Integrand size = 32, antiderivative size = 108 \[ \int \frac {A+B x^2+C x^4+D x^6}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {A}{a}-\frac {b^2 B-a b C+a^2 D}{b^3}}{\sqrt {a+b x^2}}+\frac {(b C-2 a D) \sqrt {a+b x^2}}{b^3}+\frac {D \left (a+b x^2\right )^{3/2}}{3 b^3}-\frac {A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \] Output:
(A/a-(B*b^2-C*a*b+D*a^2)/b^3)/(b*x^2+a)^(1/2)+(C*b-2*D*a)*(b*x^2+a)^(1/2)/ b^3+1/3*D*(b*x^2+a)^(3/2)/b^3-A*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3/2)
Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x^2+C x^4+D x^6}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {3 A b^3+a \left (-8 a^2 D+a b \left (6 C-4 D x^2\right )+b^2 \left (-3 B+3 C x^2+D x^4\right )\right )}{3 a b^3 \sqrt {a+b x^2}}-\frac {A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x*(a + b*x^2)^(3/2)),x]
Output:
(3*A*b^3 + a*(-8*a^2*D + a*b*(6*C - 4*D*x^2) + b^2*(-3*B + 3*C*x^2 + D*x^4 )))/(3*a*b^3*Sqrt[a + b*x^2]) - (A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/ 2)
Time = 0.45 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2331, 2122, 2009}
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.
\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x \left (a+b x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2331 |
\(\displaystyle \frac {1}{2} \int \frac {D x^6+C x^4+B x^2+A}{x^2 \left (b x^2+a\right )^{3/2}}dx^2\) |
\(\Big \downarrow \) 2122 |
\(\displaystyle \frac {1}{2} \int \left (\frac {D x^2}{b \sqrt {b x^2+a}}+\frac {b C-a D}{b^2 \sqrt {b x^2+a}}+\frac {a \left (D a^2-b C a+b^2 B\right )-A b^3}{a b^2 \left (b x^2+a\right )^{3/2}}+\frac {A}{a \sqrt {b x^2+a} x^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {2 \left (\frac {A}{a}-\frac {a^2 D-a b C+b^2 B}{b^3}\right )}{\sqrt {a+b x^2}}+\frac {2 \sqrt {a+b x^2} (b C-a D)}{b^3}+\frac {2 D \left (a+b x^2\right )^{3/2}}{3 b^3}-\frac {2 a D \sqrt {a+b x^2}}{b^3}\right )\) |
Input:
Int[(A + B*x^2 + C*x^4 + D*x^6)/(x*(a + b*x^2)^(3/2)),x]
Output:
((2*(A/a - (b^2*B - a*b*C + a^2*D)/b^3))/Sqrt[a + b*x^2] - (2*a*D*Sqrt[a + b*x^2])/b^3 + (2*(b*C - a*D)*Sqrt[a + b*x^2])/b^3 + (2*D*(a + b*x^2)^(3/2 ))/(3*b^3) - (2*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/2))/2
Int[((Px_)*((c_.) + (d_.)*(x_))^(n_))/((a_.) + (b_.)*(x_)), x_Symbol] :> In t[ExpandIntegrand[1/Sqrt[c + d*x], Px*((c + d*x)^(n + 1/2)/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[n + 1/2, 0]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Time = 0.63 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(\frac {-\frac {8 D a^{\frac {9}{2}}}{3}+2 \left (-\frac {2 D x^{2}}{3}+C \right ) b \,a^{\frac {7}{2}}-\left (-\frac {1}{3} D x^{4}-C \,x^{2}+B \right ) b^{2} a^{\frac {5}{2}}+A \,a^{\frac {3}{2}} b^{3}-A \,b^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right ) a \sqrt {b \,x^{2}+a}}{\sqrt {b \,x^{2}+a}\, a^{\frac {5}{2}} b^{3}}\) | \(98\) |
