Integrand size = 30, antiderivative size = 303 \[ \int \frac {c+d x+e x^2+f x^3}{x^4 \sqrt {a+b x^4}} \, dx=-\frac {c \sqrt {a+b x^4}}{3 a x^3}-\frac {d \sqrt {a+b x^4}}{2 a x^2}-\frac {e \sqrt {a+b x^4}}{\sqrt {a} x \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {f \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt [4]{b} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} \left (\sqrt {b} c-3 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 a^{5/4} \sqrt {a+b x^4}} \] Output:
-1/3*c*(b*x^4+a)^(1/2)/a/x^3-1/2*d*(b*x^4+a)^(1/2)/a/x^2-e*(b*x^4+a)^(1/2) /a^(1/2)/x/(a^(1/2)+b^(1/2)*x^2)-1/2*f*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^ (1/2)-b^(1/4)*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)-1/6*b^(1/4)*(b^(1/2)*c-3*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 = 10.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.49 \[ \int \frac {c+d x+e x^2+f x^3}{x^4 \sqrt {a+b x^4}} \, dx=\frac {-2 a c \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\frac {b x^4}{a}\right )-3 x \left (a d+b d x^4+\sqrt {a} f x^2 \sqrt {a+b x^4} \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )+2 a e x \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {b x^4}{a}\right )\right )}{6 a x^3 \sqrt {a+b x^4}} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3)/(x^4*Sqrt[a + b*x^4]),x]
Output:
(-2*a*c*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-3/4, 1/2, 1/4, -((b*x^4)/a) ] - 3*x*(a*d + b*d*x^4 + Sqrt[a]*f*x^2*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b* x^4]/Sqrt[a]] + 2*a*e*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/4, 1/2, 3 /4, -((b*x^4)/a)]))/(6*a*x^3*Sqrt[a + b*x^4])
Time = 0.86 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3}{x^4 \sqrt {a+b x^4}} \, dx\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \int \left (\frac {c+e x^2}{x^4 \sqrt {a+b x^4}}+\frac {d+f x^2}{x^3 \sqrt {a+b x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {b} c-3 \sqrt {a} e\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 a^{5/4} \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {f \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {c \sqrt {a+b x^4}}{3 a x^3}-\frac {d \sqrt {a+b x^4}}{2 a x^2}-\frac {e \sqrt {a+b x^4}}{a x}+\frac {\sqrt {b} e x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3)/(x^4*Sqrt[a + b*x^4]),x]
Output:
-1/3*(c*Sqrt[a + b*x^4])/(a*x^3) - (d*Sqrt[a + b*x^4])/(2*a*x^2) - (e*Sqrt [a + b*x^4])/(a*x) + (Sqrt[b]*e*x*Sqrt[a + b*x^4])/(a*(Sqrt[a] + Sqrt[b]*x ^2)) - (f*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(2*Sqrt[a]) - (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]) - (b^(1/4)*( Sqrt[b]*c - 3*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])/(6*a^(5/ 4)*Sqrt[a + b*x^4])
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 = 1.58 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (6 e \,x^{2}+3 d x +2 c \right )}{6 a \,x^{3}}+\frac {-\frac {c b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 \sqrt {a}\, f \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}+\frac {3 i \sqrt {b}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}}{3 a}\) | \(237\) |
| elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {d \sqrt {b \,x^{4}+a}}{2 a \,x^{2}}-\frac {e \sqrt {b \,x^{4}+a}}{a x}-\frac {b c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {i e \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {f \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{2 \sqrt {a}}\) | \(248\) |
| default | \(c \left (-\frac {\sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {d \sqrt {b \,x^{4}+a}}{2 a \,x^{2}}+e \left (-\frac {\sqrt {b \,x^{4}+a}}{a x}+\frac {i \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {f \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2 \sqrt {a}}\) | \(259\) |
Input:
int((f*x^3+e*x^2+d*x+c)/x^4/(b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/6*(b*x^4+a)^(1/2)*(6*e*x^2+3*d*x+2*c)/a/x^3+1/3/a*(-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)-3/2*a^(1/2)*f*ln ((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2)+3*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 = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.45 \[ \int \frac {c+d x+e x^2+f x^3}{x^4 \sqrt {a+b x^4}} \, dx=-\frac {12 \, \sqrt {a} e x^{3} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 4 \, \sqrt {a} {\left (c + 3 \, e\right )} x^{3} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 3 \, \sqrt {a} f x^{3} \log \left (-\frac {b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) + 2 \, \sqrt {b x^{4} + a} {\left (6 \, e x^{2} + 3 \, d x + 2 \, c\right )}}{12 \, a x^{3}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^4/(b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
-1/12*(12*sqrt(a)*e*x^3*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1 ) - 4*sqrt(a)*(c + 3*e)*x^3*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a)^(1/4)) , -1) - 3*sqrt(a)*f*x^3*log(-(b*x^4 - 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x^4 ) + 2*sqrt(b*x^4 + a)*(6*e*x^2 + 3*d*x + 2*c))/(a*x^3)
Result contains complex when optimal does not.
