Integrand size = 26, antiderivative size = 133 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}-\frac {\sqrt {b} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 d^{3/2}} \]
-1/2*(-3*a*d+b*c)*arctanh(d^(1/2)*(b*x^2+a)^(1/2)/b^(1/2)/(d*x^2+c)^(1/2)) *b^(1/2)/d^(3/2)-a^(3/2)*arctanh(c^(1/2)*(b*x^2+a)^(1/2)/a^(1/2)/(d*x^2+c) ^(1/2))/c^(1/2)+1/2*b*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d
Time = 1.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\frac {1}{2} \left (\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{d}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c}}-\frac {\sqrt {b} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{d^{3/2}}\right ) \]
((b*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/d - (2*a^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[ c + d*x^2])/(Sqrt[c]*Sqrt[a + b*x^2])])/Sqrt[c] - (Sqrt[b]*(b*c - 3*a*d)*A rcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/(Sqrt[d]*Sqrt[a + b*x^2])])/d^(3/2))/2
Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {354, 113, 27, 175, 66, 104, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^{3/2}}{x^2 \sqrt {d x^2+c}}dx^2\) |
| \(\Big \downarrow \) 113 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 a^2 d-b (b c-3 a d) x^2}{2 x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx^2}{d}+\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 a^2 d-b (b c-3 a d) x^2}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx^2}{2 d}+\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{d}\right )\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 a^2 d \int \frac {1}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx^2-b (b c-3 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx^2}{2 d}+\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{d}\right )\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 a^2 d \int \frac {1}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx^2-2 b (b c-3 a d) \int \frac {1}{b-d x^4}d\frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c}}}{2 d}+\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{d}\right )\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{2} \left (\frac {4 a^2 d \int \frac {1}{c x^4-a}d\frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c}}-2 b (b c-3 a d) \int \frac {1}{b-d x^4}d\frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c}}}{2 d}+\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{d}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {4 a^{3/2} d \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}-\frac {2 \sqrt {b} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}}}{2 d}+\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{d}\right )\) |
((b*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/d + ((-4*a^(3/2)*d*ArcTanh[(Sqrt[c]*S qrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[c] - (2*Sqrt[b]*(b*c - 3* a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/Sqrt[d] )/(2*d))/2
3.10.47.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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[(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]
Leaf count of result is larger than twice the leaf count of optimal. \(242\) vs. \(2(101)=202\).
Time = 3.08 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.83
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{2 d}+\frac {3 a b \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{4 \sqrt {b d}}-\frac {b^{2} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c}{4 d \sqrt {b d}}-\frac {a^{2} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right )}{2 \sqrt {a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(243\) |
| default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (2 \sqrt {b d}\, \ln \left (\frac {a d \,x^{2}+c b \,x^{2}+2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a^{2} d -3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a b d +\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{2} c -2 \sqrt {b d}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a c}\, b \right )}{4 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, \sqrt {a c}\, d}\) | \(251\) |
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/2*b/d*(b*d* x^4+(a*d+b*c)*x^2+a*c)^(1/2)+3/4*a*b*ln((1/2*a*d+1/2*b*c+b*d*x^2)/(b*d)^(1 /2)+(b*d*x^4+(a*d+b*c)*x^2+a*c)^(1/2))/(b*d)^(1/2)-1/4*b^2/d*ln((1/2*a*d+1 /2*b*c+b*d*x^2)/(b*d)^(1/2)+(b*d*x^4+(a*d+b*c)*x^2+a*c)^(1/2))/(b*d)^(1/2) *c-1/2*a^2/(a*c)^(1/2)*ln((2*a*c+(a*d+b*c)*x^2+2*(a*c)^(1/2)*(b*d*x^4+(a*d +b*c)*x^2+a*c)^(1/2))/x^2))
Time = 0.72 (sec) , antiderivative size = 918, normalized size of antiderivative = 6.90 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\left [\frac {2 \, a d \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ) - {\left (b c - 3 \, a d\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) + 4 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} b}{8 \, d}, \frac {a d \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ) + {\left (b c - 3 \, a d\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) + 2 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} b}{4 \, d}, \frac {4 \, a d \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) - {\left (b c - 3 \, a d\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) + 4 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} b}{8 \, d}, \frac {2 \, a d \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) + {\left (b c - 3 \, a d\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) + 2 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} b}{4 \, d}\right ] \]
[1/8*(2*a*d*sqrt(a/c)*log(((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^4 + 8*a^2*c^2 + 8*(a*b*c^2 + a^2*c*d)*x^2 - 4*(2*a*c^2 + (b*c^2 + a*c*d)*x^2)*sqrt(b*x^ 2 + a)*sqrt(d*x^2 + c)*sqrt(a/c))/x^4) - (b*c - 3*a*d)*sqrt(b/d)*log(8*b^2 *d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x^2 + 4*( 2*b*d^2*x^2 + b*c*d + a*d^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b/d)) + 4*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*b)/d, 1/4*(a*d*sqrt(a/c)*log(((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^4 + 8*a^2*c^2 + 8*(a*b*c^2 + a^2*c*d)*x^2 - 4*(2*a *c^2 + (b*c^2 + a*c*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(a/c))/x^4 ) + (b*c - 3*a*d)*sqrt(-b/d)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b/d)/(b^2*d*x^4 + a*b*c + (b^2*c + a*b*d)*x^2) ) + 2*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*b)/d, 1/8*(4*a*d*sqrt(-a/c)*arctan(1 /2*((b*c + a*d)*x^2 + 2*a*c)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-a/c)/(a *b*d*x^4 + a^2*c + (a*b*c + a^2*d)*x^2)) - (b*c - 3*a*d)*sqrt(b/d)*log(8*b ^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x^2 + 4 *(2*b*d^2*x^2 + b*c*d + a*d^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b/d)) + 4*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*b)/d, 1/4*(2*a*d*sqrt(-a/c)*arctan(1/2 *((b*c + a*d)*x^2 + 2*a*c)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-a/c)/(a*b *d*x^4 + a^2*c + (a*b*c + a^2*d)*x^2)) + (b*c - 3*a*d)*sqrt(-b/d)*arctan(1 /2*(2*b*d*x^2 + b*c + a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b/d)/(b^2 *d*x^4 + a*b*c + (b^2*c + a*b*d)*x^2)) + 2*sqrt(b*x^2 + a)*sqrt(d*x^2 +...
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{x \sqrt {c + d x^{2}}}\, dx \]
Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x\,\sqrt {d\,x^2+c}} \,d x \]