Integrand size = 29, antiderivative size = 29 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}} \, dx=\text {Int}\left (\frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}},x\right ) \] Output:
Defer(Int)((-c^2*d*x^2+d)^2/x^2/(a+b*arcsin(c*x))^(3/2),x)
Not integrable
Time = 10.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}} \, dx \] Input:
Integrate[(d - c^2*d*x^2)^2/(x^2*(a + b*ArcSin[c*x])^(3/2)),x]
Output:
Integrate[(d - c^2*d*x^2)^2/(x^2*(a + b*ArcSin[c*x])^(3/2)), x]
Not integrable
Time = 2.47 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5214 |
| \(\displaystyle -\frac {6 c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x \sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {4 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 \sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x^2 \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 5226 |
| \(\displaystyle -\frac {4 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 \sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {6 c d^2 \int \left (\frac {x^3 c^4}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}-\frac {2 x c^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}+\frac {1}{x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}\right )dx}{b}-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x^2 \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 c d^2 \left (\int \frac {1}{x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}dx+\frac {2 \sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {3 \sqrt {\frac {\pi }{2}} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 \sqrt {b}}-\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 \sqrt {b}}\right )}{b}-\frac {4 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 \sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x^2 \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 5234 |
| \(\displaystyle -\frac {6 c d^2 \left (\int \frac {1}{x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}dx+\frac {2 \sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {3 \sqrt {\frac {\pi }{2}} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 \sqrt {b}}-\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 \sqrt {b}}\right )}{b}-\frac {4 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^3 \sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x^2 \sqrt {a+b \arcsin (c x)}}\) |
Input:
Int[(d - c^2*d*x^2)^2/(x^2*(a + b*ArcSin[c*x])^(3/2)),x]
Output:
$Aborted
Not integrable
Time = 0.63 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{2}}{x^{2} \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}d x\]
Input:
int((-c^2*d*x^2+d)^2/x^2/(a+b*arcsin(c*x))^(3/2),x)
Output:
int((-c^2*d*x^2+d)^2/x^2/(a+b*arcsin(c*x))^(3/2),x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^2/x^2/(a+b*arcsin(c*x))^(3/2),x, algorithm="frica s")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Not integrable
Time = 5.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.93 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}} \, dx=d^{2} \left (\int \left (- \frac {2 c^{2} x^{2}}{a x^{2} \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b x^{2} \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{4}}{a x^{2} \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b x^{2} \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \frac {1}{a x^{2} \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b x^{2} \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)**2/x**2/(a+b*asin(c*x))**(3/2),x)
Output:
d**2*(Integral(-2*c**2*x**2/(a*x**2*sqrt(a + b*asin(c*x)) + b*x**2*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(c**4*x**4/(a*x**2*sqrt(a + b*asin (c*x)) + b*x**2*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(1/(a*x**2* sqrt(a + b*asin(c*x)) + b*x**2*sqrt(a + b*asin(c*x))*asin(c*x)), x))
Not integrable
Time = 1.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^2/x^2/(a+b*arcsin(c*x))^(3/2),x, algorithm="maxim a")
Output:
integrate((c^2*d*x^2 - d)^2/((b*arcsin(c*x) + a)^(3/2)*x^2), x)
Not integrable
Time = 1.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^2/x^2/(a+b*arcsin(c*x))^(3/2),x, algorithm="giac" )
Output:
integrate((c^2*d*x^2 - d)^2/((b*arcsin(c*x) + a)^(3/2)*x^2), x)
Not integrable
Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {{\left (d-c^2\,d\,x^2\right )}^2}{x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:
int((d - c^2*d*x^2)^2/(x^2*(a + b*asin(c*x))^(3/2)),x)
Output:
int((d - c^2*d*x^2)^2/(x^2*(a + b*asin(c*x))^(3/2)), x)
Not integrable
Time = 0.34 (sec) , antiderivative size = 347, normalized size of antiderivative = 11.97 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x^2 (a+b \arcsin (c x))^{3/2}} \, dx=\frac {d^{2} \left (\mathit {asin} \left (c x \right ) \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}}{\mathit {asin} \left (c x \right )^{2} b^{2} x^{2}+2 \mathit {asin} \left (c x \right ) a b \,x^{2}+a^{2} x^{2}}d x \right ) b^{2}+\mathit {asin} \left (c x \right ) \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{2}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) b^{2} c^{4}+4 \mathit {asin} \left (c x \right ) \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x}{\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b +\sqrt {-c^{2} x^{2}+1}\, a}d x \right ) b \,c^{3}+4 \sqrt {\mathit {asin} \left (c x \right ) b +a}\, \sqrt {-c^{2} x^{2}+1}\, c +\left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}}{\mathit {asin} \left (c x \right )^{2} b^{2} x^{2}+2 \mathit {asin} \left (c x \right ) a b \,x^{2}+a^{2} x^{2}}d x \right ) a b +\left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{2}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) a b \,c^{4}+4 \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x}{\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b +\sqrt {-c^{2} x^{2}+1}\, a}d x \right ) a \,c^{3}\right )}{b \left (\mathit {asin} \left (c x \right ) b +a \right )} \] Input:
int((-c^2*d*x^2+d)^2/x^2/(a+b*asin(c*x))^(3/2),x)
Output:
(d**2*(asin(c*x)*int(sqrt(asin(c*x)*b + a)/(asin(c*x)**2*b**2*x**2 + 2*asi n(c*x)*a*b*x**2 + a**2*x**2),x)*b**2 + asin(c*x)*int((sqrt(asin(c*x)*b + a )*x**2)/(asin(c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)*b**2*c**4 + 4*asin (c*x)*int((sqrt(asin(c*x)*b + a)*x)/(sqrt( - c**2*x**2 + 1)*asin(c*x)*b + sqrt( - c**2*x**2 + 1)*a),x)*b*c**3 + 4*sqrt(asin(c*x)*b + a)*sqrt( - c**2 *x**2 + 1)*c + int(sqrt(asin(c*x)*b + a)/(asin(c*x)**2*b**2*x**2 + 2*asin( c*x)*a*b*x**2 + a**2*x**2),x)*a*b + int((sqrt(asin(c*x)*b + a)*x**2)/(asin (c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)*a*b*c**4 + 4*int((sqrt(asin(c*x )*b + a)*x)/(sqrt( - c**2*x**2 + 1)*asin(c*x)*b + sqrt( - c**2*x**2 + 1)*a ),x)*a*c**3))/(b*(asin(c*x)*b + a))