Integrand size = 26, antiderivative size = 133 \[ \int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx=-\frac {A \sqrt {b x^2+c x^4}}{7 b x^8}-\frac {(7 b B-6 A c) \sqrt {b x^2+c x^4}}{35 b^2 x^6}+\frac {4 c (7 b B-6 A c) \sqrt {b x^2+c x^4}}{105 b^3 x^4}-\frac {8 c^2 (7 b B-6 A c) \sqrt {b x^2+c x^4}}{105 b^4 x^2} \] Output:
-1/7*A*(c*x^4+b*x^2)^(1/2)/b/x^8-1/35*(-6*A*c+7*B*b)*(c*x^4+b*x^2)^(1/2)/b ^2/x^6+4/105*c*(-6*A*c+7*B*b)*(c*x^4+b*x^2)^(1/2)/b^3/x^4-8/105*c^2*(-6*A* c+7*B*b)*(c*x^4+b*x^2)^(1/2)/b^4/x^2
Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.67 \[ \int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx=-\frac {\sqrt {x^2 \left (b+c x^2\right )} \left (7 b B x^2 \left (3 b^2-4 b c x^2+8 c^2 x^4\right )+3 A \left (5 b^3-6 b^2 c x^2+8 b c^2 x^4-16 c^3 x^6\right )\right )}{105 b^4 x^8} \] Input:
Integrate[(A + B*x^2)/(x^7*Sqrt[b*x^2 + c*x^4]),x]
Output:
-1/105*(Sqrt[x^2*(b + c*x^2)]*(7*b*B*x^2*(3*b^2 - 4*b*c*x^2 + 8*c^2*x^4) + 3*A*(5*b^3 - 6*b^2*c*x^2 + 8*b*c^2*x^4 - 16*c^3*x^6)))/(b^4*x^8)
Time = 0.53 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1940, 1220, 1129, 1129, 1123}
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.
\(\displaystyle \int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx\) |
\(\Big \downarrow \) 1940 |
\(\displaystyle \frac {1}{2} \int \frac {B x^2+A}{x^8 \sqrt {c x^4+b x^2}}dx^2\) |
\(\Big \downarrow \) 1220 |
\(\displaystyle \frac {1}{2} \left (\frac {(7 b B-6 A c) \int \frac {1}{x^6 \sqrt {c x^4+b x^2}}dx^2}{7 b}-\frac {2 A \sqrt {b x^2+c x^4}}{7 b x^8}\right )\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle \frac {1}{2} \left (\frac {(7 b B-6 A c) \left (-\frac {4 c \int \frac {1}{x^4 \sqrt {c x^4+b x^2}}dx^2}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^6}\right )}{7 b}-\frac {2 A \sqrt {b x^2+c x^4}}{7 b x^8}\right )\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle \frac {1}{2} \left (\frac {(7 b B-6 A c) \left (-\frac {4 c \left (-\frac {2 c \int \frac {1}{x^2 \sqrt {c x^4+b x^2}}dx^2}{3 b}-\frac {2 \sqrt {b x^2+c x^4}}{3 b x^4}\right )}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^6}\right )}{7 b}-\frac {2 A \sqrt {b x^2+c x^4}}{7 b x^8}\right )\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (-\frac {4 c \left (\frac {4 c \sqrt {b x^2+c x^4}}{3 b^2 x^2}-\frac {2 \sqrt {b x^2+c x^4}}{3 b x^4}\right )}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^6}\right ) (7 b B-6 A c)}{7 b}-\frac {2 A \sqrt {b x^2+c x^4}}{7 b x^8}\right )\) |
Input:
Int[(A + B*x^2)/(x^7*Sqrt[b*x^2 + c*x^4]),x]
Output:
((-2*A*Sqrt[b*x^2 + c*x^4])/(7*b*x^8) + ((7*b*B - 6*A*c)*((-2*Sqrt[b*x^2 + c*x^4])/(5*b*x^6) - (4*c*((-2*Sqrt[b*x^2 + c*x^4])/(3*b*x^4) + (4*c*Sqrt[ b*x^2 + c*x^4])/(3*b^2*x^2)))/(5*b)))/(7*b))/2
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_) ^(n_))^(q_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1) *(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] && !IntegerQ[p] && NeQ[k, j] && I ntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]
Time = 0.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(-\frac {\left (c \,x^{2}+b \right ) \left (\left (\frac {7 B \,x^{2}}{5}+A \right ) b^{3}-\frac {6 c \,x^{2} \left (\frac {14 B \,x^{2}}{9}+A \right ) b^{2}}{5}+\frac {8 \left (\frac {7 B \,x^{2}}{3}+A \right ) c^{2} x^{4} b}{5}-\frac {16 A \,c^{3} x^{6}}{5}\right )}{7 \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, x^{6} b^{4}}\) | \(85\) |
