Integrand size = 24, antiderivative size = 102 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c \sqrt {c+d x^2}}{a x}-\frac {(b c-a d)^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} b}+\frac {d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b} \]
-(-a*d+b*c)^(3/2)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a^(3/ 2)/b+d^(3/2)*arctanh(x*d^(1/2)/(d*x^2+c)^(1/2))/b-c*(d*x^2+c)^(1/2)/a/x
Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.26 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\frac {(b c-a d)^{3/2} x \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )-\sqrt {a} \left (b c \sqrt {c+d x^2}+a d^{3/2} x \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )\right )}{a^{3/2} b x} \]
((b*c - a*d)^(3/2)*x*ArcTan[(a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]) )/(Sqrt[a]*Sqrt[b*c - a*d])] - Sqrt[a]*(b*c*Sqrt[c + d*x^2] + a*d^(3/2)*x* Log[-(Sqrt[d]*x) + Sqrt[c + d*x^2]]))/(a^(3/2)*b*x)
Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {376, 25, 398, 224, 219, 291, 218}
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.
\(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 376 |
\(\displaystyle \frac {\int -\frac {c (b c-2 a d)-a d^2 x^2}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a}-\frac {c \sqrt {c+d x^2}}{a x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {c (b c-2 a d)-a d^2 x^2}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a}-\frac {c \sqrt {c+d x^2}}{a x}\) |
\(\Big \downarrow \) 398 |
\(\displaystyle -\frac {\frac {(b c-a d)^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}-\frac {a d^2 \int \frac {1}{\sqrt {d x^2+c}}dx}{b}}{a}-\frac {c \sqrt {c+d x^2}}{a x}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {\frac {(b c-a d)^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}-\frac {a d^2 \int \frac {1}{1-\frac {d x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}}{a}-\frac {c \sqrt {c+d x^2}}{a x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {(b c-a d)^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}-\frac {a d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b}}{a}-\frac {c \sqrt {c+d x^2}}{a x}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle -\frac {\frac {(b c-a d)^2 \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}-\frac {a d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b}}{a}-\frac {c \sqrt {c+d x^2}}{a x}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {\frac {(b c-a d)^{3/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} b}-\frac {a d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b}}{a}-\frac {c \sqrt {c+d x^2}}{a x}\) |
-((c*Sqrt[c + d*x^2])/(a*x)) - (((b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d] *x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*b) - (a*d^(3/2)*ArcTanh[(Sqrt[d]* x)/Sqrt[c + d*x^2]])/b)/a
3.7.91.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1 )/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^ 2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b*c - a*d)*(m + 1) + 2*c*(b*c*(p + 1) + a* d*(q - 1)) + d*((b*c - a*d)*(m + 1) + 2*b*c*(p + q))*x^2, x], x], x] /; Fre eQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && LtQ[m, -1] & & IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Time = 3.00 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.12
method | result | size |
pseudoelliptic | \(\frac {-x \left (a d -b c \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )+\left (\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) d^{\frac {3}{2}} a x -b c \sqrt {d \,x^{2}+c}\right ) \sqrt {\left (a d -b c \right ) a}}{\sqrt {\left (a d -b c \right ) a}\, a x b}\) | \(114\) |
risch | \(-\frac {c \sqrt {d \,x^{2}+c}}{a x}+\frac {\frac {d^{\frac {3}{2}} a \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b}-\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{a}\) | \(403\) |
default | \(\text {Expression too large to display}\) | \(1330\) |
1/((a*d-b*c)*a)^(1/2)*(-x*(a*d-b*c)^2*arctanh((d*x^2+c)^(1/2)/x*a/((a*d-b* c)*a)^(1/2))+(arctanh((d*x^2+c)^(1/2)/x/d^(1/2))*d^(3/2)*a*x-b*c*(d*x^2+c) ^(1/2))*((a*d-b*c)*a)^(1/2))/a/x/b
Time = 0.33 (sec) , antiderivative size = 718, normalized size of antiderivative = 7.04 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\left [\frac {2 \, a d^{\frac {3}{2}} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - {\left (b c - a d\right )} x \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, \sqrt {d x^{2} + c} b c}{4 \, a b x}, -\frac {4 \, a \sqrt {-d} d x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (b c - a d\right )} x \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, \sqrt {d x^{2} + c} b c}{4 \, a b x}, \frac {a d^{\frac {3}{2}} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - {\left (b c - a d\right )} x \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) - 2 \, \sqrt {d x^{2} + c} b c}{2 \, a b x}, -\frac {2 \, a \sqrt {-d} d x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (b c - a d\right )} x \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + 2 \, \sqrt {d x^{2} + c} b c}{2 \, a b x}\right ] \]
[1/4*(2*a*d^(3/2)*x*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - (b*c - a*d)*x*sqrt(-(b*c - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*(a^2*c*x - (a*b*c - 2*a^2*d) *x^3)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*sqrt(d*x^2 + c)*b*c)/(a*b*x), -1/4*(4*a*sqrt(-d)*d*x*arctan(sqrt(-d)*x/ sqrt(d*x^2 + c)) + (b*c - a*d)*x*sqrt(-(b*c - a*d)/a)*log(((b^2*c^2 - 8*a* b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*(a^2* c*x - (a*b*c - 2*a^2*d)*x^3)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^ 4 + 2*a*b*x^2 + a^2)) + 4*sqrt(d*x^2 + c)*b*c)/(a*b*x), 1/2*(a*d^(3/2)*x*l og(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - (b*c - a*d)*x*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^3 + (b*c^2 - a*c*d)*x)) - 2*sqrt(d*x^2 + c)*b*c )/(a*b*x), -1/2*(2*a*sqrt(-d)*d*x*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (b* c - a*d)*x*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d *x^2 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^3 + (b*c^2 - a*c*d)*x)) + 2*sqrt(d*x^2 + c)*b*c)/(a*b*x)]
\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}}}{x^{2} \left (a + b x^{2}\right )}\, dx \]
\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )} x^{2}} \,d x } \]
\[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}}{x^2\,\left (b\,x^2+a\right )} \,d x \]