Integrand size = 26, antiderivative size = 256 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=-\frac {d (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^3}-\frac {a (b c-a d)^2 e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {(5 b c-9 a d) (b c-a d) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {3 (b c-5 a d) (b c-a d) e^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{8 \sqrt {a} c^{7/2}} \]
-3/8*(-5*a*d+b*c)*(-a*d+b*c)*e^(3/2)*arctanh(c^(1/2)*(e*(b*x^2+a)/(d*x^2+c ))^(1/2)/a^(1/2)/e^(1/2))/c^(7/2)/a^(1/2)-d*(-a*d+b*c)*e*(e*(b*x^2+a)/(d*x ^2+c))^(1/2)/c^3-1/4*a*(-a*d+b*c)^2*e^3*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^3/ (a*e-c*e*(b*x^2+a)/(d*x^2+c))^2+1/8*(-9*a*d+5*b*c)*(-a*d+b*c)*e^2*(e*(b*x^ 2+a)/(d*x^2+c))^(1/2)/c^3/(a*e-c*e*(b*x^2+a)/(d*x^2+c))
Time = 4.30 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.73 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=-\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \left (b c x^2 \left (5 c+13 d x^2\right )+a \left (2 c^2-5 c d x^2-15 d^2 x^4\right )\right )+3 \left (b^2 c^2-6 a b c d+5 a^2 d^2\right ) x^4 \sqrt {c+d x^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )\right )}{8 \sqrt {a} c^{7/2} x^4 \sqrt {a+b x^2}} \]
-1/8*(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2] *(b*c*x^2*(5*c + 13*d*x^2) + a*(2*c^2 - 5*c*d*x^2 - 15*d^2*x^4)) + 3*(b^2* c^2 - 6*a*b*c*d + 5*a^2*d^2)*x^4*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(Sqrt[a]*c^(7/2)*x^4*Sqrt[a + b*x^2] )
Time = 0.41 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.84, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2053, 2052, 25, 360, 25, 1471, 27, 299, 221}
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.
| \(\displaystyle \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (\frac {e \left (b x^2+a\right )}{d x^2+c}\right )^{3/2}}{x^6}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle e (b c-a d) \int -\frac {x^8 \left (b e-d x^4\right )}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\left (e (b c-a d) \int \frac {x^8 \left (b e-d x^4\right )}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\right )\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle e (b c-a d) \left (-\frac {\int -\frac {-4 c^2 d x^8+4 c (b c-a d) e x^4+a (b c-a d) e^2}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle e (b c-a d) \left (\frac {\int \frac {-4 c^2 d x^8+4 c (b c-a d) e x^4+a (b c-a d) e^2}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle e (b c-a d) \left (\frac {\frac {e (5 b c-9 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {\int \frac {a e \left ((3 b c-7 a d) e-8 c d x^4\right )}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 a e}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e (b c-a d) \left (\frac {\frac {e (5 b c-9 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {1}{2} \int \frac {(3 b c-7 a d) e-8 c d x^4}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle e (b c-a d) \left (\frac {\frac {1}{2} \left (-3 e (b c-5 a d) \int \frac {1}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}-8 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )+\frac {e (5 b c-9 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle e (b c-a d) \left (\frac {\frac {1}{2} \left (-\frac {3 \sqrt {e} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} \sqrt {c}}-8 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )+\frac {e (5 b c-9 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}}{4 c^3}-\frac {a e^2 (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-c x^4\right )^2}\right )\) |
