Integrand size = 17, antiderivative size = 198 \[ \int (d+e x) \left (a+c x^4\right )^{3/2} \, dx=\frac {2}{7} a d x \sqrt {a+c x^4}+\frac {3}{16} a e x^2 \sqrt {a+c x^4}+\frac {1}{7} d x \left (a+c x^4\right )^{3/2}+\frac {1}{8} e x^2 \left (a+c x^4\right )^{3/2}+\frac {3 a^2 e \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{16 \sqrt {c}}+\frac {2 a^{7/4} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{7 \sqrt [4]{c} \sqrt {a+c x^4}} \] Output:
2/7*a*d*x*(c*x^4+a)^(1/2)+3/16*a*e*x^2*(c*x^4+a)^(1/2)+1/7*d*x*(c*x^4+a)^( 3/2)+1/8*e*x^2*(c*x^4+a)^(3/2)+3/16*a^2*e*arctanh(c^(1/2)*x^2/(c*x^4+a)^(1 /2))/c^(1/2)+2/7*a^(7/4)*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 = 10.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.61 \[ \int (d+e x) \left (a+c x^4\right )^{3/2} \, dx=\frac {\sqrt {a+c x^4} \left (\sqrt {c} e x^2 \left (5 a+2 c x^4\right ) \sqrt {1+\frac {c x^4}{a}}+3 a^{3/2} e \text {arcsinh}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )+16 a \sqrt {c} d x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^4}{a}\right )\right )}{16 \sqrt {c} \sqrt {1+\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x)*(a + c*x^4)^(3/2),x]
Output:
(Sqrt[a + c*x^4]*(Sqrt[c]*e*x^2*(5*a + 2*c*x^4)*Sqrt[1 + (c*x^4)/a] + 3*a^ (3/2)*e*ArcSinh[(Sqrt[c]*x^2)/Sqrt[a]] + 16*a*Sqrt[c]*d*x*Hypergeometric2F 1[-3/2, 1/4, 5/4, -((c*x^4)/a)]))/(16*Sqrt[c]*Sqrt[1 + (c*x^4)/a])
Time = 0.52 (sec) , antiderivative size = 198, 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.
| \(\displaystyle \int \left (a+c x^4\right )^{3/2} (d+e x) \, dx\) |
| \(\Big \downarrow \) 2424 |
| \(\displaystyle \int \left (d \left (a+c x^4\right )^{3/2}+e x \left (a+c x^4\right )^{3/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^{7/4} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{7 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {3 a^2 e \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{16 \sqrt {c}}+\frac {2}{7} a d x \sqrt {a+c x^4}+\frac {1}{7} d x \left (a+c x^4\right )^{3/2}+\frac {3}{16} a e x^2 \sqrt {a+c x^4}+\frac {1}{8} e x^2 \left (a+c x^4\right )^{3/2}\) |
Input:
Int[(d + e*x)*(a + c*x^4)^(3/2),x]
Output:
(2*a*d*x*Sqrt[a + c*x^4])/7 + (3*a*e*x^2*Sqrt[a + c*x^4])/16 + (d*x*(a + c *x^4)^(3/2))/7 + (e*x^2*(a + c*x^4)^(3/2))/8 + (3*a^2*e*ArcTanh[(Sqrt[c]*x ^2)/Sqrt[a + c*x^4]])/(16*Sqrt[c]) + (2*a^(7/4)*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])/(7*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 = 0.65 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(\frac {x \left (14 c e \,x^{5}+16 c d \,x^{4}+35 a e x +48 a d \right ) \sqrt {c \,x^{4}+a}}{112}+\frac {4 a^{2} d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 a^{2} e \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{16 \sqrt {c}}\) | \(139\) |
| default | \(d \left (\frac {c \,x^{5} \sqrt {c \,x^{4}+a}}{7}+\frac {3 a x \sqrt {c \,x^{4}+a}}{7}+\frac {4 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+e \left (\frac {3 a^{2} \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{16 \sqrt {c}}+\frac {c \,x^{6} \sqrt {c \,x^{4}+a}}{8}+\frac {5 a \,x^{2} \sqrt {c \,x^{4}+a}}{16}\right )\) | \(165\) |
| elliptic | \(\frac {c e \,x^{6} \sqrt {c \,x^{4}+a}}{8}+\frac {c d \,x^{5} \sqrt {c \,x^{4}+a}}{7}+\frac {5 a e \,x^{2} \sqrt {c \,x^{4}+a}}{16}+\frac {3 a d x \sqrt {c \,x^{4}+a}}{7}+\frac {4 a^{2} d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 a^{2} e \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{16 \sqrt {c}}\) | \(168\) |
Input:
int((e*x+d)*(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/112*x*(14*c*e*x^5+16*c*d*x^4+35*a*e*x+48*a*d)*(c*x^4+a)^(1/2)+4/7*a^2*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)+3 /16*a^2*e*ln(c^(1/2)*x^2+(c*x^4+a)^(1/2))/c^(1/2)
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.59 \[ \int (d+e x) \left (a+c x^4\right )^{3/2} \, dx=\frac {128 \, a c^{\frac {3}{2}} d \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 21 \, a^{2} \sqrt {c} e \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) + 2 \, {\left (14 \, c^{2} e x^{6} + 16 \, c^{2} d x^{5} + 35 \, a c e x^{2} + 48 \, a c d x\right )} \sqrt {c x^{4} + a}}{224 \, c} \] Input:
integrate((e*x+d)*(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/224*(128*a*c^(3/2)*d*(-a/c)^(3/4)*elliptic_f(arcsin((-a/c)^(1/4)/x), -1) + 21*a^2*sqrt(c)*e*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)*x^2 - a) + 2* (14*c^2*e*x^6 + 16*c^2*d*x^5 + 35*a*c*e*x^2 + 48*a*c*d*x)*sqrt(c*x^4 + a)) /c
Result contains complex when optimal does not.
