.68 \[ \int \frac {c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=-\frac {6 \, {\left (b^{2} c x^{5} + a b c x\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 2 \, {\left ({\left (3 \, b^{2} c - a b e\right )} x^{5} + {\left (3 \, a b c - a^{2} e\right )} x\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (b^{2} d x^{5} + a b d x\right )} \sqrt {a} \log \left (-\frac {b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) + 2 \, {\left (3 \, b^{2} c x^{4} - a b f x^{3} - a b e x^{2} - a b d x + 2 \, a b c\right )} \sqrt {b x^{4} + a}}{4 \, {\left (a^{2} b^{2} x^{5} + a^{3} b x\right )}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^2/(b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/4*(6*(b^2*c*x^5 + a*b*c*x)*sqrt(a)*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b /a)^(1/4)), -1) - 2*((3*b^2*c - a*b*e)*x^5 + (3*a*b*c - a^2*e)*x)*sqrt(a)* (-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) - (b^2*d*x^5 + a*b*d*x )*sqrt(a)*log(-(b*x^4 - 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x^4) + 2*(3*b^2*c *x^4 - a*b*f*x^3 - a*b*e*x^2 - a*b*d*x + 2*a*b*c)*sqrt(b*x^4 + a))/(a^2*b^ 2*x^5 + a^3*b*x)
Result contains complex when optimal does not.
Time = 8.04 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=d \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{4}}{a}}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {a^{3} \log {\left (\frac {b x^{4}}{a} \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{4}}{a}} + 1 \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {a^{2} b x^{4} \log {\left (\frac {b x^{4}}{a} \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 a^{2} b x^{4} \log {\left (\sqrt {1 + \frac {b x^{4}}{a}} + 1 \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}}\right ) + \frac {c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x \Gamma \left (\frac {3}{4}\right )} + \frac {e x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {f x^{2}}{2 a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{4}}{a}}} \] Input:
integrate((f*x**3+e*x**2+d*x+c)/x**2/(b*x**4+a)**(3/2),x)
Output:
d*(2*a**3*sqrt(1 + b*x**4/a)/(4*a**(9/2) + 4*a**(7/2)*b*x**4) + a**3*log(b *x**4/a)/(4*a**(9/2) + 4*a**(7/2)*b*x**4) - 2*a**3*log(sqrt(1 + b*x**4/a) + 1)/(4*a**(9/2) + 4*a**(7/2)*b*x**4) + a**2*b*x**4*log(b*x**4/a)/(4*a**(9 /2) + 4*a**(7/2)*b*x**4) - 2*a**2*b*x**4*log(sqrt(1 + b*x**4/a) + 1)/(4*a* *(9/2) + 4*a**(7/2)*b*x**4)) + c*gamma(-1/4)*hyper((-1/4, 3/2), (3/4,), b* x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*x*gamma(3/4)) + e*x*gamma(1/4)*hyper(( 1/4, 3/2), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(5/4)) + f*x **2/(2*a**(3/2)*sqrt(1 + b*x**4/a))
\[ \int \frac {c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {f x^{3} + e x^{2} + d x + c}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^2/(b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((f*x^3 + e*x^2 + d*x + c)/((b*x^4 + a)^(3/2)*x^2), x)