trager | \(-\frac {\left (-48 A \,c^{3} x^{6}+56 B b \,c^{2} x^{6}+24 A b \,c^{2} x^{4}-28 x^{4} B \,b^{2} c -18 A \,b^{2} c \,x^{2}+21 x^{2} B \,b^{3}+15 A \,b^{3}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{105 b^{4} x^{8}}\) | \(87\) |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (-48 A \,c^{3} x^{6}+56 B b \,c^{2} x^{6}+24 A b \,c^{2} x^{4}-28 x^{4} B \,b^{2} c -18 A \,b^{2} c \,x^{2}+21 x^{2} B \,b^{3}+15 A \,b^{3}\right )}{105 x^{6} b^{4} \sqrt {c \,x^{4}+b \,x^{2}}}\) | \(94\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (-48 A \,c^{3} x^{6}+56 B b \,c^{2} x^{6}+24 A b \,c^{2} x^{4}-28 x^{4} B \,b^{2} c -18 A \,b^{2} c \,x^{2}+21 x^{2} B \,b^{3}+15 A \,b^{3}\right )}{105 x^{6} b^{4} \sqrt {c \,x^{4}+b \,x^{2}}}\) | \(94\) |
risch | \(-\frac {\left (c \,x^{2}+b \right ) \left (-48 A \,c^{3} x^{6}+56 B b \,c^{2} x^{6}+24 A b \,c^{2} x^{4}-28 x^{4} B \,b^{2} c -18 A \,b^{2} c \,x^{2}+21 x^{2} B \,b^{3}+15 A \,b^{3}\right )}{105 x^{6} \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, b^{4}}\) | \(94\) |
orering | \(-\frac {\left (c \,x^{2}+b \right ) \left (-48 A \,c^{3} x^{6}+56 B b \,c^{2} x^{6}+24 A b \,c^{2} x^{4}-28 x^{4} B \,b^{2} c -18 A \,b^{2} c \,x^{2}+21 x^{2} B \,b^{3}+15 A \,b^{3}\right )}{105 x^{6} b^{4} \sqrt {c \,x^{4}+b \,x^{2}}}\) | \(94\) |
Input:
int((B*x^2+A)/x^7/(c*x^4+b*x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/7*(c*x^2+b)*((7/5*B*x^2+A)*b^3-6/5*c*x^2*(14/9*B*x^2+A)*b^2+8/5*(7/3*B* x^2+A)*c^2*x^4*b-16/5*A*c^3*x^6)/(x^2*(c*x^2+b))^(1/2)/x^6/b^4
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.65 \[ \int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx=-\frac {{\left (8 \, {\left (7 \, B b c^{2} - 6 \, A c^{3}\right )} x^{6} - 4 \, {\left (7 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{4} + 15 \, A b^{3} + 3 \, {\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{105 \, b^{4} x^{8}} \] Input:
integrate((B*x^2+A)/x^7/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")
Output:
-1/105*(8*(7*B*b*c^2 - 6*A*c^3)*x^6 - 4*(7*B*b^2*c - 6*A*b*c^2)*x^4 + 15*A *b^3 + 3*(7*B*b^3 - 6*A*b^2*c)*x^2)*sqrt(c*x^4 + b*x^2)/(b^4*x^8)
\[ \int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx=\int \frac {A + B x^{2}}{x^{7} \sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \] Input:
integrate((B*x**2+A)/x**7/(c*x**4+b*x**2)**(1/2),x)
Output:
Integral((A + B*x**2)/(x**7*sqrt(x**2*(b + c*x**2))), x)
Time = 0.04 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx=-\frac {1}{15} \, B {\left (\frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{3} x^{2}} - \frac {4 \, \sqrt {c x^{4} + b x^{2}} c}{b^{2} x^{4}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}}}{b x^{6}}\right )} + \frac {1}{35} \, A {\left (\frac {16 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{4} x^{2}} - \frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{3} x^{4}} + \frac {6 \, \sqrt {c x^{4} + b x^{2}} c}{b^{2} x^{6}} - \frac {5 \, \sqrt {c x^{4} + b x^{2}}}{b x^{8}}\right )} \] Input:
integrate((B*x^2+A)/x^7/(c*x^4+b*x^2)^(1/2),x, algorithm="maxima")
Output:
-1/15*B*(8*sqrt(c*x^4 + b*x^2)*c^2/(b^3*x^2) - 4*sqrt(c*x^4 + b*x^2)*c/(b^ 2*x^4) + 3*sqrt(c*x^4 + b*x^2)/(b*x^6)) + 1/35*A*(16*sqrt(c*x^4 + b*x^2)*c ^3/(b^4*x^2) - 8*sqrt(c*x^4 + b*x^2)*c^2/(b^3*x^4) + 6*sqrt(c*x^4 + b*x^2) *c/(b^2*x^6) - 5*sqrt(c*x^4 + b*x^2)/(b*x^8))
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (117) = 234\).
