Integrand size = 24, antiderivative size = 101 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{3 d \left (d+e x^2\right )^{3/2}}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) x}{3 d^2 e^2 \sqrt {d+e x^2}}+\frac {c \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{5/2}} \]
1/3*(a+d*(-b*e+c*d)/e^2)*x/d/(e*x^2+d)^(3/2)+c*arctanh(x*e^(1/2)/(e*x^2+d) ^(1/2))/e^(5/2)-1/3*(4*c*d^2-e*(2*a*e+b*d))*x/d^2/e^2/(e*x^2+d)^(1/2)
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {-c d^2 x \left (3 d+4 e x^2\right )+e^2 x \left (3 a d+b d x^2+2 a e x^2\right )}{3 d^2 e^2 \left (d+e x^2\right )^{3/2}}-\frac {c \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{e^{5/2}} \]
(-(c*d^2*x*(3*d + 4*e*x^2)) + e^2*x*(3*a*d + b*d*x^2 + 2*a*e*x^2))/(3*d^2* e^2*(d + e*x^2)^(3/2)) - (c*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]])/e^(5/2)
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1471, 25, 27, 298, 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.
\(\displaystyle \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1471 |
\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {3 c d x^2+e \left (2 a-\frac {d (c d-b e)}{e^2}\right )}{e \left (e x^2+d\right )^{3/2}}dx}{3 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {-\frac {c d^2}{e}+3 c x^2 d+b d+2 a e}{e \left (e x^2+d\right )^{3/2}}dx}{3 d}+\frac {x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {-\frac {c d^2}{e}+3 c x^2 d+b d+2 a e}{\left (e x^2+d\right )^{3/2}}dx}{3 d e}+\frac {x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle \frac {\frac {3 c d \int \frac {1}{\sqrt {e x^2+d}}dx}{e}+\frac {x \left (2 a e+b d-\frac {4 c d^2}{e}\right )}{d \sqrt {d+e x^2}}}{3 d e}+\frac {x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {3 c d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{e}+\frac {x \left (2 a e+b d-\frac {4 c d^2}{e}\right )}{d \sqrt {d+e x^2}}}{3 d e}+\frac {x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {x \left (2 a e+b d-\frac {4 c d^2}{e}\right )}{d \sqrt {d+e x^2}}+\frac {3 c d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{3/2}}}{3 d e}+\frac {x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^2\right )^{3/2}}\) |
((c*d^2 - b*d*e + a*e^2)*x)/(3*d*e^2*(d + e*x^2)^(3/2)) + (((b*d - (4*c*d^ 2)/e + 2*a*e)*x)/(d*Sqrt[d + e*x^2]) + (3*c*d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e^(3/2))/(3*d*e)
3.3.81.3.1 Defintions of rubi rules used
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)^(-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[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} c \,d^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+x \left (d \left (\frac {b \,x^{2}}{3}+a \right ) e^{\frac {5}{2}}-\frac {4 c \,d^{2} e^{\frac {3}{2}} x^{2}}{3}-c \,d^{3} \sqrt {e}+\frac {2 a \,e^{\frac {7}{2}} x^{2}}{3}\right )}{e^{\frac {5}{2}} \left (e \,x^{2}+d \right )^{\frac {3}{2}} d^{2}}\) | \(95\) |
default | \(a \left (\frac {x}{3 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {e \,x^{2}+d}}\right )+c \left (-\frac {x^{3}}{3 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {-\frac {x}{e \sqrt {e \,x^{2}+d}}+\frac {\ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{e^{\frac {3}{2}}}}{e}\right )+b \left (-\frac {x}{2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {d \left (\frac {x}{3 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {e \,x^{2}+d}}\right )}{2 e}\right )\) | \(150\) |
1/e^(5/2)/(e*x^2+d)^(3/2)*((e*x^2+d)^(3/2)*c*d^2*arctanh((e*x^2+d)^(1/2)/x /e^(1/2))+x*(d*(1/3*b*x^2+a)*e^(5/2)-4/3*c*d^2*e^(3/2)*x^2-c*d^3*e^(1/2)+2 /3*a*e^(7/2)*x^2))/d^2
Time = 0.31 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.86 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (c d^{2} e^{2} x^{4} + 2 \, c d^{3} e x^{2} + c d^{4}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - 2 \, {\left ({\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} x^{3} + 3 \, {\left (c d^{3} e - a d e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{6 \, {\left (d^{2} e^{5} x^{4} + 2 \, d^{3} e^{4} x^{2} + d^{4} e^{3}\right )}}, -\frac {3 \, {\left (c d^{2} e^{2} x^{4} + 2 \, c d^{3} e x^{2} + c d^{4}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left ({\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} x^{3} + 3 \, {\left (c d^{3} e - a d e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d^{2} e^{5} x^{4} + 2 \, d^{3} e^{4} x^{2} + d^{4} e^{3}\right )}}\right ] \]
[1/6*(3*(c*d^2*e^2*x^4 + 2*c*d^3*e*x^2 + c*d^4)*sqrt(e)*log(-2*e*x^2 - 2*s qrt(e*x^2 + d)*sqrt(e)*x - d) - 2*((4*c*d^2*e^2 - b*d*e^3 - 2*a*e^4)*x^3 + 3*(c*d^3*e - a*d*e^3)*x)*sqrt(e*x^2 + d))/(d^2*e^5*x^4 + 2*d^3*e^4*x^2 + d^4*e^3), -1/3*(3*(c*d^2*e^2*x^4 + 2*c*d^3*e*x^2 + c*d^4)*sqrt(-e)*arctan( sqrt(-e)*x/sqrt(e*x^2 + d)) + ((4*c*d^2*e^2 - b*d*e^3 - 2*a*e^4)*x^3 + 3*( c*d^3*e - a*d*e^3)*x)*sqrt(e*x^2 + d))/(d^2*e^5*x^4 + 2*d^3*e^4*x^2 + d^4* e^3)]
Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (94) = 188\).
