Integrand size = 30, antiderivative size = 418 \[ \int x^4 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}-\frac {2 a^2 e x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac {\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}-\frac {a^2 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}+\frac {2 a^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^{7/4} \left (5 \sqrt {b} c+7 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 b^{7/4} \sqrt {a+b x^4}} \]
1/10*f*x^4*(b*x^4+a)^(3/2)/b-1/120*(-15*b*d*x^2+8*a*f)*(b*x^4+a)^(3/2)/b^2 -1/16*a^2*d*arctanh(x^2*b^(1/2)/(b*x^4+a)^(1/2))/b^(3/2)+2/21*a*c*x*(b*x^4 +a)^(1/2)/b-1/16*a*d*x^2*(b*x^4+a)^(1/2)/b+2/45*a*e*x^3*(b*x^4+a)^(1/2)/b+ 1/63*x^5*(7*e*x^2+9*c)*(b*x^4+a)^(1/2)-2/15*a^2*e*x*(b*x^4+a)^(1/2)/b^(3/2 )/(a^(1/2)+x^2*b^(1/2))+2/15*a^(9/4)*e*(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)/b^(7/4)/(b*x^4+a)^(1/2)-1/105*a^(7/4)*(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*arcta n(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(7*e*a^(1/2)+5*c*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)/b^(7/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.66 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.48 \[ \int x^4 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {\sqrt {a+b x^4} \left (720 b c x \left (a+b x^4\right )+560 b e x^3 \left (a+b x^4\right )+315 b d x^2 \left (a+2 b x^4\right )+168 f \left (a+b x^4\right ) \left (-2 a+3 b x^4\right )-\frac {315 a^{3/2} \sqrt {b} d \text {arcsinh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {1+\frac {b x^4}{a}}}-\frac {720 a b c x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}-\frac {560 a b e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}\right )}{5040 b^2} \]
(Sqrt[a + b*x^4]*(720*b*c*x*(a + b*x^4) + 560*b*e*x^3*(a + b*x^4) + 315*b* d*x^2*(a + 2*b*x^4) + 168*f*(a + b*x^4)*(-2*a + 3*b*x^4) - (315*a^(3/2)*Sq rt[b]*d*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[1 + (b*x^4)/a] - (720*a*b*c*x *Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a] - (5 60*a*b*e*x^3*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b* x^4)/a]))/(5040*b^2)
Time = 0.66 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.03, 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.
| \(\displaystyle \int x^4 \sqrt {a+b x^4} \left (c+d x+e x^2+f x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \int \left (x^4 \sqrt {a+b x^4} \left (c+e x^2\right )+x^5 \sqrt {a+b x^4} \left (d+f x^2\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (7 \sqrt {a} e+5 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 b^{7/4} \sqrt {a+b x^4}}+\frac {2 a^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^2 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}-\frac {2 a^2 e x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {a f \left (a+b x^4\right )^{3/2}}{15 b^2}+\frac {1}{63} x^5 \sqrt {a+b x^4} \left (9 c+7 e x^2\right )+\frac {2 a c x \sqrt {a+b x^4}}{21 b}+\frac {d x^2 \left (a+b x^4\right )^{3/2}}{8 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}\) |
(2*a*c*x*Sqrt[a + b*x^4])/(21*b) - (a*d*x^2*Sqrt[a + b*x^4])/(16*b) + (2*a *e*x^3*Sqrt[a + b*x^4])/(45*b) - (2*a^2*e*x*Sqrt[a + b*x^4])/(15*b^(3/2)*( Sqrt[a] + Sqrt[b]*x^2)) + (x^5*(9*c + 7*e*x^2)*Sqrt[a + b*x^4])/63 - (a*f* (a + b*x^4)^(3/2))/(15*b^2) + (d*x^2*(a + b*x^4)^(3/2))/(8*b) + (f*x^4*(a + b*x^4)^(3/2))/(10*b) - (a^2*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(1 6*b^(3/2)) + (2*a^(9/4)*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])/(15*b^( 7/4)*Sqrt[a + b*x^4]) - (a^(7/4)*(5*Sqrt[b]*c + 7*Sqrt[a]*e)*(Sqrt[a] + Sq rt[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])/(105*b^(7/4)*Sqrt[a + b*x^4])
3.5.95.3.1 Defintions of rubi rules used
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 = 2.04 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(-\frac {\left (-504 b^{2} f \,x^{8}-560 b^{2} e \,x^{7}-630 b^{2} d \,x^{6}-720 b^{2} c \,x^{5}-168 a b f \,x^{4}-224 a b e \,x^{3}-315 x^{2} a b d -480 a b c x +336 a^{2} f \right ) \sqrt {b \,x^{4}+a}}{5040 b^{2}}-\frac {a^{2} \left (\frac {80 c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {112 i e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {105 d \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2 \sqrt {b}}\right )}{840 b}\) | \(290\) |
