Integrand size = 17, antiderivative size = 158 \[ \int (d+e x) \sqrt {a+c x^4} \, dx=\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\frac {a e \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}}+\frac {a^{3/4} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}} \] Output:
1/3*d*x*(c*x^4+a)^(1/2)+1/4*e*x^2*(c*x^4+a)^(1/2)+1/4*a*e*arctanh(c^(1/2)* x^2/(c*x^4+a)^(1/2))/c^(1/2)+1/3*a^(3/4)*d*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a )/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4 )),1/2*2^(1/2))/c^(1/4)/(c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.47 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.69 \[ \int (d+e x) \sqrt {a+c x^4} \, dx=\frac {\sqrt {a+c x^4} \left (\sqrt {c} e x^2 \sqrt {1+\frac {c x^4}{a}}+\sqrt {a} e \text {arcsinh}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )+4 \sqrt {c} d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^4}{a}\right )\right )}{4 \sqrt {c} \sqrt {1+\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x)*Sqrt[a + c*x^4],x]
Output:
(Sqrt[a + c*x^4]*(Sqrt[c]*e*x^2*Sqrt[1 + (c*x^4)/a] + Sqrt[a]*e*ArcSinh[(S qrt[c]*x^2)/Sqrt[a]] + 4*Sqrt[c]*d*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -(( c*x^4)/a)]))/(4*Sqrt[c]*Sqrt[1 + (c*x^4)/a])
Time = 0.46 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2424, 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.
\(\displaystyle \int \sqrt {a+c x^4} (d+e x) \, dx\) |
\(\Big \downarrow \) 2424 |
\(\displaystyle \int \left (d \sqrt {a+c x^4}+e x \sqrt {a+c x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^{3/4} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {a e \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}}+\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}\) |
Input:
Int[(d + e*x)*Sqrt[a + c*x^4],x]
Output:
(d*x*Sqrt[a + c*x^4])/3 + (e*x^2*Sqrt[a + c*x^4])/4 + (a*e*ArcTanh[(Sqrt[c ]*x^2)/Sqrt[a + c*x^4]])/(4*Sqrt[c]) + (a^(3/4)*d*(Sqrt[a] + Sqrt[c]*x^2)* Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x) /a^(1/4)], 1/2])/(3*c^(1/4)*Sqrt[a + c*x^4])
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^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, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && !PolyQ[Pq, x^(n/2)]
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {x \left (3 e x +4 d \right ) \sqrt {c \,x^{4}+a}}{12}+\frac {2 a d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {a e \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{4 \sqrt {c}}\) | \(119\) |
default | \(d \left (\frac {x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+e \left (\frac {x^{2} \sqrt {c \,x^{4}+a}}{4}+\frac {a \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{4 \sqrt {c}}\right )\) | \(129\) |
elliptic | \(\frac {e \,x^{2} \sqrt {c \,x^{4}+a}}{4}+\frac {d x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {a e \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{4 \sqrt {c}}\) | \(130\) |
Input:
int((e*x+d)*(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/12*x*(3*e*x+4*d)*(c*x^4+a)^(1/2)+2/3*a*d/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I* c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2) *EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)+1/4*a*e*ln(c^(1/2)*x^2+(c*x^4+a) ^(1/2))/c^(1/2)
Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int (d+e x) \sqrt {a+c x^4} \, dx=\frac {16 \, c^{\frac {3}{2}} d \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 3 \, a \sqrt {c} e \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) + 2 \, \sqrt {c x^{4} + a} {\left (3 \, c e x^{2} + 4 \, c d x\right )}}{24 \, c} \] Input:
integrate((e*x+d)*(c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
1/24*(16*c^(3/2)*d*(-a/c)^(3/4)*elliptic_f(arcsin((-a/c)^(1/4)/x), -1) + 3 *a*sqrt(c)*e*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)*x^2 - a) + 2*sqrt(c* x^4 + a)*(3*c*e*x^2 + 4*c*d*x))/c
Result contains complex when optimal does not.
Time = 2.00 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.56 \[ \int (d+e x) \sqrt {a+c x^4} \, dx=\frac {\sqrt {a} d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {a} e x^{2} \sqrt {1 + \frac {c x^{4}}{a}}}{4} + \frac {a e \operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{4 \sqrt {c}} \] Input:
integrate((e*x+d)*(c*x**4+a)**(1/2),x)
Output:
sqrt(a)*d*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(I*pi)/a )/(4*gamma(5/4)) + sqrt(a)*e*x**2*sqrt(1 + c*x**4/a)/4 + a*e*asinh(sqrt(c) *x**2/sqrt(a))/(4*sqrt(c))
\[ \int (d+e x) \sqrt {a+c x^4} \, dx=\int { \sqrt {c x^{4} + a} {\left (e x + d\right )} \,d x } \] Input:
integrate((e*x+d)*(c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(c*x^4 + a)*(e*x + d), x)
\[ \int (d+e x) \sqrt {a+c x^4} \, dx=\int { \sqrt {c x^{4} + a} {\left (e x + d\right )} \,d x } \] Input:
integrate((e*x+d)*(c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(c*x^4 + a)*(e*x + d), x)
Timed out. \[ \int (d+e x) \sqrt {a+c x^4} \, dx=\int \sqrt {c\,x^4+a}\,\left (d+e\,x\right ) \,d x \] Input:
int((a + c*x^4)^(1/2)*(d + e*x),x)
Output:
int((a + c*x^4)^(1/2)*(d + e*x), x)
\[ \int (d+e x) \sqrt {a+c x^4} \, dx=\frac {8 \sqrt {c \,x^{4}+a}\, c d x +6 \sqrt {c \,x^{4}+a}\, c e \,x^{2}-3 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+a}-\sqrt {c}\, x^{2}\right ) a e +3 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}\right ) a e +16 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a c d}{24 c} \] Input:
int((e*x+d)*(c*x^4+a)^(1/2),x)
Output:
(8*sqrt(a + c*x**4)*c*d*x + 6*sqrt(a + c*x**4)*c*e*x**2 - 3*sqrt(c)*log(sq rt(a + c*x**4) - sqrt(c)*x**2)*a*e + 3*sqrt(c)*log(sqrt(a + c*x**4) + sqrt (c)*x**2)*a*e + 16*int(sqrt(a + c*x**4)/(a + c*x**4),x)*a*c*d)/(24*c)