Time = 4.24 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.08 \[ \int (d+e x) \left (a+c x^4\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{\frac {3}{2}} e x^{2} \sqrt {1 + \frac {c x^{4}}{a}}}{4} + \frac {a^{\frac {3}{2}} e x^{2}}{16 \sqrt {1 + \frac {c x^{4}}{a}}} + \frac {\sqrt {a} c d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {3 \sqrt {a} c e x^{6}}{16 \sqrt {1 + \frac {c x^{4}}{a}}} + \frac {3 a^{2} e \operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{16 \sqrt {c}} + \frac {c^{2} e x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {c x^{4}}{a}}} \] Input:
integrate((e*x+d)*(c*x**4+a)**(3/2),x)
Output:
a**(3/2)*d*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(I*pi)/ a)/(4*gamma(5/4)) + a**(3/2)*e*x**2*sqrt(1 + c*x**4/a)/4 + a**(3/2)*e*x**2 /(16*sqrt(1 + c*x**4/a)) + sqrt(a)*c*d*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4)) + 3*sqrt(a)*c*e*x**6/(16* sqrt(1 + c*x**4/a)) + 3*a**2*e*asinh(sqrt(c)*x**2/sqrt(a))/(16*sqrt(c)) + c**2*e*x**10/(8*sqrt(a)*sqrt(1 + c*x**4/a))
\[ \int (d+e x) \left (a+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )} \,d x } \] Input:
integrate((e*x+d)*(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + a)^(3/2)*(e*x + d), x)
\[ \int (d+e x) \left (a+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )} \,d x } \] Input:
integrate((e*x+d)*(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + a)^(3/2)*(e*x + d), x)
Timed out. \[ \int (d+e x) \left (a+c x^4\right )^{3/2} \, dx=\int {\left (c\,x^4+a\right )}^{3/2}\,\left (d+e\,x\right ) \,d x \] Input:
int((a + c*x^4)^(3/2)*(d + e*x),x)
Output:
int((a + c*x^4)^(3/2)*(d + e*x), x)
\[ \int (d+e x) \left (a+c x^4\right )^{3/2} \, dx=\frac {96 \sqrt {c \,x^{4}+a}\, a c d x +70 \sqrt {c \,x^{4}+a}\, a c e \,x^{2}+32 \sqrt {c \,x^{4}+a}\, c^{2} d \,x^{5}+28 \sqrt {c \,x^{4}+a}\, c^{2} e \,x^{6}-21 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+a}-\sqrt {c}\, x^{2}\right ) a^{2} e +21 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}\right ) a^{2} e +128 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a^{2} c d}{224 c} \] Input:
int((e*x+d)*(c*x^4+a)^(3/2),x)
Output:
(96*sqrt(a + c*x**4)*a*c*d*x + 70*sqrt(a + c*x**4)*a*c*e*x**2 + 32*sqrt(a + c*x**4)*c**2*d*x**5 + 28*sqrt(a + c*x**4)*c**2*e*x**6 - 21*sqrt(c)*log(s qrt(a + c*x**4) - sqrt(c)*x**2)*a**2*e + 21*sqrt(c)*log(sqrt(a + c*x**4) + sqrt(c)*x**2)*a**2*e + 128*int(sqrt(a + c*x**4)/(a + c*x**4),x)*a**2*c*d) /(224*c)