Integrand size = 30, antiderivative size = 352 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^7} \, dx=-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}\right ) \sqrt {a+b x^4}-\frac {b c \sqrt {a+b x^4}}{6 a x^2}-\frac {2 b d \sqrt {a+b x^4}}{5 a x}+\frac {2 b^{3/2} d x \sqrt {a+b x^4}}{5 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {b e \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {2 b^{5/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{3/4} \sqrt {a+b x^4}}+\frac {b^{3/4} \left (3 \sqrt {b} d+5 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}} \]
-1/4*b*e*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(1/2)-1/60*(10*c/x^6+12*d/x^5+ 15*e/x^4+20*f/x^3)*(b*x^4+a)^(1/2)-1/6*b*c*(b*x^4+a)^(1/2)/a/x^2-2/5*b*d*( b*x^4+a)^(1/2)/a/x+2/5*b^(3/2)*d*x*(b*x^4+a)^(1/2)/a/(a^(1/2)+x^2*b^(1/2)) -2/5*b^(5/4)*d*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^( 1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*( a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(3/4)/(b* x^4+a)^(1/2)+1/15*b^(3/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2 *arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2 *2^(1/2))*(5*f*a^(1/2)+3*d*b^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1 /2)+x^2*b^(1/2))^2)^(1/2)/a^(3/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.41 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^7} \, dx=-\frac {\sqrt {a+b x^4} \left (12 a d x \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{2},-\frac {1}{4},-\frac {b x^4}{a}\right )+5 \left (\sqrt {1+\frac {b x^4}{a}} \left (2 a c+3 a e x^2+2 b c x^4\right )+3 b e x^6 \text {arctanh}\left (\sqrt {1+\frac {b x^4}{a}}\right )+4 a f x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{4},-\frac {b x^4}{a}\right )\right )\right )}{60 a x^6 \sqrt {1+\frac {b x^4}{a}}} \]
-1/60*(Sqrt[a + b*x^4]*(12*a*d*x*Hypergeometric2F1[-5/4, -1/2, -1/4, -((b* x^4)/a)] + 5*(Sqrt[1 + (b*x^4)/a]*(2*a*c + 3*a*e*x^2 + 2*b*c*x^4) + 3*b*e* x^6*ArcTanh[Sqrt[1 + (b*x^4)/a]] + 4*a*f*x^3*Hypergeometric2F1[-3/4, -1/2, 1/4, -((b*x^4)/a)])))/(a*x^6*Sqrt[1 + (b*x^4)/a])
Time = 0.62 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^4} \left (c+d x+e x^2+f x^3\right )}{x^7} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -2 b \int -\frac {20 f x^3+15 e x^2+12 d x+10 c}{60 x^3 \sqrt {b x^4+a}}dx-\frac {1}{60} \sqrt {a+b x^4} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} b \int \frac {20 f x^3+15 e x^2+12 d x+10 c}{x^3 \sqrt {b x^4+a}}dx-\frac {1}{60} \sqrt {a+b x^4} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}\right )\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{30} b \int \left (\frac {15 e x^2+10 c}{x^3 \sqrt {b x^4+a}}+\frac {20 f x^2+12 d}{x^2 \sqrt {b x^4+a}}\right )dx-\frac {1}{60} \sqrt {a+b x^4} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{30} b \left (\frac {2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} f+3 \sqrt {b} d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{a^{3/4} \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {12 \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {15 e \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {5 c \sqrt {a+b x^4}}{a x^2}-\frac {12 d \sqrt {a+b x^4}}{a x}+\frac {12 \sqrt {b} d x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}\right )-\frac {1}{60} \sqrt {a+b x^4} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}\right )\) |
-1/60*(((10*c)/x^6 + (12*d)/x^5 + (15*e)/x^4 + (20*f)/x^3)*Sqrt[a + b*x^4] ) + (b*((-5*c*Sqrt[a + b*x^4])/(a*x^2) - (12*d*Sqrt[a + b*x^4])/(a*x) + (1 2*Sqrt[b]*d*x*Sqrt[a + b*x^4])/(a*(Sqrt[a] + Sqrt[b]*x^2)) - (15*e*ArcTanh [Sqrt[a + b*x^4]/Sqrt[a]])/(2*Sqrt[a]) - (12*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]) + (2*(3*Sqrt[b]*d + 5*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^(3/4)*b^(1/4)*Sqrt[a + b*x^4])))/30
3.6.6.3.1 Defintions of rubi rules used
