Integrand size = 19, antiderivative size = 113 \[ \int \sqrt {d+e x^2} \left (a+c x^4\right ) \, dx=\frac {1}{16} \left (8 a+\frac {c d^2}{e^2}\right ) x \sqrt {d+e x^2}-\frac {c d x \left (d+e x^2\right )^{3/2}}{8 e^2}+\frac {c x^3 \left (d+e x^2\right )^{3/2}}{6 e}+\frac {d \left (c d^2+8 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{16 e^{5/2}} \] Output:
1/16*(8*a+c*d^2/e^2)*x*(e*x^2+d)^(1/2)-1/8*c*d*x*(e*x^2+d)^(3/2)/e^2+1/6*c *x^3*(e*x^2+d)^(3/2)/e+1/16*d*(8*a*e^2+c*d^2)*arctanh(e^(1/2)*x/(e*x^2+d)^ (1/2))/e^(5/2)
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \sqrt {d+e x^2} \left (a+c x^4\right ) \, dx=\frac {x \sqrt {d+e x^2} \left (-3 c d^2+24 a e^2+2 c d e x^2+8 c e^2 x^4\right )}{48 e^2}-\frac {d \left (c d^2+8 a e^2\right ) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{16 e^{5/2}} \] Input:
Integrate[Sqrt[d + e*x^2]*(a + c*x^4),x]
Output:
(x*Sqrt[d + e*x^2]*(-3*c*d^2 + 24*a*e^2 + 2*c*d*e*x^2 + 8*c*e^2*x^4))/(48* e^2) - (d*(c*d^2 + 8*a*e^2)*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]])/(16*e^(5/ 2))
Time = 0.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1474, 27, 299, 211, 224, 219}
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.
\(\displaystyle \int \left (a+c x^4\right ) \sqrt {d+e x^2} \, dx\) |
\(\Big \downarrow \) 1474 |
\(\displaystyle \frac {\int 3 \left (2 a e-c d x^2\right ) \sqrt {e x^2+d}dx}{6 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2}}{6 e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \left (2 a e-c d x^2\right ) \sqrt {e x^2+d}dx}{2 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2}}{6 e}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\frac {\left (8 a e^2+c d^2\right ) \int \sqrt {e x^2+d}dx}{4 e}-\frac {c d x \left (d+e x^2\right )^{3/2}}{4 e}}{2 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2}}{6 e}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\frac {\left (8 a e^2+c d^2\right ) \left (\frac {1}{2} d \int \frac {1}{\sqrt {e x^2+d}}dx+\frac {1}{2} x \sqrt {d+e x^2}\right )}{4 e}-\frac {c d x \left (d+e x^2\right )^{3/2}}{4 e}}{2 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2}}{6 e}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {\left (8 a e^2+c d^2\right ) \left (\frac {1}{2} d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {1}{2} x \sqrt {d+e x^2}\right )}{4 e}-\frac {c d x \left (d+e x^2\right )^{3/2}}{4 e}}{2 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2}}{6 e}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\left (8 a e^2+c d^2\right ) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )}{4 e}-\frac {c d x \left (d+e x^2\right )^{3/2}}{4 e}}{2 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2}}{6 e}\) |
Input:
Int[Sqrt[d + e*x^2]*(a + c*x^4),x]
Output:
(c*x^3*(d + e*x^2)^(3/2))/(6*e) + (-1/4*(c*d*x*(d + e*x^2)^(3/2))/e + ((c* d^2 + 8*a*e^2)*((x*Sqrt[d + e*x^2])/2 + (d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e* x^2]])/(2*Sqrt[e])))/(4*e))/(2*e)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si mp[c^p*x^(4*p - 1)*((d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))), x] + Simp[1/( e*(4*p + 2*q + 1)) Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 2*q + 1)*(a + c *x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x], x] /; FreeQ[{a, c, d, e, q}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && !LtQ[q, -1]
Time = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(\frac {d \left (a \,e^{2}+\frac {c \,d^{2}}{8}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+x \sqrt {e \,x^{2}+d}\, \left (\left (\frac {c \,x^{4}}{3}+a \right ) e^{\frac {5}{2}}-\frac {d c \left (-\frac {2 e^{\frac {3}{2}} x^{2}}{3}+\sqrt {e}\, d \right )}{8}\right )}{2 e^{\frac {5}{2}}}\) | \(80\) |
risch | \(\frac {x \left (8 c \,x^{4} e^{2}+2 d e \,x^{2} c +24 a \,e^{2}-3 c \,d^{2}\right ) \sqrt {e \,x^{2}+d}}{48 e^{2}}+\frac {d \left (8 a \,e^{2}+c \,d^{2}\right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{16 e^{\frac {5}{2}}}\) | \(81\) |
default | \(a \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )+c \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 e}\right )\) | \(122\) |