(b*c - a*d)*e*(-1/4*(a*(b*c - a*d)*e^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/ (c^3*(a*e - c*x^4)^2) + (((5*b*c - 9*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^ 2)])/(2*(a*e - c*x^4)) + (-8*d*Sqrt[(e*(a + b*x^2))/(c + d*x^2)] - (3*(b*c - 5*a*d)*Sqrt[e]*ArcTanh[(Sqrt[c]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqr t[a]*Sqrt[e])])/(Sqrt[a]*Sqrt[c]))/2)/(4*c^3))
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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 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]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.24 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (-7 a d \,x^{2}+5 b c \,x^{2}+2 a c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{8 c^{3} x^{4}}+\frac {\left (-\frac {\left (15 a^{2} d^{2}-18 a b c d +3 b^{2} c^{2}\right ) \ln \left (\frac {2 a c e +\left (e d a +e b c \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {b d e \,x^{4}+\left (e d a +e b c \right ) x^{2}+a c e}}{x^{2}}\right )}{2 \sqrt {a c e}}+\frac {8 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b \,x^{2}+a \right )}{\left (a d -b c \right ) \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{8 c^{3} \left (b \,x^{2}+a \right )}\) | \(270\) |
| default | \(\text {Expression too large to display}\) | \(1042\) |
-1/8*(d*x^2+c)*(-7*a*d*x^2+5*b*c*x^2+2*a*c)/c^3/x^4*e*(e*(b*x^2+a)/(d*x^2+ c))^(1/2)+1/8/c^3*(-1/2*(15*a^2*d^2-18*a*b*c*d+3*b^2*c^2)/(a*c*e)^(1/2)*ln ((2*a*c*e+(a*d*e+b*c*e)*x^2+2*(a*c*e)^(1/2)*(b*d*e*x^4+(a*d*e+b*c*e)*x^2+a *c*e)^(1/2))/x^2)+8*d*(a^2*d^2-2*a*b*c*d+b^2*c^2)*(b*x^2+a)/(a*d-b*c)/(b*d *e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2))*e/(b*x^2+a)*(e*(b*x^2+a)/(d*x^2+c ))^(1/2)*((d*x^2+c)*e*(b*x^2+a))^(1/2)
Time = 2.54 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.70 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\left [\frac {3 \, {\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} e x^{4} \sqrt {\frac {e}{a c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} - 4 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d + a^{2} c d^{2}\right )} x^{4} + {\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{a c}}}{x^{4}}\right ) - 4 \, {\left ({\left (13 \, b c d - 15 \, a d^{2}\right )} e x^{4} + 2 \, a c^{2} e + 5 \, {\left (b c^{2} - a c d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{32 \, c^{3} x^{4}}, \frac {3 \, {\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} e x^{4} \sqrt {-\frac {e}{a c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{a c}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) - 2 \, {\left ({\left (13 \, b c d - 15 \, a d^{2}\right )} e x^{4} + 2 \, a c^{2} e + 5 \, {\left (b c^{2} - a c d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{16 \, c^{3} x^{4}}\right ] \]
[1/32*(3*(b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*e*x^4*sqrt(e/(a*c))*log(((b^2*c ^2 + 6*a*b*c*d + a^2*d^2)*e*x^4 + 8*a^2*c^2*e + 8*(a*b*c^2 + a^2*c*d)*e*x^ 2 - 4*(2*a^2*c^3 + (a*b*c^2*d + a^2*c*d^2)*x^4 + (a*b*c^3 + 3*a^2*c^2*d)*x ^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))*sqrt(e/(a*c)))/x^4) - 4*((13*b*c*d - 15*a*d^2)*e*x^4 + 2*a*c^2*e + 5*(b*c^2 - a*c*d)*e*x^2)*sqrt((b*e*x^2 + a* e)/(d*x^2 + c)))/(c^3*x^4), 1/16*(3*(b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*e*x^ 4*sqrt(-e/(a*c))*arctan(1/2*((b*c + a*d)*x^2 + 2*a*c)*sqrt((b*e*x^2 + a*e) /(d*x^2 + c))*sqrt(-e/(a*c))/(b*e*x^2 + a*e)) - 2*((13*b*c*d - 15*a*d^2)*e *x^4 + 2*a*c^2*e + 5*(b*c^2 - a*c*d)*e*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(c^3*x^4)]
Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\text {Exception raised: ValueError} \]
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
Exception generated. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{2,[4,1,4]%%%},[2,7,0]%%%}+%%%{%%%{-8,[3,2,4]%%%},[2,6, 1]%%%}+%%
Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^5} \,d x \]