Integrand size = 26, antiderivative size = 366 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\frac {d^2 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac {(b c-a d)^2 (b c+11 a d) e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{3/2} c^{9/2}} \]
1/6*(-a*d+b*c)^3*e^2*(e*(b*x^2+a)/(d*x^2+c))^(5/2)/a/c^2/(a*e-c*e*(b*x^2+a )/(d*x^2+c))^3+1/16*(-a*d+b*c)*(-35*a^2*d^2+10*a*b*c*d+b^2*c^2)*e^(3/2)*ar ctanh(c^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a^(1/2)/e^(1/2))/a^(3/2)/c^(9/ 2)+d^2*(-a*d+b*c)*e*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^4+1/24*(-a*d+b*c)^2*(1 1*a*d+b*c)*e^3*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^4/(a*e-c*e*(b*x^2+a)/(d*x^2 +c))^2-1/48*(-a*d+b*c)*(-79*a^2*d^2+50*a*b*c*d+5*b^2*c^2)*e^2*(e*(b*x^2+a) /(d*x^2+c))^(1/2)/a/c^4/(a*e-c*e*(b*x^2+a)/(d*x^2+c))
Time = 4.76 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.67 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \left (3 b^2 c^2 x^4 \left (c+d x^2\right )+2 a b c x^2 \left (7 c^2-19 c d x^2-50 d^2 x^4\right )+a^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )\right )+3 \left (b^3 c^3+9 a b^2 c^2 d-45 a^2 b c d^2+35 a^3 d^3\right ) x^6 \sqrt {c+d x^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )\right )}{48 a^{3/2} c^{9/2} x^6 \sqrt {a+b x^2}} \]
(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(-(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2]*(3 *b^2*c^2*x^4*(c + d*x^2) + 2*a*b*c*x^2*(7*c^2 - 19*c*d*x^2 - 50*d^2*x^4) + a^2*(8*c^3 - 14*c^2*d*x^2 + 35*c*d^2*x^4 + 105*d^3*x^6))) + 3*(b^3*c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*x^6*Sqrt[c + d*x^2]*ArcTanh[( Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(48*a^(3/2)*c^(9/2)* x^6*Sqrt[a + b*x^2])
Time = 0.54 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.82, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2053, 2052, 366, 25, 27, 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.
\(\displaystyle \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx\) |
\(\Big \downarrow \) 2053 |
\(\displaystyle \frac {1}{2} \int \frac {\left (\frac {e \left (b x^2+a\right )}{d x^2+c}\right )^{3/2}}{x^8}dx^2\) |
\(\Big \downarrow \) 2052 |
\(\displaystyle e (b c-a d) \int \frac {x^8 \left (b e-d x^4\right )^2}{\left (a e-c x^4\right )^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\) |
\(\Big \downarrow \) 366 |
\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}-\frac {\int -\frac {x^8 \left (-6 a c d^2 e x^4+6 b^2 c^2 e^2-5 (b c e-a d e)^2\right )}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{6 a c^2 e}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle e (b c-a d) \left (\frac {\int \frac {e x^8 \left (\left (b^2 c^2+10 a b d c-5 a^2 d^2\right ) e-6 a c d^2 x^4\right )}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{6 a c^2 e}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e (b c-a d) \left (\frac {\int \frac {x^8 \left (\left (b^2 c^2+10 a b d c-5 a^2 d^2\right ) e-6 a c d^2 x^4\right )}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\) |
\(\Big \downarrow \) 360 |
\(\displaystyle e (b c-a d) \left (\frac {\frac {\int -\frac {-24 a c^3 d^2 x^8+4 c^2 (b c-a d) (b c+11 a d) e x^4+a c (b c-a d) (b c+11 a d) e^2}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}+\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle e (b c-a d) \left (\frac {\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}-\frac {\int \frac {-24 a c^3 d^2 x^8+4 c^2 (b c-a d) (b c+11 a d) e x^4+a c (b c-a d) (b c+11 a d) e^2}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\) |
\(\Big \downarrow \) 1471 |
\(\displaystyle e (b c-a d) \left (\frac {\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}-\frac {\frac {c e \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {\int \frac {3 a c e \left (\left (b^2 c^2+10 a b d c-19 a^2 d^2\right ) e-16 a c d^2 x^4\right )}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 a e}}{4 c^3}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e (b c-a d) \left (\frac {\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}-\frac {\frac {c e \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {3}{2} c \int \frac {\left (b^2 c^2+10 a b d c-19 a^2 d^2\right ) e-16 a c d^2 x^4}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 c^3}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\) |
\(\Big \downarrow \) 299 |
