Integrand size = 30, antiderivative size = 280 \[ \int \frac {c+d x+e x^2+f x^3}{x^3 \sqrt {a+b x^4}} \, dx=-\frac {c \sqrt {a+b x^4}}{2 a x^2}-\frac {d \sqrt {a+b x^4}}{\sqrt {a} x \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {e \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} d+\sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt [4]{b} \sqrt {a+b x^4}} \] Output:
-1/2*c*(b*x^4+a)^(1/2)/a/x^2-d*(b*x^4+a)^(1/2)/a^(1/2)/x/(a^(1/2)+b^(1/2)* x^2)-1/2*e*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(1/2)-b^(1/4)*d*(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)+1/2*(b^(1/2)*d+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^(3/4)/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 = 10.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.53 \[ \int \frac {c+d x+e x^2+f x^3}{x^3 \sqrt {a+b x^4}} \, dx=-\frac {c \sqrt {a+b x^4}}{2 a x^2}-\frac {e \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {d \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {b x^4}{a}\right )}{x \sqrt {a+b x^4}}+\frac {f x \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )}{\sqrt {a+b x^4}} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3)/(x^3*Sqrt[a + b*x^4]),x]
Output:
-1/2*(c*Sqrt[a + b*x^4])/(a*x^2) - (e*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(2 *Sqrt[a]) - (d*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((b* x^4)/a)])/(x*Sqrt[a + b*x^4]) + (f*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1 [1/4, 1/2, 5/4, -((b*x^4)/a)])/Sqrt[a + b*x^4]
Time = 0.80 (sec) , antiderivative size = 300, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3}{x^3 \sqrt {a+b x^4}} \, dx\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \int \left (\frac {c+e x^2}{x^3 \sqrt {a+b x^4}}+\frac {d+f x^2}{x^2 \sqrt {a+b x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} f+\sqrt {b} d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {e \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {c \sqrt {a+b x^4}}{2 a x^2}-\frac {d \sqrt {a+b x^4}}{a x}+\frac {\sqrt {b} d x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3)/(x^3*Sqrt[a + b*x^4]),x]
Output:
-1/2*(c*Sqrt[a + b*x^4])/(a*x^2) - (d*Sqrt[a + b*x^4])/(a*x) + (Sqrt[b]*d* x*Sqrt[a + b*x^4])/(a*(Sqrt[a] + Sqrt[b]*x^2)) - (e*ArcTanh[Sqrt[a + b*x^4 ]/Sqrt[a]])/(2*Sqrt[a]) - (b^(1/4)*d*(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]) + ((Sqrt[b]*d + 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])/(2*a^(3/4)*b^(1/4)*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 = 1.22 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.80
| method | result | size |
| elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{2 a \,x^{2}}-\frac {d \sqrt {b \,x^{4}+a}}{a x}+\frac {f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {i d \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {e \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{2 \sqrt {a}}\) | \(225\) |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (2 d x +c \right )}{2 a \,x^{2}}+\frac {\frac {a f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\sqrt {a}\, e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}+\frac {i \sqrt {b}\, d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}}{a}\) | \(227\) |
| default | \(\frac {f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {c \sqrt {b \,x^{4}+a}}{2 a \,x^{2}}+d \left (-\frac {\sqrt {b \,x^{4}+a}}{a x}+\frac {i \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2 \sqrt {a}}\) | \(235\) |
Input:
int((f*x^3+e*x^2+d*x+c)/x^3/(b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*c*(b*x^4+a)^(1/2)/a/x^2-d/a*(b*x^4+a)^(1/2)/x+f/(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)+I/a^(1/2)*d*b^(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)-El lipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))-1/2*e/a^(1/2)*arctanh(a^(1/2)/(b*x ^4+a)^(1/2))
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.50 \[ \int \frac {c+d x+e x^2+f x^3}{x^3 \sqrt {a+b x^4}} \, dx=-\frac {4 \, \sqrt {a} b d x^{2} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - \sqrt {a} b e x^{2} \log \left (-\frac {b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 4 \, {\left (b d - a f\right )} \sqrt {a} x^{2} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, \sqrt {b x^{4} + a} {\left (2 \, b d x + b c\right )}}{4 \, a b x^{2}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^3/(b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
-1/4*(4*sqrt(a)*b*d*x^2*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1 ) - sqrt(a)*b*e*x^2*log(-(b*x^4 - 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x^4) - 4*(b*d - a*f)*sqrt(a)*x^2*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + 2*sqrt(b*x^4 + a)*(2*b*d*x + b*c))/(a*b*x^2)
Result contains complex when optimal does not.
