Integrand size = 28, antiderivative size = 356 \[ \int x \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {1}{4} c x^2 \sqrt {a+b x^4}+\frac {2 a d x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {x \left (5 a f-21 b d x^2\right ) \sqrt {a+b x^4}}{105 b}+\frac {e \left (a+b x^4\right )^{3/2}}{6 b}+\frac {f x \left (a+b x^4\right )^{3/2}}{7 b}+\frac {a c \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{4 \sqrt {b}}-\frac {2 a^{5/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a+b x^4}}+\frac {a^{5/4} \left (21 \sqrt {b} d-5 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 b^{5/4} \sqrt {a+b x^4}} \] Output:
1/4*c*x^2*(b*x^4+a)^(1/2)+2/5*a*d*x*(b*x^4+a)^(1/2)/b^(1/2)/(a^(1/2)+b^(1/ 2)*x^2)-1/105*x*(-21*b*d*x^2+5*a*f)*(b*x^4+a)^(1/2)/b+1/6*e*(b*x^4+a)^(3/2 )/b+1/7*f*x*(b*x^4+a)^(3/2)/b+1/4*a*c*arctanh(b^(1/2)*x^2/(b*x^4+a)^(1/2)) /b^(1/2)-2/5*a^(5/4)*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))/b^(3/ 4)/(b*x^4+a)^(1/2)+1/105*a^(5/4)*(21*b^(1/2)*d-5*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))/b^(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 = 211, normalized size of antiderivative = 0.59 \[ \int x \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {\sqrt {a+b x^4} \left (14 a e \sqrt {1+\frac {b x^4}{a}}+12 a f x \sqrt {1+\frac {b x^4}{a}}+21 b c x^2 \sqrt {1+\frac {b x^4}{a}}+14 b e x^4 \sqrt {1+\frac {b x^4}{a}}+12 b f x^5 \sqrt {1+\frac {b x^4}{a}}+21 \sqrt {a} \sqrt {b} c \text {arcsinh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )-12 a f x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )+28 b d x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )\right )}{84 b \sqrt {1+\frac {b x^4}{a}}} \] Input:
Integrate[x*(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]
Output:
(Sqrt[a + b*x^4]*(14*a*e*Sqrt[1 + (b*x^4)/a] + 12*a*f*x*Sqrt[1 + (b*x^4)/a ] + 21*b*c*x^2*Sqrt[1 + (b*x^4)/a] + 14*b*e*x^4*Sqrt[1 + (b*x^4)/a] + 12*b *f*x^5*Sqrt[1 + (b*x^4)/a] + 21*Sqrt[a]*Sqrt[b]*c*ArcSinh[(Sqrt[b]*x^2)/Sq rt[a]] - 12*a*f*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)] + 28*b*d *x^3*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^4)/a)]))/(84*b*Sqrt[1 + (b*x ^4)/a])
Time = 0.88 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
| \(\displaystyle \int x \sqrt {a+b x^4} \left (c+d x+e x^2+f x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \int \left (x \sqrt {a+b x^4} \left (c+e x^2\right )+x^2 \sqrt {a+b x^4} \left (d+f x^2\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (21 \sqrt {b} d-5 \sqrt {a} f\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 b^{5/4} \sqrt {a+b x^4}}-\frac {2 a^{5/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a+b x^4}}+\frac {a c \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{4 \sqrt {b}}+\frac {1}{4} c x^2 \sqrt {a+b x^4}+\frac {1}{35} x^3 \sqrt {a+b x^4} \left (7 d+5 f x^2\right )+\frac {2 a d x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {e \left (a+b x^4\right )^{3/2}}{6 b}+\frac {2 a f x \sqrt {a+b x^4}}{21 b}\) |
Input:
Int[x*(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]
Output:
(2*a*f*x*Sqrt[a + b*x^4])/(21*b) + (c*x^2*Sqrt[a + b*x^4])/4 + (2*a*d*x*Sq rt[a + b*x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (x^3*(7*d + 5*f*x^2)* Sqrt[a + b*x^4])/35 + (e*(a + b*x^4)^(3/2))/(6*b) + (a*c*ArcTanh[(Sqrt[b]* x^2)/Sqrt[a + b*x^4]])/(4*Sqrt[b]) - (2*a^(5/4)*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])/(5*b^(3/4)*Sqrt[a + b*x^4]) + (a^(5/4)*(21*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])/(105*b^(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 = 1.57 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(\frac {\left (60 b f \,x^{5}+70 b e \,x^{4}+84 b d \,x^{3}+105 b c \,x^{2}+40 a f x +70 a e \right ) \sqrt {b \,x^{4}+a}}{420 b}-\frac {a \left (\frac {20 a f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {105 c \sqrt {b}\, \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{2}-\frac {84 i \sqrt {b}\, d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )}{210 b}\) | \(254\) |
| default | \(c \left (\frac {x^{2} \sqrt {b \,x^{4}+a}}{4}+\frac {a \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{4 \sqrt {b}}\right )+d \left (\frac {x^{3} \sqrt {b \,x^{4}+a}}{5}+\frac {2 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )+f \left (\frac {x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {2 a x \sqrt {b \,x^{4}+a}}{21 b}-\frac {2 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+\frac {e \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{6 b}\) | \(280\) |