\(\displaystyle e (b c-a d) \left (\frac {\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}-\frac {\frac {c e \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {3}{2} c \left (e \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \int \frac {1}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}+16 a d^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{4 c^3}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle e (b c-a d) \left (\frac {\frac {a e^2 (b c-a d) (11 a d+b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^2 \left (a e-c x^4\right )^2}-\frac {\frac {c e \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (a e-c x^4\right )}-\frac {3}{2} c \left (\frac {\sqrt {e} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} \sqrt {c}}+16 a d^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{4 c^3}}{6 a c^2}+\frac {e x^{10} (b c-a d)^2}{6 a c^2 \left (a e-c x^4\right )^3}\right )\) |
(b*c - a*d)*e*(((b*c - a*d)^2*e*x^10)/(6*a*c^2*(a*e - c*x^4)^3) + ((a*(b*c - a*d)*(b*c + 11*a*d)*e^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(4*c^2*(a*e - c*x^4)^2) - ((c*(5*b^2*c^2 + 50*a*b*c*d - 79*a^2*d^2)*e*Sqrt[(e*(a + b*x ^2))/(c + d*x^2)])/(2*(a*e - c*x^4)) - (3*c*(16*a*d^2*Sqrt[(e*(a + b*x^2)) /(c + d*x^2)] + ((b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*Sqrt[e]*ArcTanh[(Sqrt [c]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*Sqrt[c ])))/2)/(4*c^3))/(6*a*c^2))
3.3.82.3.1 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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a* b^2*e*(p + 1))), x] + Simp[1/(2*a*b^2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + 2*b^2*c^2*(p + 1) + 2*a*b*d^2*(p + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p , -1]
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 = 0.27 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (57 a^{2} d^{2} x^{4}-52 b d a c \,x^{4}+3 b^{2} c^{2} x^{4}-22 a^{2} c d \,x^{2}+14 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{48 c^{4} x^{6} a}-\frac {\left (-\frac {\left (35 a^{3} d^{3}-45 a^{2} b c \,d^{2}+9 d \,c^{2} b^{2} a +b^{3} c^{3}\right ) \ln \left (\frac {2 a c e +\left (e d a +e b c \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {b d e \,x^{4}+\left (e d a +e b c \right ) x^{2}+a c e}}{x^{2}}\right )}{2 \sqrt {a c e}}+\frac {16 a \,d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b \,x^{2}+a \right )}{\left (a d -b c \right ) \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{16 c^{4} a \left (b \,x^{2}+a \right )}\) | \(333\) |
default | \(\text {Expression too large to display}\) | \(1498\) |
-1/48*(d*x^2+c)*(57*a^2*d^2*x^4-52*a*b*c*d*x^4+3*b^2*c^2*x^4-22*a^2*c*d*x^ 2+14*a*b*c^2*x^2+8*a^2*c^2)/c^4/x^6/a*e*(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/16 /c^4/a*(-1/2*(35*a^3*d^3-45*a^2*b*c*d^2+9*a*b^2*c^2*d+b^3*c^3)/(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)+16*a*d^2*(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 = 9.70 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.57 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\left [\frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} e x^{6} \sqrt {\frac {e}{a c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} + 4 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d + a^{2} c d^{2}\right )} x^{4} + {\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{a c}}}{x^{4}}\right ) - 4 \, {\left ({\left (3 \, b^{2} c^{2} d - 100 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} e x^{6} + 8 \, a^{2} c^{3} e + {\left (3 \, b^{2} c^{3} - 38 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} e x^{4} + 14 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{192 \, a c^{4} x^{6}}, -\frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} e x^{6} \sqrt {-\frac {e}{a c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{a c}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) + 2 \, {\left ({\left (3 \, b^{2} c^{2} d - 100 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} e x^{6} + 8 \, a^{2} c^{3} e + {\left (3 \, b^{2} c^{3} - 38 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} e x^{4} + 14 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{96 \, a c^{4} x^{6}}\right ] \]
[1/192*(3*(b^3*c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*e*x^6*sq rt(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*((3*b^2*c^2*d - 100*a*b*c*d^2 + 105*a^2*d^3)*e*x^6 + 8*a^2*c^3* e + (3*b^2*c^3 - 38*a*b*c^2*d + 35*a^2*c*d^2)*e*x^4 + 14*(a*b*c^3 - a^2*c^ 2*d)*e*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a*c^4*x^6), -1/96*(3*(b^3* c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*e*x^6*sqrt(-e/(a*c))*ar ctan(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*((3*b^2*c^2*d - 100*a*b*c*d^2 + 105*a^2*d^3 )*e*x^6 + 8*a^2*c^3*e + (3*b^2*c^3 - 38*a*b*c^2*d + 35*a^2*c*d^2)*e*x^4 + 14*(a*b*c^3 - a^2*c^2*d)*e*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a*c^4* x^6)]
Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\text {Exception raised: TypeError} \]
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,[5,1,5]%%%},[2,9,0]%%%}+%%%{%%%{-10,[4,2,5]%%%},[2,8 ,1]%%%}+%
Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^7} \,d x \]