Integrand size = 21, antiderivative size = 90 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 x}{a b^2 \sqrt {a+b x^2}}+\frac {d^2 x \sqrt {a+b x^2}}{2 b^2}+\frac {d (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \]
1/2*d*(-3*a*d+4*b*c)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(5/2)+(-a*d+b*c) ^2*x/a/b^2/(b*x^2+a)^(1/2)+1/2*d^2*x*(b*x^2+a)^(1/2)/b^2
Time = 0.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (2 b^2 c^2-4 a b c d+3 a^2 d^2+a b d^2 x^2\right )}{2 a b^2 \sqrt {a+b x^2}}-\frac {d (4 b c-3 a d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{5/2}} \]
(x*(2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2 + a*b*d^2*x^2))/(2*a*b^2*Sqrt[a + b* x^2]) - (d*(4*b*c - 3*a*d)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(2*b^(5/2) )
Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {315, 27, 299, 224, 219}
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.
\(\displaystyle \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 315 |
\(\displaystyle \frac {\int \frac {d \left (a c-(2 b c-3 a d) x^2\right )}{\sqrt {b x^2+a}}dx}{a b}+\frac {x \left (c+d x^2\right ) (b c-a d)}{a b \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {d \int \frac {a c-(2 b c-3 a d) x^2}{\sqrt {b x^2+a}}dx}{a b}+\frac {x \left (c+d x^2\right ) (b c-a d)}{a b \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {d \left (\frac {a (4 b c-3 a d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}-\frac {x \sqrt {a+b x^2} (2 b c-3 a d)}{2 b}\right )}{a b}+\frac {x \left (c+d x^2\right ) (b c-a d)}{a b \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {d \left (\frac {a (4 b c-3 a d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}-\frac {x \sqrt {a+b x^2} (2 b c-3 a d)}{2 b}\right )}{a b}+\frac {x \left (c+d x^2\right ) (b c-a d)}{a b \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {d \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (4 b c-3 a d)}{2 b^{3/2}}-\frac {x \sqrt {a+b x^2} (2 b c-3 a d)}{2 b}\right )}{a b}+\frac {x \left (c+d x^2\right ) (b c-a d)}{a b \sqrt {a+b x^2}}\) |
((b*c - a*d)*x*(c + d*x^2))/(a*b*Sqrt[a + b*x^2]) + (d*(-1/2*((2*b*c - 3*a *d)*x*Sqrt[a + b*x^2])/b + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/(a*b)
3.1.83.3.1 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[((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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 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]
Time = 2.39 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(-\frac {3 \left (\sqrt {b \,x^{2}+a}\, a d \left (a d -\frac {4 b c}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-x \left (-\frac {4 \left (-\frac {d \,x^{2}}{4}+c \right ) d a \,b^{\frac {3}{2}}}{3}+\sqrt {b}\, a^{2} d^{2}+\frac {2 b^{\frac {5}{2}} c^{2}}{3}\right )\right )}{2 \sqrt {b \,x^{2}+a}\, b^{\frac {5}{2}} a}\) | \(93\) |
risch | \(\frac {d^{2} x \sqrt {b \,x^{2}+a}}{2 b^{2}}-\frac {\frac {a \,d^{2} x}{\sqrt {b \,x^{2}+a}}-\frac {2 b^{2} c^{2} x}{a \sqrt {b \,x^{2}+a}}+\left (3 a b \,d^{2}-4 b^{2} c d \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b^{2}}\) | \(114\) |
default | \(\frac {c^{2} x}{a \sqrt {b \,x^{2}+a}}+d^{2} \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+2 c d \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )\) | \(123\) |
-3/2/(b*x^2+a)^(1/2)/b^(5/2)*((b*x^2+a)^(1/2)*a*d*(a*d-4/3*b*c)*arctanh((b *x^2+a)^(1/2)/x/b^(1/2))-x*(-4/3*(-1/4*d*x^2+c)*d*a*b^(3/2)+b^(1/2)*a^2*d^ 2+2/3*b^(5/2)*c^2))/a
Time = 0.27 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.07 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (a b^{2} d^{2} x^{3} + {\left (2 \, b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, -\frac {{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (a b^{2} d^{2} x^{3} + {\left (2 \, b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \]
[-1/4*((4*a^2*b*c*d - 3*a^3*d^2 + (4*a*b^2*c*d - 3*a^2*b*d^2)*x^2)*sqrt(b) *log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(a*b^2*d^2*x^3 + (2*b ^3*c^2 - 4*a*b^2*c*d + 3*a^2*b*d^2)*x)*sqrt(b*x^2 + a))/(a*b^4*x^2 + a^2*b ^3), -1/2*((4*a^2*b*c*d - 3*a^3*d^2 + (4*a*b^2*c*d - 3*a^2*b*d^2)*x^2)*sqr t(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (a*b^2*d^2*x^3 + (2*b^3*c^2 - 4 *a*b^2*c*d + 3*a^2*b*d^2)*x)*sqrt(b*x^2 + a))/(a*b^4*x^2 + a^2*b^3)]
\[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.20 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {d^{2} x^{3}}{2 \, \sqrt {b x^{2} + a} b} + \frac {c^{2} x}{\sqrt {b x^{2} + a} a} - \frac {2 \, c d x}{\sqrt {b x^{2} + a} b} + \frac {3 \, a d^{2} x}{2 \, \sqrt {b x^{2} + a} b^{2}} + \frac {2 \, c d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {3 \, a d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} \]
1/2*d^2*x^3/(sqrt(b*x^2 + a)*b) + c^2*x/(sqrt(b*x^2 + a)*a) - 2*c*d*x/(sqr t(b*x^2 + a)*b) + 3/2*a*d^2*x/(sqrt(b*x^2 + a)*b^2) + 2*c*d*arcsinh(b*x/sq rt(a*b))/b^(3/2) - 3/2*a*d^2*arcsinh(b*x/sqrt(a*b))/b^(5/2)
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (\frac {d^{2} x^{2}}{b} + \frac {2 \, b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}}{a b^{3}}\right )} x}{2 \, \sqrt {b x^{2} + a}} - \frac {{\left (4 \, b c d - 3 \, a d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \]
1/2*(d^2*x^2/b + (2*b^3*c^2 - 4*a*b^2*c*d + 3*a^2*b*d^2)/(a*b^3))*x/sqrt(b *x^2 + a) - 1/2*(4*b*c*d - 3*a*d^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a))) /b^(5/2)
Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]