Integrand size = 21, antiderivative size = 131 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2} \, dx=-\frac {(b c-a d) x \sqrt {a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^2}-\frac {\sqrt {b c-a d} (2 b c+a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} d^2} \]
b^(3/2)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/d^2-1/2*(a*d+2*b*c)*arctanh(x*( -a*d+b*c)^(1/2)/c^(1/2)/(b*x^2+a)^(1/2))*(-a*d+b*c)^(1/2)/c^(3/2)/d^2-1/2* (-a*d+b*c)*x*(b*x^2+a)^(1/2)/c/d/(d*x^2+c)
Time = 0.47 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2} \, dx=\frac {\frac {d (-b c+a d) x \sqrt {a+b x^2}}{c \left (c+d x^2\right )}-\frac {\sqrt {-b c+a d} (2 b c+a d) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{c^{3/2}}-2 b^{3/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 d^2} \]
((d*(-(b*c) + a*d)*x*Sqrt[a + b*x^2])/(c*(c + d*x^2)) - (Sqrt[-(b*c) + a*d ]*(2*b*c + a*d)*ArcTan[(-(d*x*Sqrt[a + b*x^2]) + Sqrt[b]*(c + d*x^2))/(Sqr t[c]*Sqrt[-(b*c) + a*d])])/c^(3/2) - 2*b^(3/2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(2*d^2)
Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {315, 398, 224, 219, 291, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\int \frac {2 b^2 c x^2+a (b c+a d)}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c d}-\frac {x \sqrt {a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\frac {2 b^2 c \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) (a d+2 b c) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}}{2 c d}-\frac {x \sqrt {a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {2 b^2 c \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) (a d+2 b c) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}}{2 c d}-\frac {x \sqrt {a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {2 b^{3/2} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) (a d+2 b c) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}}{2 c d}-\frac {x \sqrt {a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {2 b^{3/2} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) (a d+2 b c) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{2 c d}-\frac {x \sqrt {a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {2 b^{3/2} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} (a d+2 b c) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}}{2 c d}-\frac {x \sqrt {a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )}\) |
-1/2*((b*c - a*d)*x*Sqrt[a + b*x^2])/(c*d*(c + d*x^2)) + ((2*b^(3/2)*c*Arc Tanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d - (Sqrt[b*c - a*d]*(2*b*c + a*d)*ArcT anh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d))/(2*c*d)
3.1.58.3.1 Defintions of rubi rules used
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[((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[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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 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 = 2.43 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.12
| method | result | size |
| pseudoelliptic | \(\frac {-\left (d \,x^{2}+c \right ) \left (a d +2 b c \right ) \left (a d -b c \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\sqrt {\left (a d -b c \right ) c}\, \left (2 c \,b^{\frac {3}{2}} \left (d \,x^{2}+c \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+d x \sqrt {b \,x^{2}+a}\, \left (a d -b c \right )\right )}{2 \sqrt {\left (a d -b c \right ) c}\, d^{2} c \left (d \,x^{2}+c \right )}\) | \(147\) |
| default | \(\text {Expression too large to display}\) | \(3349\) |
1/2/((a*d-b*c)*c)^(1/2)*(-(d*x^2+c)*(a*d+2*b*c)*(a*d-b*c)*arctan(c*(b*x^2+ a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+((a*d-b*c)*c)^(1/2)*(2*c*b^(3/2)*(d*x^2+c) *arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+d*x*(b*x^2+a)^(1/2)*(a*d-b*c)))/d^2/c/ (d*x^2+c)