\[ \int \frac {c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {f x^{3} + e x^{2} + d x + c}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^2/(b*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((f*x^3 + e*x^2 + d*x + c)/((b*x^4 + a)^(3/2)*x^2), x)
Time = 7.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.42 \[ \int \frac {c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=\frac {d}{2\,a\,\sqrt {b\,x^4+a}}-\frac {d\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^4+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}+\frac {f\,x^2}{2\,a\,\sqrt {b\,x^4+a}}-\frac {c\,{\left (\frac {a}{b\,x^4}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {7}{4};\ \frac {11}{4};\ -\frac {a}{b\,x^4}\right )}{7\,x\,{\left (b\,x^4+a\right )}^{3/2}}+\frac {e\,x\,{\left (\frac {b\,x^4}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{2};\ \frac {5}{4};\ -\frac {b\,x^4}{a}\right )}{{\left (b\,x^4+a\right )}^{3/2}} \] Input:
int((c + d*x + e*x^2 + f*x^3)/(x^2*(a + b*x^4)^(3/2)),x)
Output:
d/(2*a*(a + b*x^4)^(1/2)) - (d*atanh((a + b*x^4)^(1/2)/a^(1/2)))/(2*a^(3/2 )) + (f*x^2)/(2*a*(a + b*x^4)^(1/2)) - (c*(a/(b*x^4) + 1)^(3/2)*hypergeom( [3/2, 7/4], 11/4, -a/(b*x^4)))/(7*x*(a + b*x^4)^(3/2)) + (e*x*((b*x^4)/a + 1)^(3/2)*hypergeom([1/4, 3/2], 5/4, -(b*x^4)/a))/(a + b*x^4)^(3/2)
\[ \int \frac {c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((f*x^3+e*x^2+d*x+c)/x^2/(b*x^4+a)^(3/2),x)
Output:
(2*sqrt(a)*sqrt(a + b*x**4)*log(sqrt(a + b*x**4) - sqrt(a))*a*b*d*x**2 + 2 *sqrt(a)*sqrt(a + b*x**4)*log(sqrt(a + b*x**4) - sqrt(a))*b**2*d*x**6 - 2* sqrt(a)*sqrt(a + b*x**4)*log(sqrt(a + b*x**4) + sqrt(a))*a*b*d*x**2 - 2*sq rt(a)*sqrt(a + b*x**4)*log(sqrt(a + b*x**4) + sqrt(a))*b**2*d*x**6 + 2*sqr t(b)*sqrt(a + b*x**4)*a**2*d + 6*sqrt(b)*sqrt(a + b*x**4)*a**2*f*x**2 + 4* sqrt(b)*sqrt(a + b*x**4)*a*b*d*x**4 + 8*sqrt(b)*sqrt(a + b*x**4)*a*b*f*x** 6 + 8*sqrt(a + b*x**4)*int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x**8 ),x)*a**3*b*e*x**2 + 8*sqrt(a + b*x**4)*int(sqrt(a + b*x**4)/(a**2 + 2*a*b *x**4 + b**2*x**8),x)*a**2*b**2*e*x**6 + 8*sqrt(a + b*x**4)*int(sqrt(a + b *x**4)/(a**2*x**2 + 2*a*b*x**6 + b**2*x**10),x)*a**3*b*c*x**2 + 8*sqrt(a + b*x**4)*int(sqrt(a + b*x**4)/(a**2*x**2 + 2*a*b*x**6 + b**2*x**10),x)*a** 2*b**2*c*x**6 + sqrt(b)*sqrt(a)*log(sqrt(a + b*x**4) - sqrt(a))*a**2*d + 3 *sqrt(b)*sqrt(a)*log(sqrt(a + b*x**4) - sqrt(a))*a*b*d*x**4 + 2*sqrt(b)*sq rt(a)*log(sqrt(a + b*x**4) - sqrt(a))*b**2*d*x**8 - sqrt(b)*sqrt(a)*log(sq rt(a + b*x**4) + sqrt(a))*a**2*d - 3*sqrt(b)*sqrt(a)*log(sqrt(a + b*x**4) + sqrt(a))*a*b*d*x**4 - 2*sqrt(b)*sqrt(a)*log(sqrt(a + b*x**4) + sqrt(a))* b**2*d*x**8 + 4*sqrt(b)*int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x** 8),x)*a**4*e + 12*sqrt(b)*int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x **8),x)*a**3*b*e*x**4 + 8*sqrt(b)*int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*a**2*b**2*e*x**8 + 4*sqrt(b)*int(sqrt(a + b*x**4)/(a**2...