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 = 2.11 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (24 b d \,x^{5}+10 b c \,x^{4}+20 a f \,x^{3}+15 a e \,x^{2}+12 a d x +10 a c \right )}{60 x^{6} a}+\frac {b \left (\frac {20 a f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {12 i \sqrt {b}\, d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {15 \sqrt {a}\, e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}\right )}{30 a}\) | \(262\) |
| elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {d \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {e \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {f \sqrt {b \,x^{4}+a}}{3 x^{3}}-\frac {b c \sqrt {b \,x^{4}+a}}{6 a \,x^{2}}-\frac {2 b d \sqrt {b \,x^{4}+a}}{5 a x}+\frac {2 b f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {2 i d \,b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {b e \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4 \sqrt {a}}\) | \(290\) |
| default | \(-\frac {c \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}+f \left (-\frac {\sqrt {b \,x^{4}+a}}{3 x^{3}}+\frac {2 b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+e \left (-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4 \sqrt {a}}+\frac {b \sqrt {b \,x^{4}+a}}{4 a}\right )+d \left (-\frac {\sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {2 b \sqrt {b \,x^{4}+a}}{5 a x}+\frac {2 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(303\) |
-1/60*(b*x^4+a)^(1/2)*(24*b*d*x^5+10*b*c*x^4+20*a*f*x^3+15*a*e*x^2+12*a*d* x+10*a*c)/x^6/a+1/30*b/a*(20*a*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)+12*I*b^(1/2)*d*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))-15/2*a^(1/2)*e*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x ^2))
Time = 0.13 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.47 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^7} \, dx=-\frac {48 \, \sqrt {a} b d x^{6} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 15 \, \sqrt {a} b e x^{6} \log \left (-\frac {b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 16 \, {\left (3 \, b d - 5 \, a f\right )} \sqrt {a} x^{6} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (24 \, b d x^{5} + 10 \, b c x^{4} + 20 \, a f x^{3} + 15 \, a e x^{2} + 12 \, a d x + 10 \, a c\right )} \sqrt {b x^{4} + a}}{120 \, a x^{6}} \]
-1/120*(48*sqrt(a)*b*d*x^6*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1) - 15*sqrt(a)*b*e*x^6*log(-(b*x^4 - 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x ^4) - 16*(3*b*d - 5*a*f)*sqrt(a)*x^6*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/ a)^(1/4)), -1) + 2*(24*b*d*x^5 + 10*b*c*x^4 + 20*a*f*x^3 + 15*a*e*x^2 + 12 *a*d*x + 10*a*c)*sqrt(b*x^4 + a))/(a*x^6)
Result contains complex when optimal does not.
Time = 2.72 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.54 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^7} \, dx=\frac {\sqrt {a} d \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {\sqrt {a} f \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {\sqrt {b} c \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {\sqrt {b} e \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} - \frac {b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{4}} + 1}}{6 a} - \frac {b e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 \sqrt {a}} \]
sqrt(a)*d*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(a)*f*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,) , b*x**4*exp_polar(I*pi)/a)/(4*x**3*gamma(1/4)) - sqrt(b)*c*sqrt(a/(b*x**4 ) + 1)/(6*x**4) - sqrt(b)*e*sqrt(a/(b*x**4) + 1)/(4*x**2) - b**(3/2)*c*sqr t(a/(b*x**4) + 1)/(6*a) - b*e*asinh(sqrt(a)/(sqrt(b)*x**2))/(4*sqrt(a))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^7} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{7}} \,d x } \]
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^7} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^7} \, dx=\int \frac {\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^7} \,d x \]