Integrand size = 24, antiderivative size = 118 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {(b c-a d)^2}{3 c^2 d \left (c+d x^2\right )^{3/2}}+\frac {2 a (b c-a d)}{c^3 \sqrt {c+d x^2}}-\frac {a^2 \sqrt {c+d x^2}}{2 c^3 x^2}-\frac {a (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 c^{7/2}} \] Output:
-1/3*(-a*d+b*c)^2/c^2/d/(d*x^2+c)^(3/2)+2*a*(-a*d+b*c)/c^3/(d*x^2+c)^(1/2) -1/2*a^2*(d*x^2+c)^(1/2)/c^3/x^2-1/2*a*(-5*a*d+4*b*c)*arctanh((d*x^2+c)^(1 /2)/c^(1/2))/c^(7/2)
Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{5/2}} \, dx=\frac {-2 b^2 c^3 x^2+4 a b c d x^2 \left (4 c+3 d x^2\right )-a^2 d \left (3 c^2+20 c d x^2+15 d^2 x^4\right )}{6 c^3 d x^2 \left (c+d x^2\right )^{3/2}}+\frac {a (-4 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 c^{7/2}} \] Input:
Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)^(5/2)),x]
Output:
(-2*b^2*c^3*x^2 + 4*a*b*c*d*x^2*(4*c + 3*d*x^2) - a^2*d*(3*c^2 + 20*c*d*x^ 2 + 15*d^2*x^4))/(6*c^3*d*x^2*(c + d*x^2)^(3/2)) + (a*(-4*b*c + 5*a*d)*Arc Tanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*c^(7/2))
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {354, 100, 27, 87, 61, 73, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^2}{x^4 \left (d x^2+c\right )^{5/2}}dx^2\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 b^2 c x^2+a (4 b c-5 a d)}{2 x^2 \left (d x^2+c\right )^{5/2}}dx^2}{c}-\frac {a^2}{c x^2 \left (c+d x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 b^2 c x^2+a (4 b c-5 a d)}{x^2 \left (d x^2+c\right )^{5/2}}dx^2}{2 c}-\frac {a^2}{c x^2 \left (c+d x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {a (4 b c-5 a d) \int \frac {1}{x^2 \left (d x^2+c\right )^{3/2}}dx^2}{c}-\frac {2 \left (\frac {2 b^2 c}{d}-\frac {a (4 b c-5 a d)}{c}\right )}{3 \left (c+d x^2\right )^{3/2}}}{2 c}-\frac {a^2}{c x^2 \left (c+d x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {a (4 b c-5 a d) \left (\frac {\int \frac {1}{x^2 \sqrt {d x^2+c}}dx^2}{c}+\frac {2}{c \sqrt {c+d x^2}}\right )}{c}-\frac {2 \left (\frac {2 b^2 c}{d}-\frac {a (4 b c-5 a d)}{c}\right )}{3 \left (c+d x^2\right )^{3/2}}}{2 c}-\frac {a^2}{c x^2 \left (c+d x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {a (4 b c-5 a d) \left (\frac {2 \int \frac {1}{\frac {x^4}{d}-\frac {c}{d}}d\sqrt {d x^2+c}}{c d}+\frac {2}{c \sqrt {c+d x^2}}\right )}{c}-\frac {2 \left (\frac {2 b^2 c}{d}-\frac {a (4 b c-5 a d)}{c}\right )}{3 \left (c+d x^2\right )^{3/2}}}{2 c}-\frac {a^2}{c x^2 \left (c+d x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {a (4 b c-5 a d) \left (\frac {2}{c \sqrt {c+d x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{3/2}}\right )}{c}-\frac {2 \left (\frac {2 b^2 c}{d}-\frac {a (4 b c-5 a d)}{c}\right )}{3 \left (c+d x^2\right )^{3/2}}}{2 c}-\frac {a^2}{c x^2 \left (c+d x^2\right )^{3/2}}\right )\) |
Input:
Int[(a + b*x^2)^2/(x^3*(c + d*x^2)^(5/2)),x]
Output:
(-(a^2/(c*x^2*(c + d*x^2)^(3/2))) + ((-2*((2*b^2*c)/d - (a*(4*b*c - 5*a*d) )/c))/(3*(c + d*x^2)^(3/2)) + (a*(4*b*c - 5*a*d)*(2/(c*Sqrt[c + d*x^2]) - (2*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/c^(3/2)))/c)/(2*c))/2
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 0.60 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(-\frac {-5 a d \left (a d -\frac {4 b c}{5}\right ) \left (x^{2} d +c \right )^{\frac {3}{2}} x^{2} \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{\sqrt {c}}\right )+\frac {20 a \,d^{2} x^{2} \left (-\frac {3 b \,x^{2}}{5}+a \right ) c^{\frac {3}{2}}}{3}+a d \left (-\frac {16 b \,x^{2}}{3}+a \right ) c^{\frac {5}{2}}+5 \sqrt {c}\, a^{2} d^{3} x^{4}+\frac {2 c^{\frac {7}{2}} b^{2} x^{2}}{3}}{2 \left (x^{2} d +c \right )^{\frac {3}{2}} c^{\frac {7}{2}} d \,x^{2}}\) | \(119\) |
