Integrand size = 22, antiderivative size = 216 \[ \int \frac {1}{x^2 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {b^2 \left (2 a c d+\left (b c^2-a d^2\right ) x\right )}{a^2 \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}-\frac {\sqrt {a+b x^2}}{a^2 c^2 x}-\frac {d^5 \sqrt {a+b x^2}}{c^2 \left (b c^2+a d^2\right )^2 (c+d x)}-\frac {d^4 \left (5 b c^2+2 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^3 \left (b c^2+a d^2\right )^{5/2}}+\frac {2 d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2} c^3} \] Output:
-b^2*(2*a*c*d+(-a*d^2+b*c^2)*x)/a^2/(a*d^2+b*c^2)^2/(b*x^2+a)^(1/2)-(b*x^2 +a)^(1/2)/a^2/c^2/x-d^5*(b*x^2+a)^(1/2)/c^2/(a*d^2+b*c^2)^2/(d*x+c)-d^4*(2 *a*d^2+5*b*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/ c^3/(a*d^2+b*c^2)^(5/2)+2*d*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3/2)/c^3
Time = 1.70 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^2 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\frac {c \left (2 b^3 c^4 x^2 (c+d x)+a b^2 c^2 (c+d x)^3+a^3 d^4 (c+2 d x)+a^2 b d^2 \left (2 c^3+2 c^2 d x+c d^2 x^2+2 d^3 x^3\right )\right )}{a^2 \left (b c^2+a d^2\right )^2 x (c+d x) \sqrt {a+b x^2}}+\frac {2 d^4 \left (5 b c^2+2 a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}}+\frac {4 d \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{c^3} \] Input:
Integrate[1/(x^2*(c + d*x)^2*(a + b*x^2)^(3/2)),x]
Output:
-(((c*(2*b^3*c^4*x^2*(c + d*x) + a*b^2*c^2*(c + d*x)^3 + a^3*d^4*(c + 2*d* x) + a^2*b*d^2*(2*c^3 + 2*c^2*d*x + c*d^2*x^2 + 2*d^3*x^3)))/(a^2*(b*c^2 + a*d^2)^2*x*(c + d*x)*Sqrt[a + b*x^2]) + (2*d^4*(5*b*c^2 + 2*a*d^2)*ArcTan [(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/(-(b*c^2 ) - a*d^2)^(5/2) + (4*d*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/a^ (3/2))/c^3)
Time = 0.91 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.66, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2} (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 617 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a+b x^2\right )^{3/2} (c+d x)}-\frac {2 d}{c^3 x \left (a+b x^2\right )^{3/2}}+\frac {d^2}{c^2 \left (a+b x^2\right )^{3/2} (c+d x)^2}+\frac {1}{c^2 x^2 \left (a+b x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2} c^3}-\frac {2 b x}{a^2 c^2 \sqrt {a+b x^2}}-\frac {3 b d^4 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c \left (a d^2+b c^2\right )^{5/2}}-\frac {2 d^4 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^3 \left (a d^2+b c^2\right )^{3/2}}-\frac {2 d}{a c^3 \sqrt {a+b x^2}}+\frac {d^2 (a d+b c x)}{a c^2 \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}+\frac {d^3 \sqrt {a+b x^2} \left (b c^2-2 a d^2\right )}{a c^2 (c+d x) \left (a d^2+b c^2\right )^2}-\frac {1}{a c^2 x \sqrt {a+b x^2}}+\frac {2 d^2 (a d+b c x)}{a c^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}\) |
Input:
Int[1/(x^2*(c + d*x)^2*(a + b*x^2)^(3/2)),x]
Output:
(-2*d)/(a*c^3*Sqrt[a + b*x^2]) - 1/(a*c^2*x*Sqrt[a + b*x^2]) - (2*b*x)/(a^ 2*c^2*Sqrt[a + b*x^2]) + (2*d^2*(a*d + b*c*x))/(a*c^3*(b*c^2 + a*d^2)*Sqrt [a + b*x^2]) + (d^2*(a*d + b*c*x))/(a*c^2*(b*c^2 + a*d^2)*(c + d*x)*Sqrt[a + b*x^2]) + (d^3*(b*c^2 - 2*a*d^2)*Sqrt[a + b*x^2])/(a*c^2*(b*c^2 + a*d^2 )^2*(c + d*x)) - (3*b*d^4*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[ a + b*x^2])])/(c*(b*c^2 + a*d^2)^(5/2)) - (2*d^4*ArcTanh[(a*d - b*c*x)/(Sq rt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c^3*(b*c^2 + a*d^2)^(3/2)) + (2*d*Ar cTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(a^(3/2)*c^3)
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(641\) vs. \(2(198)=396\).