| default | \(-\frac {f \left (b \,x^{4}+a \right )^{\frac {3}{2}} \left (-3 b \,x^{4}+2 a \right )}{30 b^{2}}+e \left (\frac {x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {2 a \,x^{3} \sqrt {b \,x^{4}+a}}{45 b}-\frac {2 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {x^{2} \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{8 b}-\frac {a \,x^{2} \sqrt {b \,x^{4}+a}}{16 b}-\frac {a^{2} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{16 b^{\frac {3}{2}}}\right )+c \left (\frac {x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {2 a x \sqrt {b \,x^{4}+a}}{21 b}-\frac {2 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(331\) |
| elliptic | \(\frac {f \,x^{8} \sqrt {b \,x^{4}+a}}{10}+\frac {e \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {d \,x^{6} \sqrt {b \,x^{4}+a}}{8}+\frac {c \,x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {a f \,x^{4} \sqrt {b \,x^{4}+a}}{30 b}+\frac {2 a e \,x^{3} \sqrt {b \,x^{4}+a}}{45 b}+\frac {a d \,x^{2} \sqrt {b \,x^{4}+a}}{16 b}+\frac {2 a c x \sqrt {b \,x^{4}+a}}{21 b}-\frac {a^{2} f \sqrt {b \,x^{4}+a}}{15 b^{2}}-\frac {2 a^{2} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {a^{2} d \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{16 b^{\frac {3}{2}}}-\frac {2 i a^{\frac {5}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(358\) |
-1/5040*(-504*b^2*f*x^8-560*b^2*e*x^7-630*b^2*d*x^6-720*b^2*c*x^5-168*a*b* f*x^4-224*a*b*e*x^3-315*a*b*d*x^2-480*a*b*c*x+336*a^2*f)/b^2*(b*x^4+a)^(1/ 2)-1/840*a^2/b*(80*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)+112*I*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)/b^(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))+105/2*d*ln(x^2*b^(1/2)+(b*x^4+a)^(1/2))/b^(1/2))
Time = 0.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.51 \[ \int x^4 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=-\frac {1344 \, a^{2} \sqrt {b} e x \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 315 \, a^{2} \sqrt {b} d x \log \left (-2 \, b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right ) + 192 \, {\left (5 \, a b c - 7 \, a^{2} e\right )} \sqrt {b} x \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 2 \, {\left (504 \, b^{2} f x^{9} + 560 \, b^{2} e x^{8} + 630 \, b^{2} d x^{7} + 720 \, b^{2} c x^{6} + 168 \, a b f x^{5} + 224 \, a b e x^{4} + 315 \, a b d x^{3} + 480 \, a b c x^{2} - 336 \, a^{2} f x - 672 \, a^{2} e\right )} \sqrt {b x^{4} + a}}{10080 \, b^{2} x} \]
-1/10080*(1344*a^2*sqrt(b)*e*x*(-a/b)^(3/4)*elliptic_e(arcsin((-a/b)^(1/4) /x), -1) - 315*a^2*sqrt(b)*d*x*log(-2*b*x^4 + 2*sqrt(b*x^4 + a)*sqrt(b)*x^ 2 - a) + 192*(5*a*b*c - 7*a^2*e)*sqrt(b)*x*(-a/b)^(3/4)*elliptic_f(arcsin( (-a/b)^(1/4)/x), -1) - 2*(504*b^2*f*x^9 + 560*b^2*e*x^8 + 630*b^2*d*x^7 + 720*b^2*c*x^6 + 168*a*b*f*x^5 + 224*a*b*e*x^4 + 315*a*b*d*x^3 + 480*a*b*c* x^2 - 336*a^2*f*x - 672*a^2*e)*sqrt(b*x^4 + a))/(b^2*x)
Time = 3.38 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.60 \[ \int x^4 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {a^{\frac {3}{2}} d x^{2}}{16 b \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} c x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {3 \sqrt {a} d x^{6}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} - \frac {a^{2} d \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{16 b^{\frac {3}{2}}} + f \left (\begin {cases} - \frac {a^{2} \sqrt {a + b x^{4}}}{15 b^{2}} + \frac {a x^{4} \sqrt {a + b x^{4}}}{30 b} + \frac {x^{8} \sqrt {a + b x^{4}}}{10} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) + \frac {b d x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
a**(3/2)*d*x**2/(16*b*sqrt(1 + b*x**4/a)) + sqrt(a)*c*x**5*gamma(5/4)*hype r((-1/2, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4)) + 3*sqrt(a )*d*x**6/(16*sqrt(1 + b*x**4/a)) + sqrt(a)*e*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(11/4)) - a**2*d*asinh(sq rt(b)*x**2/sqrt(a))/(16*b**(3/2)) + f*Piecewise((-a**2*sqrt(a + b*x**4)/(1 5*b**2) + a*x**4*sqrt(a + b*x**4)/(30*b) + x**8*sqrt(a + b*x**4)/10, Ne(b, 0)), (sqrt(a)*x**8/8, True)) + b*d*x**10/(8*sqrt(a)*sqrt(1 + b*x**4/a))
\[ \int x^4 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\int { \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{4} \,d x } \]
\[ \int x^4 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\int { \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{4} \,d x } \]
Timed out. \[ \int x^4 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\int x^4\,\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]