Time = 0.35 (sec) , antiderivative size = 907, normalized size of antiderivative = 6.92 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2} \, dx=\left [-\frac {4 \, {\left (b c d - a d^{2}\right )} \sqrt {b x^{2} + a} x - 4 \, {\left (b c d x^{2} + b c^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (2 \, b c^{2} + a c d + {\left (2 \, b c d + a d^{2}\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a c^{2} x + {\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{8 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}}, -\frac {4 \, {\left (b c d - a d^{2}\right )} \sqrt {b x^{2} + a} x + 8 \, {\left (b c d x^{2} + b c^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, b c^{2} + a c d + {\left (2 \, b c d + a d^{2}\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a c^{2} x + {\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{8 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}}, -\frac {2 \, {\left (b c d - a d^{2}\right )} \sqrt {b x^{2} + a} x - {\left (2 \, b c^{2} + a c d + {\left (2 \, b c d + a d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{c}} \arctan \left (\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {b c - a d}{c}}}{2 \, {\left ({\left (b^{2} c - a b d\right )} x^{3} + {\left (a b c - a^{2} d\right )} x\right )}}\right ) - 2 \, {\left (b c d x^{2} + b c^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{4 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}}, -\frac {2 \, {\left (b c d - a d^{2}\right )} \sqrt {b x^{2} + a} x + 4 \, {\left (b c d x^{2} + b c^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, b c^{2} + a c d + {\left (2 \, b c d + a d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{c}} \arctan \left (\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {b c - a d}{c}}}{2 \, {\left ({\left (b^{2} c - a b d\right )} x^{3} + {\left (a b c - a^{2} d\right )} x\right )}}\right )}{4 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}}\right ] \]
[-1/8*(4*(b*c*d - a*d^2)*sqrt(b*x^2 + a)*x - 4*(b*c*d*x^2 + b*c^2)*sqrt(b) *log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - (2*b*c^2 + a*c*d + (2*b *c*d + a*d^2)*x^2)*sqrt((b*c - a*d)/c)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d ^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 - 4*(a*c^2*x + (2*b*c^2 - a*c*d)*x^3)*sqrt(b*x^2 + a)*sqrt((b*c - a*d)/c))/(d^2*x^4 + 2*c*d*x^2 + c^2)))/(c*d^3*x^2 + c^2*d^2), -1/8*(4*(b*c*d - a*d^2)*sqrt(b*x^2 + a)*x + 8*(b*c*d*x^2 + b*c^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (2*b*c ^2 + a*c*d + (2*b*c*d + a*d^2)*x^2)*sqrt((b*c - a*d)/c)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 - 4*(a* c^2*x + (2*b*c^2 - a*c*d)*x^3)*sqrt(b*x^2 + a)*sqrt((b*c - a*d)/c))/(d^2*x ^4 + 2*c*d*x^2 + c^2)))/(c*d^3*x^2 + c^2*d^2), -1/4*(2*(b*c*d - a*d^2)*sqr t(b*x^2 + a)*x - (2*b*c^2 + a*c*d + (2*b*c*d + a*d^2)*x^2)*sqrt(-(b*c - a* d)/c)*arctan(1/2*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + a)*sqrt(-(b*c - a* d)/c)/((b^2*c - a*b*d)*x^3 + (a*b*c - a^2*d)*x)) - 2*(b*c*d*x^2 + b*c^2)*s qrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a))/(c*d^3*x^2 + c^2*d ^2), -1/4*(2*(b*c*d - a*d^2)*sqrt(b*x^2 + a)*x + 4*(b*c*d*x^2 + b*c^2)*sqr t(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (2*b*c^2 + a*c*d + (2*b*c*d + a *d^2)*x^2)*sqrt(-(b*c - a*d)/c)*arctan(1/2*((2*b*c - a*d)*x^2 + a*c)*sqrt( b*x^2 + a)*sqrt(-(b*c - a*d)/c)/((b^2*c - a*b*d)*x^3 + (a*b*c - a^2*d)*x)) )/(c*d^3*x^2 + c^2*d^2)]
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{\left (c + d x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (109) = 218\).
Time = 0.30 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.42 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2} \, dx=-\frac {b^{\frac {3}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{2 \, d^{2}} + \frac {{\left (2 \, b^{\frac {5}{2}} c^{2} - a b^{\frac {3}{2}} c d - a^{2} \sqrt {b} d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{2 \, \sqrt {-b^{2} c^{2} + a b c d} c d^{2}} - \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} d^{2} + a^{2} b^{\frac {3}{2}} c d - a^{3} \sqrt {b} d^{2}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )} c d^{2}} \]
-1/2*b^(3/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2)/d^2 + 1/2*(2*b^(5/2)*c^2 - a*b^(3/2)*c*d - a^2*sqrt(b)*d^2)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/(sqrt(-b^2*c^2 + a*b*c*d) *c*d^2) - (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*c^2 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2)*c*d + (sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqr t(b)*d^2 + a^2*b^(3/2)*c*d - a^3*sqrt(b)*d^2)/(((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^ 2 + a))^2*a*d + a^2*d)*c*d^2)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^2} \,d x \]