Time = 0.71 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.77 \[ \int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx=\frac {16 \, {\left (70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B c^{\frac {5}{2}} - 175 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b c^{\frac {5}{2}} + 210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A c^{\frac {7}{2}} + 147 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{2} c^{\frac {5}{2}} - 126 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b c^{\frac {7}{2}} - 49 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{3} c^{\frac {5}{2}} + 42 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{2} c^{\frac {7}{2}} + 7 \, B b^{4} c^{\frac {5}{2}} - 6 \, A b^{3} c^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{7} \mathrm {sgn}\left (x\right )} \] Input:
integrate((B*x^2+A)/x^7/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")
Output:
16/105*(70*(sqrt(c)*x - sqrt(c*x^2 + b))^8*B*c^(5/2) - 175*(sqrt(c)*x - sq rt(c*x^2 + b))^6*B*b*c^(5/2) + 210*(sqrt(c)*x - sqrt(c*x^2 + b))^6*A*c^(7/ 2) + 147*(sqrt(c)*x - sqrt(c*x^2 + b))^4*B*b^2*c^(5/2) - 126*(sqrt(c)*x - sqrt(c*x^2 + b))^4*A*b*c^(7/2) - 49*(sqrt(c)*x - sqrt(c*x^2 + b))^2*B*b^3* c^(5/2) + 42*(sqrt(c)*x - sqrt(c*x^2 + b))^2*A*b^2*c^(7/2) + 7*B*b^4*c^(5/ 2) - 6*A*b^3*c^(7/2))/(((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^7*sgn(x))
Time = 9.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx=\frac {\left (6\,A\,c-7\,B\,b\right )\,\sqrt {c\,x^4+b\,x^2}}{35\,b^2\,x^6}-\frac {A\,\sqrt {c\,x^4+b\,x^2}}{7\,b\,x^8}-\frac {\left (24\,A\,c^2-28\,B\,b\,c\right )\,\sqrt {c\,x^4+b\,x^2}}{105\,b^3\,x^4}+\frac {\left (48\,A\,c^3-56\,B\,b\,c^2\right )\,\sqrt {c\,x^4+b\,x^2}}{105\,b^4\,x^2} \] Input:
int((A + B*x^2)/(x^7*(b*x^2 + c*x^4)^(1/2)),x)
Output:
((6*A*c - 7*B*b)*(b*x^2 + c*x^4)^(1/2))/(35*b^2*x^6) - (A*(b*x^2 + c*x^4)^ (1/2))/(7*b*x^8) - ((24*A*c^2 - 28*B*b*c)*(b*x^2 + c*x^4)^(1/2))/(105*b^3* x^4) + ((48*A*c^3 - 56*B*b*c^2)*(b*x^2 + c*x^4)^(1/2))/(105*b^4*x^2)
Time = 0.20 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^2}{x^7 \sqrt {b x^2+c x^4}} \, dx=\frac {-15 \sqrt {c \,x^{2}+b}\, a \,b^{3}+18 \sqrt {c \,x^{2}+b}\, a \,b^{2} c \,x^{2}-24 \sqrt {c \,x^{2}+b}\, a b \,c^{2} x^{4}+48 \sqrt {c \,x^{2}+b}\, a \,c^{3} x^{6}-21 \sqrt {c \,x^{2}+b}\, b^{4} x^{2}+28 \sqrt {c \,x^{2}+b}\, b^{3} c \,x^{4}-56 \sqrt {c \,x^{2}+b}\, b^{2} c^{2} x^{6}-48 \sqrt {c}\, a \,c^{3} x^{7}+56 \sqrt {c}\, b^{2} c^{2} x^{7}}{105 b^{4} x^{7}} \] Input:
int((B*x^2+A)/x^7/(c*x^4+b*x^2)^(1/2),x)
Output:
( - 15*sqrt(b + c*x**2)*a*b**3 + 18*sqrt(b + c*x**2)*a*b**2*c*x**2 - 24*sq rt(b + c*x**2)*a*b*c**2*x**4 + 48*sqrt(b + c*x**2)*a*c**3*x**6 - 21*sqrt(b + c*x**2)*b**4*x**2 + 28*sqrt(b + c*x**2)*b**3*c*x**4 - 56*sqrt(b + c*x** 2)*b**2*c**2*x**6 - 48*sqrt(c)*a*c**3*x**7 + 56*sqrt(c)*b**2*c**2*x**7)/(1 05*b**4*x**7)