Time = 1.64 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.45 \[ \int \frac {c+d x+e x^2+f x^3}{x^3 \sqrt {a+b x^4}} \, dx=- \frac {\sqrt {b} c \sqrt {\frac {a}{b x^{4}} + 1}}{2 a} + \frac {d \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x \Gamma \left (\frac {3}{4}\right )} - \frac {e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{2 \sqrt {a}} + \frac {f x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((f*x**3+e*x**2+d*x+c)/x**3/(b*x**4+a)**(1/2),x)
Output:
-sqrt(b)*c*sqrt(a/(b*x**4) + 1)/(2*a) + d*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)) - e*asinh(sqrt(a )/(sqrt(b)*x**2))/(2*sqrt(a)) + f*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b *x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(5/4))
\[ \int \frac {c+d x+e x^2+f x^3}{x^3 \sqrt {a+b x^4}} \, dx=\int { \frac {f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{4} + a} x^{3}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^3/(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^3), x)
\[ \int \frac {c+d x+e x^2+f x^3}{x^3 \sqrt {a+b x^4}} \, dx=\int { \frac {f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{4} + a} x^{3}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)/x^3/(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^3), x)
Time = 7.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.42 \[ \int \frac {c+d x+e x^2+f x^3}{x^3 \sqrt {a+b x^4}} \, dx=\frac {f\,x\,\sqrt {\frac {b\,x^4}{a}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ -\frac {b\,x^4}{a}\right )}{\sqrt {b\,x^4+a}}-\frac {c\,\sqrt {b\,x^4+a}}{2\,a\,x^2}-\frac {d\,\sqrt {\frac {a}{b\,x^4}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{4};\ \frac {7}{4};\ -\frac {a}{b\,x^4}\right )}{3\,x\,\sqrt {b\,x^4+a}}-\frac {e\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^4+a}}{\sqrt {a}}\right )}{2\,\sqrt {a}} \] Input:
int((c + d*x + e*x^2 + f*x^3)/(x^3*(a + b*x^4)^(1/2)),x)
Output:
(f*x*((b*x^4)/a + 1)^(1/2)*hypergeom([1/4, 1/2], 5/4, -(b*x^4)/a))/(a + b* x^4)^(1/2) - (c*(a + b*x^4)^(1/2))/(2*a*x^2) - (d*(a/(b*x^4) + 1)^(1/2)*hy pergeom([1/2, 3/4], 7/4, -a/(b*x^4)))/(3*x*(a + b*x^4)^(1/2)) - (e*atanh(( a + b*x^4)^(1/2)/a^(1/2)))/(2*a^(1/2))
\[ \int \frac {c+d x+e x^2+f x^3}{x^3 \sqrt {a+b x^4}} \, dx=\frac {\sqrt {a}\, \sqrt {b \,x^{4}+a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {a}\right ) e \,x^{2}-\sqrt {a}\, \sqrt {b \,x^{4}+a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {a}\right ) e \,x^{2}-4 \sqrt {b}\, \sqrt {b \,x^{4}+a}\, c \,x^{2}+4 \sqrt {b \,x^{4}+a}\, \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{6}+a \,x^{2}}d x \right ) a d \,x^{2}+4 \sqrt {b \,x^{4}+a}\, \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{4}+a}d x \right ) a f \,x^{2}+\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {a}\right ) e \,x^{4}-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {a}\right ) e \,x^{4}+4 \sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{6}+a \,x^{2}}d x \right ) a d \,x^{4}+4 \sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{4}+a}d x \right ) a f \,x^{4}-2 a c -4 b c \,x^{4}}{4 a \,x^{2} \left (\sqrt {b \,x^{4}+a}+\sqrt {b}\, x^{2}\right )} \] Input:
int((f*x^3+e*x^2+d*x+c)/x^3/(b*x^4+a)^(1/2),x)
Output:
(sqrt(a)*sqrt(a + b*x**4)*log(sqrt(a + b*x**4) - sqrt(a))*e*x**2 - sqrt(a) *sqrt(a + b*x**4)*log(sqrt(a + b*x**4) + sqrt(a))*e*x**2 - 4*sqrt(b)*sqrt( a + b*x**4)*c*x**2 + 4*sqrt(a + b*x**4)*int(sqrt(a + b*x**4)/(a*x**2 + b*x **6),x)*a*d*x**2 + 4*sqrt(a + b*x**4)*int(sqrt(a + b*x**4)/(a + b*x**4),x) *a*f*x**2 + sqrt(b)*sqrt(a)*log(sqrt(a + b*x**4) - sqrt(a))*e*x**4 - sqrt( b)*sqrt(a)*log(sqrt(a + b*x**4) + sqrt(a))*e*x**4 + 4*sqrt(b)*int(sqrt(a + b*x**4)/(a*x**2 + b*x**6),x)*a*d*x**4 + 4*sqrt(b)*int(sqrt(a + b*x**4)/(a + b*x**4),x)*a*f*x**4 - 2*a*c - 4*b*c*x**4)/(4*a*x**2*(sqrt(a + b*x**4) + sqrt(b)*x**2))