Time = 6.51 (sec) , antiderivative size = 450, normalized size of antiderivative = 4.46 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=a \left (\frac {3 d x}{3 d^{\frac {7}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {5}{2}} e x^{2} \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {2 e x^{3}}{3 d^{\frac {7}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {5}{2}} e x^{2} \sqrt {1 + \frac {e x^{2}}{d}}}\right ) + \frac {b x^{3}}{3 d^{\frac {5}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {3}{2}} e x^{2} \sqrt {1 + \frac {e x^{2}}{d}}} + c \left (\frac {3 d^{\frac {39}{2}} e^{11} \sqrt {1 + \frac {e x^{2}}{d}} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{3 d^{\frac {39}{2}} e^{\frac {27}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {37}{2}} e^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {3 d^{\frac {37}{2}} e^{12} x^{2} \sqrt {1 + \frac {e x^{2}}{d}} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{3 d^{\frac {39}{2}} e^{\frac {27}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {37}{2}} e^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {3 d^{19} e^{\frac {23}{2}} x}{3 d^{\frac {39}{2}} e^{\frac {27}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {37}{2}} e^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {4 d^{18} e^{\frac {25}{2}} x^{3}}{3 d^{\frac {39}{2}} e^{\frac {27}{2}} \sqrt {1 + \frac {e x^{2}}{d}} + 3 d^{\frac {37}{2}} e^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {e x^{2}}{d}}}\right ) \]
a*(3*d*x/(3*d**(7/2)*sqrt(1 + e*x**2/d) + 3*d**(5/2)*e*x**2*sqrt(1 + e*x** 2/d)) + 2*e*x**3/(3*d**(7/2)*sqrt(1 + e*x**2/d) + 3*d**(5/2)*e*x**2*sqrt(1 + e*x**2/d))) + b*x**3/(3*d**(5/2)*sqrt(1 + e*x**2/d) + 3*d**(3/2)*e*x**2 *sqrt(1 + e*x**2/d)) + c*(3*d**(39/2)*e**11*sqrt(1 + e*x**2/d)*asinh(sqrt( e)*x/sqrt(d))/(3*d**(39/2)*e**(27/2)*sqrt(1 + e*x**2/d) + 3*d**(37/2)*e**( 29/2)*x**2*sqrt(1 + e*x**2/d)) + 3*d**(37/2)*e**12*x**2*sqrt(1 + e*x**2/d) *asinh(sqrt(e)*x/sqrt(d))/(3*d**(39/2)*e**(27/2)*sqrt(1 + e*x**2/d) + 3*d* *(37/2)*e**(29/2)*x**2*sqrt(1 + e*x**2/d)) - 3*d**19*e**(23/2)*x/(3*d**(39 /2)*e**(27/2)*sqrt(1 + e*x**2/d) + 3*d**(37/2)*e**(29/2)*x**2*sqrt(1 + e*x **2/d)) - 4*d**18*e**(25/2)*x**3/(3*d**(39/2)*e**(27/2)*sqrt(1 + e*x**2/d) + 3*d**(37/2)*e**(29/2)*x**2*sqrt(1 + e*x**2/d)))
Exception generated. \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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 = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {x {\left (\frac {{\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} x^{2}}{d^{2} e^{3}} + \frac {3 \, {\left (c d^{3} e - a d e^{3}\right )}}{d^{2} e^{3}}\right )}}{3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}} - \frac {c \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{e^{\frac {5}{2}}} \]
-1/3*x*((4*c*d^2*e^2 - b*d*e^3 - 2*a*e^4)*x^2/(d^2*e^3) + 3*(c*d^3*e - a*d *e^3)/(d^2*e^3))/(e*x^2 + d)^(3/2) - c*log(abs(-sqrt(e)*x + sqrt(e*x^2 + d )))/e^(5/2)
Timed out. \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {c\,x^4+b\,x^2+a}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]