default | \(A \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )-\frac {B}{b \sqrt {b \,x^{2}+a}}+C \left (\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )+D \left (\frac {x^{4}}{3 b \sqrt {b \,x^{2}+a}}-\frac {4 a \left (\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )}{3 b}\right )\) | \(153\) |
Input:
int((D*x^6+C*x^4+B*x^2+A)/x/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/(b*x^2+a)^(1/2)*(-8/3*D*a^(9/2)+2*(-2/3*D*x^2+C)*b*a^(7/2)-(-1/3*D*x^4-C *x^2+B)*b^2*a^(5/2)+A*a^(3/2)*b^3-A*b^3*arctanh((b*x^2+a)^(1/2)/a^(1/2))*a *(b*x^2+a)^(1/2))/a^(5/2)/b^3
Time = 0.09 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.63 \[ \int \frac {A+B x^2+C x^4+D x^6}{x \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (A b^{4} x^{2} + A a b^{3}\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (D a^{2} b^{2} x^{4} - 8 \, D a^{4} + 6 \, C a^{3} b - 3 \, B a^{2} b^{2} + 3 \, A a b^{3} - {\left (4 \, D a^{3} b - 3 \, C a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}, \frac {3 \, {\left (A b^{4} x^{2} + A a b^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (D a^{2} b^{2} x^{4} - 8 \, D a^{4} + 6 \, C a^{3} b - 3 \, B a^{2} b^{2} + 3 \, A a b^{3} - {\left (4 \, D a^{3} b - 3 \, C a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}\right ] \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[1/6*(3*(A*b^4*x^2 + A*a*b^3)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt (a) + 2*a)/x^2) + 2*(D*a^2*b^2*x^4 - 8*D*a^4 + 6*C*a^3*b - 3*B*a^2*b^2 + 3 *A*a*b^3 - (4*D*a^3*b - 3*C*a^2*b^2)*x^2)*sqrt(b*x^2 + a))/(a^2*b^4*x^2 + a^3*b^3), 1/3*(3*(A*b^4*x^2 + A*a*b^3)*sqrt(-a)*arctan(sqrt(b*x^2 + a)*sqr t(-a)/a) + (D*a^2*b^2*x^4 - 8*D*a^4 + 6*C*a^3*b - 3*B*a^2*b^2 + 3*A*a*b^3 - (4*D*a^3*b - 3*C*a^2*b^2)*x^2)*sqrt(b*x^2 + a))/(a^2*b^4*x^2 + a^3*b^3)]
Time = 12.98 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.31 \[ \int \frac {A+B x^2+C x^4+D x^6}{x \left (a+b x^2\right )^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {A b \operatorname {atan}{\left (\frac {\sqrt {a + b x^{2}}}{\sqrt {- a}} \right )}}{2 a \sqrt {- a}} + \frac {D \left (a + b x^{2}\right )^{\frac {3}{2}}}{6 b^{2}} + \frac {\sqrt {a + b x^{2}} \left (C b - 2 D a\right )}{2 b^{2}} - \frac {- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}}{2 a b^{2} \sqrt {a + b x^{2}}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {A \log {\left (x^{2} \right )} + B x^{2} + \frac {C x^{4}}{2} + \frac {D x^{6}}{3}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((D*x**6+C*x**4+B*x**2+A)/x/(b*x**2+a)**(3/2),x)
Output:
Piecewise((2*(A*b*atan(sqrt(a + b*x**2)/sqrt(-a))/(2*a*sqrt(-a)) + D*(a + b*x**2)**(3/2)/(6*b**2) + sqrt(a + b*x**2)*(C*b - 2*D*a)/(2*b**2) - (-A*b* *3 + B*a*b**2 - C*a**2*b + D*a**3)/(2*a*b**2*sqrt(a + b*x**2)))/b, Ne(b, 0 )), ((A*log(x**2) + B*x**2 + C*x**4/2 + D*x**6/3)/(2*a**(3/2)), True))
Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x^2+C x^4+D x^6}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {D x^{4}}{3 \, \sqrt {b x^{2} + a} b} - \frac {4 \, D a x^{2}}{3 \, \sqrt {b x^{2} + a} b^{2}} + \frac {C x^{2}}{\sqrt {b x^{2} + a} b} - \frac {A \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {3}{2}}} + \frac {A}{\sqrt {b x^{2} + a} a} - \frac {8 \, D a^{2}}{3 \, \sqrt {b x^{2} + a} b^{3}} + \frac {2 \, C a}{\sqrt {b x^{2} + a} b^{2}} - \frac {B}{\sqrt {b x^{2} + a} b} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
1/3*D*x^4/(sqrt(b*x^2 + a)*b) - 4/3*D*a*x^2/(sqrt(b*x^2 + a)*b^2) + C*x^2/ (sqrt(b*x^2 + a)*b) - A*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) + A/(sqrt(b* x^2 + a)*a) - 8/3*D*a^2/(sqrt(b*x^2 + a)*b^3) + 2*C*a/(sqrt(b*x^2 + a)*b^2 ) - B/(sqrt(b*x^2 + a)*b)
Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x^2+C x^4+D x^6}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {A \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {D a^{3} - C a^{2} b + B a b^{2} - A b^{3}}{\sqrt {b x^{2} + a} a b^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} D b^{6} - 6 \, \sqrt {b x^{2} + a} D a b^{6} + 3 \, \sqrt {b x^{2} + a} C b^{7}}{3 \, b^{9}} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
A*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) - (D*a^3 - C*a^2*b + B*a*b ^2 - A*b^3)/(sqrt(b*x^2 + a)*a*b^3) + 1/3*((b*x^2 + a)^(3/2)*D*b^6 - 6*sqr t(b*x^2 + a)*D*a*b^6 + 3*sqrt(b*x^2 + a)*C*b^7)/b^9
Time = 1.94 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^2+C x^4+D x^6}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {A}{a\,\sqrt {b\,x^2+a}}-\frac {B}{b\,\sqrt {b\,x^2+a}}-\frac {A\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {C\,\left (b\,x^2+2\,a\right )}{b^2\,\sqrt {b\,x^2+a}}-\frac {D\,\left (8\,a^2+4\,a\,b\,x^2-b^2\,x^4\right )}{3\,b^3\,\sqrt {b\,x^2+a}} \] Input:
int((A + B*x^2 + C*x^4 + x^6*D)/(x*(a + b*x^2)^(3/2)),x)
Output:
A/(a*(a + b*x^2)^(1/2)) - B/(b*(a + b*x^2)^(1/2)) - (A*atanh((a + b*x^2)^( 1/2)/a^(1/2)))/a^(3/2) + (C*(2*a + b*x^2))/(b^2*(a + b*x^2)^(1/2)) - (D*(8 *a^2 - b^2*x^4 + 4*a*b*x^2))/(3*b^3*(a + b*x^2)^(1/2))
Time = 0.16 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.07 \[ \int \frac {A+B x^2+C x^4+D x^6}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {-8 \sqrt {b \,x^{2}+a}\, a^{3} d +6 \sqrt {b \,x^{2}+a}\, a^{2} b c -4 \sqrt {b \,x^{2}+a}\, a^{2} b d \,x^{2}+3 \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,x^{2}+\sqrt {b \,x^{2}+a}\, a \,b^{2} d \,x^{4}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} x^{2}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} x^{2}}{3 a \,b^{3} \left (b \,x^{2}+a \right )} \] Input:
int((D*x^6+C*x^4+B*x^2+A)/x/(b*x^2+a)^(3/2),x)
Output:
( - 8*sqrt(a + b*x**2)*a**3*d + 6*sqrt(a + b*x**2)*a**2*b*c - 4*sqrt(a + b *x**2)*a**2*b*d*x**2 + 3*sqrt(a + b*x**2)*a*b**2*c*x**2 + sqrt(a + b*x**2) *a*b**2*d*x**4 + 3*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sq rt(a))*a*b**3 + 3*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqr t(a))*b**4*x**2 - 3*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/s qrt(a))*a*b**3 - 3*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sq rt(a))*b**4*x**2)/(3*a*b**3*(a + b*x**2))