Input:
int((e*x^2+d)^(1/2)*(c*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/2*(d*(a*e^2+1/8*c*d^2)*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))+x*(e*x^2+d)^(1 /2)*((1/3*c*x^4+a)*e^(5/2)-1/8*d*c*(-2/3*e^(3/2)*x^2+e^(1/2)*d)))/e^(5/2)
Time = 0.10 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.66 \[ \int \sqrt {d+e x^2} \left (a+c x^4\right ) \, dx=\left [\frac {3 \, {\left (c d^{3} + 8 \, a d e^{2}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (8 \, c e^{3} x^{5} + 2 \, c d e^{2} x^{3} - 3 \, {\left (c d^{2} e - 8 \, a e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{96 \, e^{3}}, -\frac {3 \, {\left (c d^{3} + 8 \, a d e^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (8 \, c e^{3} x^{5} + 2 \, c d e^{2} x^{3} - 3 \, {\left (c d^{2} e - 8 \, a e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{48 \, e^{3}}\right ] \] Input:
integrate((e*x^2+d)^(1/2)*(c*x^4+a),x, algorithm="fricas")
Output:
[1/96*(3*(c*d^3 + 8*a*d*e^2)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt (e)*x - d) + 2*(8*c*e^3*x^5 + 2*c*d*e^2*x^3 - 3*(c*d^2*e - 8*a*e^3)*x)*sqr t(e*x^2 + d))/e^3, -1/48*(3*(c*d^3 + 8*a*d*e^2)*sqrt(-e)*arctan(sqrt(-e)*x /sqrt(e*x^2 + d)) - (8*c*e^3*x^5 + 2*c*d*e^2*x^3 - 3*(c*d^2*e - 8*a*e^3)*x )*sqrt(e*x^2 + d))/e^3]
Time = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \sqrt {d+e x^2} \left (a+c x^4\right ) \, dx=\begin {cases} \sqrt {d + e x^{2}} \left (\frac {c d x^{3}}{24 e} + \frac {c x^{5}}{6} + \frac {x \left (a e - \frac {c d^{2}}{8 e}\right )}{2 e}\right ) + \left (a d - \frac {d \left (a e - \frac {c d^{2}}{8 e}\right )}{2 e}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {e} \sqrt {d + e x^{2}} + 2 e x \right )}}{\sqrt {e}} & \text {for}\: d \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {e x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: e \neq 0 \\\sqrt {d} \left (a x + \frac {c x^{5}}{5}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x**2+d)**(1/2)*(c*x**4+a),x)
Output:
Piecewise((sqrt(d + e*x**2)*(c*d*x**3/(24*e) + c*x**5/6 + x*(a*e - c*d**2/ (8*e))/(2*e)) + (a*d - d*(a*e - c*d**2/(8*e))/(2*e))*Piecewise((log(2*sqrt (e)*sqrt(d + e*x**2) + 2*e*x)/sqrt(e), Ne(d, 0)), (x*log(x)/sqrt(e*x**2), True)), Ne(e, 0)), (sqrt(d)*(a*x + c*x**5/5), True))
Exception generated. \[ \int \sqrt {d+e x^2} \left (a+c x^4\right ) \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x^2+d)^(1/2)*(c*x^4+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78 \[ \int \sqrt {d+e x^2} \left (a+c x^4\right ) \, dx=\frac {1}{48} \, {\left (2 \, {\left (4 \, c x^{2} + \frac {c d}{e}\right )} x^{2} - \frac {3 \, {\left (c d^{2} e^{2} - 8 \, a e^{4}\right )}}{e^{4}}\right )} \sqrt {e x^{2} + d} x - \frac {{\left (c d^{3} + 8 \, a d e^{2}\right )} \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{16 \, e^{\frac {5}{2}}} \] Input:
integrate((e*x^2+d)^(1/2)*(c*x^4+a),x, algorithm="giac")
Output:
1/48*(2*(4*c*x^2 + c*d/e)*x^2 - 3*(c*d^2*e^2 - 8*a*e^4)/e^4)*sqrt(e*x^2 + d)*x - 1/16*(c*d^3 + 8*a*d*e^2)*log(abs(-sqrt(e)*x + sqrt(e*x^2 + d)))/e^( 5/2)
Timed out. \[ \int \sqrt {d+e x^2} \left (a+c x^4\right ) \, dx=\int \left (c\,x^4+a\right )\,\sqrt {e\,x^2+d} \,d x \] Input:
int((a + c*x^4)*(d + e*x^2)^(1/2),x)
Output:
int((a + c*x^4)*(d + e*x^2)^(1/2), x)
Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.12 \[ \int \sqrt {d+e x^2} \left (a+c x^4\right ) \, dx=\frac {24 \sqrt {e \,x^{2}+d}\, a \,e^{3} x -3 \sqrt {e \,x^{2}+d}\, c \,d^{2} e x +2 \sqrt {e \,x^{2}+d}\, c d \,e^{2} x^{3}+8 \sqrt {e \,x^{2}+d}\, c \,e^{3} x^{5}+24 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a d \,e^{2}+3 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c \,d^{3}}{48 e^{3}} \] Input:
int((e*x^2+d)^(1/2)*(c*x^4+a),x)
Output:
(24*sqrt(d + e*x**2)*a*e**3*x - 3*sqrt(d + e*x**2)*c*d**2*e*x + 2*sqrt(d + e*x**2)*c*d*e**2*x**3 + 8*sqrt(d + e*x**2)*c*e**3*x**5 + 24*sqrt(e)*log(( sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*a*d*e**2 + 3*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*c*d**3)/(48*e**3)