| elliptic | \(\frac {x^{5} f \sqrt {b \,x^{4}+a}}{7}+\frac {e \,x^{4} \sqrt {b \,x^{4}+a}}{6}+\frac {d \,x^{3} \sqrt {b \,x^{4}+a}}{5}+\frac {c \,x^{2} \sqrt {b \,x^{4}+a}}{4}+\frac {2 a f x \sqrt {b \,x^{4}+a}}{21 b}+\frac {a e \sqrt {b \,x^{4}+a}}{6 b}-\frac {2 a^{2} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {a c \ln \left (2 \sqrt {b}\, x^{2}+2 \sqrt {b \,x^{4}+a}\right )}{4 \sqrt {b}}+\frac {2 i a^{\frac {3}{2}} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\) | \(297\) |
Input:
int(x*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/420*(60*b*f*x^5+70*b*e*x^4+84*b*d*x^3+105*b*c*x^2+40*a*f*x+70*a*e)/b*(b* x^4+a)^(1/2)-1/210*a/b*(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)-105/2*c*b^(1/2)*ln(b^(1/2)*x^2+(b*x^4+a)^(1/ 2))-84*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)))
Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.48 \[ \int x \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {336 \, a \sqrt {b} d x \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 105 \, a \sqrt {b} c x \log \left (-2 \, b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right ) - 16 \, {\left (21 \, a d + 5 \, a f\right )} \sqrt {b} x \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 2 \, {\left (60 \, b f x^{6} + 70 \, b e x^{5} + 84 \, b d x^{4} + 105 \, b c x^{3} + 40 \, a f x^{2} + 70 \, a e x + 168 \, a d\right )} \sqrt {b x^{4} + a}}{840 \, b x} \] Input:
integrate(x*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
1/840*(336*a*sqrt(b)*d*x*(-a/b)^(3/4)*elliptic_e(arcsin((-a/b)^(1/4)/x), - 1) + 105*a*sqrt(b)*c*x*log(-2*b*x^4 - 2*sqrt(b*x^4 + a)*sqrt(b)*x^2 - a) - 16*(21*a*d + 5*a*f)*sqrt(b)*x*(-a/b)^(3/4)*elliptic_f(arcsin((-a/b)^(1/4) /x), -1) + 2*(60*b*f*x^6 + 70*b*e*x^5 + 84*b*d*x^4 + 105*b*c*x^3 + 40*a*f* x^2 + 70*a*e*x + 168*a*d)*sqrt(b*x^4 + a))/(b*x)
Time = 2.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.44 \[ \int x \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {\sqrt {a} c x^{2} \sqrt {1 + \frac {b x^{4}}{a}}}{4} + \frac {\sqrt {a} d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt {a} f x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {a c \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{4 \sqrt {b}} + e \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) \] Input:
integrate(x*(f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(1/2),x)
Output:
sqrt(a)*c*x**2*sqrt(1 + b*x**4/a)/4 + sqrt(a)*d*x**3*gamma(3/4)*hyper((-1/ 2, 3/4), (7/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(7/4)) + sqrt(a)*f*x**5 *gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma( 9/4)) + a*c*asinh(sqrt(b)*x**2/sqrt(a))/(4*sqrt(b)) + e*Piecewise((sqrt(a) *x**4/4, Eq(b, 0)), ((a + b*x**4)**(3/2)/(6*b), True))
\[ \int x \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\int { \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x \,d x } \] Input:
integrate(x*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2),x, algorithm="maxima")
Output:
-1/8*(a*log(-(sqrt(b) - sqrt(b*x^4 + a)/x^2)/(sqrt(b) + sqrt(b*x^4 + a)/x^ 2))/sqrt(b) + 2*sqrt(b*x^4 + a)*a/((b - (b*x^4 + a)/x^4)*x^2))*c + integra te(sqrt(b*x^4 + a)*(f*x^4 + e*x^3 + d*x^2), x)
\[ \int x \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\int { \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x \,d x } \] Input:
integrate(x*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)*x, x)
Timed out. \[ \int x \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\int x\,\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \] Input:
int(x*(a + b*x^4)^(1/2)*(c + d*x + e*x^2 + f*x^3),x)
Output:
int(x*(a + b*x^4)^(1/2)*(c + d*x + e*x^2 + f*x^3), x)
\[ \int x \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {140 \sqrt {b \,x^{4}+a}\, a e +80 \sqrt {b \,x^{4}+a}\, a f x +210 \sqrt {b \,x^{4}+a}\, b c \,x^{2}+168 \sqrt {b \,x^{4}+a}\, b d \,x^{3}+140 \sqrt {b \,x^{4}+a}\, b e \,x^{4}+120 \sqrt {b \,x^{4}+a}\, b f \,x^{5}-105 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {b}\, x^{2}\right ) a c +105 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {b}\, x^{2}\right ) a c -80 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{4}+a}d x \right ) a^{2} f +336 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b \,x^{4}+a}d x \right ) a b d}{840 b} \] Input:
int(x*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2),x)
Output:
(140*sqrt(a + b*x**4)*a*e + 80*sqrt(a + b*x**4)*a*f*x + 210*sqrt(a + b*x** 4)*b*c*x**2 + 168*sqrt(a + b*x**4)*b*d*x**3 + 140*sqrt(a + b*x**4)*b*e*x** 4 + 120*sqrt(a + b*x**4)*b*f*x**5 - 105*sqrt(b)*log(sqrt(a + b*x**4) - sqr t(b)*x**2)*a*c + 105*sqrt(b)*log(sqrt(a + b*x**4) + sqrt(b)*x**2)*a*c - 80 *int(sqrt(a + b*x**4)/(a + b*x**4),x)*a**2*f + 336*int((sqrt(a + b*x**4)*x **2)/(a + b*x**4),x)*a*b*d)/(840*b)