Time = 0.48 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.97
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}}{a^{2} c^{2} x}-\frac {\frac {b a \,d^{2} \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{\left (\sqrt {-a b}\, d +b c \right ) \left (\sqrt {-a b}\, d -b c \right )}-\frac {2 d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}+\frac {b^{2} c^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{2 \left (\sqrt {-a b}\, d +b c \right )^{2} a \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {b^{2} c^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{2 \left (\sqrt {-a b}\, d -b c \right )^{2} a \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {2 b^{2} a \,d^{3} \left (a \,d^{2}+2 b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (\sqrt {-a b}\, d +b c \right )^{2} \left (\sqrt {-a b}\, d -b c \right )^{2} c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{a \,c^{2}}\) | \(642\) |
| default | \(\frac {-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}}{c^{2}}+\frac {-\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {3 b c d \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{a \,d^{2}+b \,c^{2}}-\frac {4 b \,d^{2} \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{c^{2}}-\frac {2 d \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 )}{c^{3}}+\frac {2 d \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{c^{3}}\) | \(918\) |
Input:
int(1/x^2/(d*x+c)^2/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-(b*x^2+a)^(1/2)/a^2/c^2/x-1/a/c^2*(b*a*d^2/((-a*b)^(1/2)*d+b*c)/((-a*b)^( 1/2)*d-b*c)*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a* d^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2* (a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2 -2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))-2*d/c/a^(1/2)*ln((2*a +2*a^(1/2)*(b*x^2+a)^(1/2))/x)+1/2*b^2*c^2/((-a*b)^(1/2)*d+b*c)^2/a/(x-(-a *b)^(1/2)/b)*((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^(1 /2)+1/2*b^2*c^2/((-a*b)^(1/2)*d-b*c)^2/a/(x+(-a*b)^(1/2)/b)*((x+(-a*b)^(1/ 2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2)+2*b^2*a*d^3*(a*d^2+2*b* c^2)/((-a*b)^(1/2)*d+b*c)^2/((-a*b)^(1/2)*d-b*c)^2/c/((a*d^2+b*c^2)/d^2)^( 1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*( b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))
Leaf count of result is larger than twice the leaf count of optimal. 830 vs. \(2 (199) = 398\).
Time = 1.46 (sec) , antiderivative size = 3390, normalized size of antiderivative = 15.69 \[ \int \frac {1}{x^2 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(d*x+c)^2/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{x^2 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{2}}\, dx \] Input:
integrate(1/x**2/(d*x+c)**2/(b*x**2+a)**(3/2),x)
Output:
Integral(1/(x**2*(a + b*x**2)**(3/2)*(c + d*x)**2), x)
\[ \int \frac {1}{x^2 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{2} x^{2}} \,d x } \] Input:
integrate(1/x^2/(d*x+c)^2/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(3/2)*(d*x + c)^2*x^2), x)
Timed out. \[ \int \frac {1}{x^2 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(d*x+c)^2/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^2 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^2} \,d x \] Input:
int(1/(x^2*(a + b*x^2)^(3/2)*(c + d*x)^2),x)
Output:
int(1/(x^2*(a + b*x^2)^(3/2)*(c + d*x)^2), x)
Time = 0.96 (sec) , antiderivative size = 2168, normalized size of antiderivative = 10.04 \[ \int \frac {1}{x^2 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(d*x+c)^2/(b*x^2+a)^(3/2),x)
Output:
(2*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4*c*d**6*x + 2*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt (a*d**2 + b*c**2) - a*d + b*c*x)*a**4*d**7*x**2 + 5*sqrt(a*d**2 + b*c**2)* log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b*c**3*d**4 *x + 5*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b*c**2*d**5*x**2 + 2*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b*c*d**6*x**3 + 2*sqrt(a *d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)* a**3*b*d**7*x**4 + 5*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d** 2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c**3*d**4*x**3 + 5*sqrt(a*d**2 + b*c* *2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c* *2*d**5*x**4 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**4*c*d**6*x - 2*sqrt (a*d**2 + b*c**2)*log(c + d*x)*a**4*d**7*x**2 - 5*sqrt(a*d**2 + b*c**2)*lo g(c + d*x)*a**3*b*c**3*d**4*x - 5*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3* b*c**2*d**5*x**2 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*b*c*d**6*x**3 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*b*d**7*x**4 - 5*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b**2*c**3*d**4*x**3 - 5*sqrt(a*d**2 + b*c**2)*l og(c + d*x)*a**2*b**2*c**2*d**5*x**4 - sqrt(a + b*x**2)*a**4*c**2*d**6 - 2 *sqrt(a + b*x**2)*a**4*c*d**7*x - 3*sqrt(a + b*x**2)*a**3*b*c**4*d**4 - 4* sqrt(a + b*x**2)*a**3*b*c**3*d**5*x - sqrt(a + b*x**2)*a**3*b*c**2*d**6...