Time = 1.73 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.43 \[ \int \frac {c+d x+e x^2+f x^3}{x^4 \sqrt {a+b x^4}} \, dx=- \frac {\sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{2 a} + \frac {c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {e \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x \Gamma \left (\frac {3}{4}\right )} - \frac {f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{2 \sqrt {a}} \] Input:
integrate((f*x**3+e*x**2+d*x+c)/x**4/(b*x**4+a)**(1/2),x)
Output:
-sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(2*a) + c*gamma(-3/4)*hyper((-3/4, 1/2), ( 1/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*x**3*gamma(1/4)) + e*gamma(-1/ 4)*hyper((-1/4, 1/2), (3/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*x*gamma (3/4)) - f*asinh(sqrt(a)/(sqrt(b)*x**2))/(2*sqrt(a))
\[ \int \frac {c+d x+e x^2+f x^3}{x^4 \sqrt {a+b x^4}} \, dx=\int { \frac {f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{4} + a} x^{4}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^4/(b*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((f*x^3 + e*x^2 + d*x + c)/(sqrt(b*x^4 + a)*x^4), x)
\[ \int \frac {c+d x+e x^2+f x^3}{x^4 \sqrt {a+b x^4}} \, dx=\int { \frac {f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{4} + a} x^{4}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^4/(b*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((f*x^3 + e*x^2 + d*x + c)/(sqrt(b*x^4 + a)*x^4), x)
Timed out. \[ \int \frac {c+d x+e x^2+f x^3}{x^4 \sqrt {a+b x^4}} \, dx=\int \frac {f\,x^3+e\,x^2+d\,x+c}{x^4\,\sqrt {b\,x^4+a}} \,d x \] Input:
int((c + d*x + e*x^2 + f*x^3)/(x^4*(a + b*x^4)^(1/2)),x)
Output:
int((c + d*x + e*x^2 + f*x^3)/(x^4*(a + b*x^4)^(1/2)), x)
\[ \int \frac {c+d x+e x^2+f x^3}{x^4 \sqrt {a+b x^4}} \, dx=\frac {\sqrt {a}\, \sqrt {b \,x^{4}+a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {a}\right ) f \,x^{2}-\sqrt {a}\, \sqrt {b \,x^{4}+a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {a}\right ) f \,x^{2}-4 \sqrt {b}\, \sqrt {b \,x^{4}+a}\, d \,x^{2}+4 \sqrt {b \,x^{4}+a}\, \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{8}+a \,x^{4}}d x \right ) a c \,x^{2}+4 \sqrt {b \,x^{4}+a}\, \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{6}+a \,x^{2}}d x \right ) a e \,x^{2}+\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {a}\right ) f \,x^{4}-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {a}\right ) f \,x^{4}+4 \sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{8}+a \,x^{4}}d x \right ) a c \,x^{4}+4 \sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{6}+a \,x^{2}}d x \right ) a e \,x^{4}-2 a d -4 b d \,x^{4}}{4 a \,x^{2} \left (\sqrt {b \,x^{4}+a}+\sqrt {b}\, x^{2}\right )} \] Input:
int((f*x^3+e*x^2+d*x+c)/x^4/(b*x^4+a)^(1/2),x)
Output:
(sqrt(a)*sqrt(a + b*x**4)*log(sqrt(a + b*x**4) - sqrt(a))*f*x**2 - sqrt(a) *sqrt(a + b*x**4)*log(sqrt(a + b*x**4) + sqrt(a))*f*x**2 - 4*sqrt(b)*sqrt( a + b*x**4)*d*x**2 + 4*sqrt(a + b*x**4)*int(sqrt(a + b*x**4)/(a*x**4 + b*x **8),x)*a*c*x**2 + 4*sqrt(a + b*x**4)*int(sqrt(a + b*x**4)/(a*x**2 + b*x** 6),x)*a*e*x**2 + sqrt(b)*sqrt(a)*log(sqrt(a + b*x**4) - sqrt(a))*f*x**4 - sqrt(b)*sqrt(a)*log(sqrt(a + b*x**4) + sqrt(a))*f*x**4 + 4*sqrt(b)*int(sqr t(a + b*x**4)/(a*x**4 + b*x**8),x)*a*c*x**4 + 4*sqrt(b)*int(sqrt(a + b*x** 4)/(a*x**2 + b*x**6),x)*a*e*x**4 - 2*a*d - 4*b*d*x**4)/(4*a*x**2*(sqrt(a + b*x**4) + sqrt(b)*x**2))