default | \(a^{2} \left (-\frac {1}{2 c \,x^{2} \left (x^{2} d +c \right )^{\frac {3}{2}}}-\frac {5 d \left (\frac {1}{3 c \left (x^{2} d +c \right )^{\frac {3}{2}}}+\frac {\frac {1}{c \sqrt {x^{2} d +c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )}{c^{\frac {3}{2}}}}{c}\right )}{2 c}\right )-\frac {b^{2}}{3 d \left (x^{2} d +c \right )^{\frac {3}{2}}}+2 a b \left (\frac {1}{3 c \left (x^{2} d +c \right )^{\frac {3}{2}}}+\frac {\frac {1}{c \sqrt {x^{2} d +c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right )}{c^{\frac {3}{2}}}}{c}\right )\) | \(173\) |
risch | \(-\frac {a^{2} \sqrt {x^{2} d +c}}{2 c^{3} x^{2}}+\frac {5 a^{2} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right ) d}{2 c^{\frac {7}{2}}}-\frac {2 a \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {x^{2} d +c}}{x}\right ) b}{c^{\frac {5}{2}}}-\frac {13 d \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a^{2}}{12 c^{3} \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {7 \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a b}{6 c^{2} \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {13 d \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a^{2}}{12 c^{3} \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {7 \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a b}{6 c^{2} \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}+\frac {\sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a^{2}}{12 c^{3} \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a b}{6 c^{2} d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}+\frac {\sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, b^{2}}{12 c \,d^{2} \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, b^{2}}{12 c d \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {\sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a^{2}}{12 c^{3} \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a b}{6 c^{2} d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}+\frac {\sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, b^{2}}{12 c \,d^{2} \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}+\frac {\sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, b^{2}}{12 c d \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\) | \(886\) |
Input:
int((b*x^2+a)^2/x^3/(d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/2/(d*x^2+c)^(3/2)*(-5*a*d*(a*d-4/5*b*c)*(d*x^2+c)^(3/2)*x^2*arctanh((d* x^2+c)^(1/2)/c^(1/2))+20/3*a*d^2*x^2*(-3/5*b*x^2+a)*c^(3/2)+a*d*(-16/3*b*x ^2+a)*c^(5/2)+5*c^(1/2)*a^2*d^3*x^4+2/3*c^(7/2)*b^2*x^2)/c^(7/2)/d/x^2
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (100) = 200\).
Time = 0.11 (sec) , antiderivative size = 429, normalized size of antiderivative = 3.64 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (4 \, a b c d^{3} - 5 \, a^{2} d^{4}\right )} x^{6} + 2 \, {\left (4 \, a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x^{4} + {\left (4 \, a b c^{3} d - 5 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (3 \, a^{2} c^{3} d - 3 \, {\left (4 \, a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} - 8 \, a b c^{3} d + 10 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (c^{4} d^{3} x^{6} + 2 \, c^{5} d^{2} x^{4} + c^{6} d x^{2}\right )}}, \frac {3 \, {\left ({\left (4 \, a b c d^{3} - 5 \, a^{2} d^{4}\right )} x^{6} + 2 \, {\left (4 \, a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x^{4} + {\left (4 \, a b c^{3} d - 5 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{2} + c} \sqrt {-c}}{c}\right ) - {\left (3 \, a^{2} c^{3} d - 3 \, {\left (4 \, a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} - 8 \, a b c^{3} d + 10 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left (c^{4} d^{3} x^{6} + 2 \, c^{5} d^{2} x^{4} + c^{6} d x^{2}\right )}}\right ] \] Input:
integrate((b*x^2+a)^2/x^3/(d*x^2+c)^(5/2),x, algorithm="fricas")
Output:
[-1/12*(3*((4*a*b*c*d^3 - 5*a^2*d^4)*x^6 + 2*(4*a*b*c^2*d^2 - 5*a^2*c*d^3) *x^4 + (4*a*b*c^3*d - 5*a^2*c^2*d^2)*x^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x ^2 + c)*sqrt(c) + 2*c)/x^2) + 2*(3*a^2*c^3*d - 3*(4*a*b*c^2*d^2 - 5*a^2*c* d^3)*x^4 + 2*(b^2*c^4 - 8*a*b*c^3*d + 10*a^2*c^2*d^2)*x^2)*sqrt(d*x^2 + c) )/(c^4*d^3*x^6 + 2*c^5*d^2*x^4 + c^6*d*x^2), 1/6*(3*((4*a*b*c*d^3 - 5*a^2* d^4)*x^6 + 2*(4*a*b*c^2*d^2 - 5*a^2*c*d^3)*x^4 + (4*a*b*c^3*d - 5*a^2*c^2* d^2)*x^2)*sqrt(-c)*arctan(sqrt(d*x^2 + c)*sqrt(-c)/c) - (3*a^2*c^3*d - 3*( 4*a*b*c^2*d^2 - 5*a^2*c*d^3)*x^4 + 2*(b^2*c^4 - 8*a*b*c^3*d + 10*a^2*c^2*d ^2)*x^2)*sqrt(d*x^2 + c))/(c^4*d^3*x^6 + 2*c^5*d^2*x^4 + c^6*d*x^2)]
\[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{x^{3} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((b*x**2+a)**2/x**3/(d*x**2+c)**(5/2),x)
Output:
Integral((a + b*x**2)**2/(x**3*(c + d*x**2)**(5/2)), x)
Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {2 \, a b \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{c^{\frac {5}{2}}} + \frac {5 \, a^{2} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{2 \, c^{\frac {7}{2}}} + \frac {2 \, a b}{\sqrt {d x^{2} + c} c^{2}} + \frac {2 \, a b}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c} - \frac {b^{2}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {5 \, a^{2} d}{2 \, \sqrt {d x^{2} + c} c^{3}} - \frac {5 \, a^{2} d}{6 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{2}} - \frac {a^{2}}{2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c x^{2}} \] Input:
integrate((b*x^2+a)^2/x^3/(d*x^2+c)^(5/2),x, algorithm="maxima")
Output:
-2*a*b*arcsinh(c/(sqrt(c*d)*abs(x)))/c^(5/2) + 5/2*a^2*d*arcsinh(c/(sqrt(c *d)*abs(x)))/c^(7/2) + 2*a*b/(sqrt(d*x^2 + c)*c^2) + 2/3*a*b/((d*x^2 + c)^ (3/2)*c) - 1/3*b^2/((d*x^2 + c)^(3/2)*d) - 5/2*a^2*d/(sqrt(d*x^2 + c)*c^3) - 5/6*a^2*d/((d*x^2 + c)^(3/2)*c^2) - 1/2*a^2/((d*x^2 + c)^(3/2)*c*x^2)
Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left (4 \, a b c - 5 \, a^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, \sqrt {-c} c^{3}} - \frac {\sqrt {d x^{2} + c} a^{2}}{2 \, c^{3} x^{2}} - \frac {b^{2} c^{3} - 6 \, {\left (d x^{2} + c\right )} a b c d - 2 \, a b c^{2} d + 6 \, {\left (d x^{2} + c\right )} a^{2} d^{2} + a^{2} c d^{2}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{3} d} \] Input:
integrate((b*x^2+a)^2/x^3/(d*x^2+c)^(5/2),x, algorithm="giac")
Output:
1/2*(4*a*b*c - 5*a^2*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(sqrt(-c)*c^3) - 1/2*sqrt(d*x^2 + c)*a^2/(c^3*x^2) - 1/3*(b^2*c^3 - 6*(d*x^2 + c)*a*b*c*d - 2*a*b*c^2*d + 6*(d*x^2 + c)*a^2*d^2 + a^2*c*d^2)/((d*x^2 + c)^(3/2)*c^3*d )
Time = 1.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{5/2}} \, dx=\frac {a\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )\,\left (5\,a\,d-4\,b\,c\right )}{2\,c^{7/2}}-\frac {\frac {\left (d\,x^2+c\right )\,\left (-5\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{3\,c^2}-\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{3\,c}+\frac {d\,{\left (d\,x^2+c\right )}^2\,\left (5\,a^2\,d-4\,a\,b\,c\right )}{2\,c^3}}{d\,{\left (d\,x^2+c\right )}^{5/2}-c\,d\,{\left (d\,x^2+c\right )}^{3/2}} \] Input:
int((a + b*x^2)^2/(x^3*(c + d*x^2)^(5/2)),x)
Output:
(a*atanh((c + d*x^2)^(1/2)/c^(1/2))*(5*a*d - 4*b*c))/(2*c^(7/2)) - (((c + d*x^2)*(b^2*c^2 - 5*a^2*d^2 + 4*a*b*c*d))/(3*c^2) - (a^2*d^2 + b^2*c^2 - 2 *a*b*c*d)/(3*c) + (d*(c + d*x^2)^2*(5*a^2*d - 4*a*b*c))/(2*c^3))/(d*(c + d *x^2)^(5/2) - c*d*(c + d*x^2)^(3/2))
Time = 0.23 (sec) , antiderivative size = 582, normalized size of antiderivative = 4.93 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{5/2}} \, dx=\frac {-3 \sqrt {d \,x^{2}+c}\, a^{2} c^{3} d -20 \sqrt {d \,x^{2}+c}\, a^{2} c^{2} d^{2} x^{2}-15 \sqrt {d \,x^{2}+c}\, a^{2} c \,d^{3} x^{4}+16 \sqrt {d \,x^{2}+c}\, a b \,c^{3} d \,x^{2}+12 \sqrt {d \,x^{2}+c}\, a b \,c^{2} d^{2} x^{4}-2 \sqrt {d \,x^{2}+c}\, b^{2} c^{4} x^{2}-15 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}-\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{2} c^{2} d^{2} x^{2}-30 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}-\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{2} c \,d^{3} x^{4}-15 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}-\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{2} d^{4} x^{6}+12 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}-\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a b \,c^{3} d \,x^{2}+24 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}-\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a b \,c^{2} d^{2} x^{4}+12 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}-\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a b c \,d^{3} x^{6}+15 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{2} c^{2} d^{2} x^{2}+30 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{2} c \,d^{3} x^{4}+15 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{2} d^{4} x^{6}-12 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a b \,c^{3} d \,x^{2}-24 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a b \,c^{2} d^{2} x^{4}-12 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a b c \,d^{3} x^{6}}{6 c^{4} d \,x^{2} \left (d^{2} x^{4}+2 c d \,x^{2}+c^{2}\right )} \] Input:
int((b*x^2+a)^2/x^3/(d*x^2+c)^(5/2),x)
Output:
( - 3*sqrt(c + d*x**2)*a**2*c**3*d - 20*sqrt(c + d*x**2)*a**2*c**2*d**2*x* *2 - 15*sqrt(c + d*x**2)*a**2*c*d**3*x**4 + 16*sqrt(c + d*x**2)*a*b*c**3*d *x**2 + 12*sqrt(c + d*x**2)*a*b*c**2*d**2*x**4 - 2*sqrt(c + d*x**2)*b**2*c **4*x**2 - 15*sqrt(c)*log((sqrt(c + d*x**2) - sqrt(c) + sqrt(d)*x)/sqrt(c) )*a**2*c**2*d**2*x**2 - 30*sqrt(c)*log((sqrt(c + d*x**2) - sqrt(c) + sqrt( d)*x)/sqrt(c))*a**2*c*d**3*x**4 - 15*sqrt(c)*log((sqrt(c + d*x**2) - sqrt( c) + sqrt(d)*x)/sqrt(c))*a**2*d**4*x**6 + 12*sqrt(c)*log((sqrt(c + d*x**2) - sqrt(c) + sqrt(d)*x)/sqrt(c))*a*b*c**3*d*x**2 + 24*sqrt(c)*log((sqrt(c + d*x**2) - sqrt(c) + sqrt(d)*x)/sqrt(c))*a*b*c**2*d**2*x**4 + 12*sqrt(c)* log((sqrt(c + d*x**2) - sqrt(c) + sqrt(d)*x)/sqrt(c))*a*b*c*d**3*x**6 + 15 *sqrt(c)*log((sqrt(c + d*x**2) + sqrt(c) + sqrt(d)*x)/sqrt(c))*a**2*c**2*d **2*x**2 + 30*sqrt(c)*log((sqrt(c + d*x**2) + sqrt(c) + sqrt(d)*x)/sqrt(c) )*a**2*c*d**3*x**4 + 15*sqrt(c)*log((sqrt(c + d*x**2) + sqrt(c) + sqrt(d)* x)/sqrt(c))*a**2*d**4*x**6 - 12*sqrt(c)*log((sqrt(c + d*x**2) + sqrt(c) + sqrt(d)*x)/sqrt(c))*a*b*c**3*d*x**2 - 24*sqrt(c)*log((sqrt(c + d*x**2) + s qrt(c) + sqrt(d)*x)/sqrt(c))*a*b*c**2*d**2*x**4 - 12*sqrt(c)*log((sqrt(c + d*x**2) + sqrt(c) + sqrt(d)*x)/sqrt(c))*a*b*c*d**3*x**6)/(6*c**4*d*x**2*( c**2 + 2*c*d*x**2 + d**2*x**4))