\(\int \frac {(d-c^2 d x^2)^2}{x (a+b \arcsin (c x))^{3/2}} \, dx\) [440]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arcsin (c x)}}-\frac {d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {3 d^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {3 d^2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2}}-\frac {d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{b^{3/2}}-\frac {2 d^2 \text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}},x\right )}{b c} \]

[Out]

-1/2*d^2*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)-1/2*
d^2*FresnelS(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(4*a/b)*2^(1/2)*Pi^(1/2)/b^(3/2)-3*d^2*cos
(2*a/b)*FresnelC(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(3/2)-3*d^2*FresnelS(2*(a+b*arcsin(c*x
))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/b^(3/2)-2*d^2*(-c^2*x^2+1)^(5/2)/b/c/x/(a+b*arcsin(c*x))^(1/2)-
2*d^2*Unintegrable(1/x^2/(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x))^(1/2),x)/b/c

Rubi [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arcsin (c x))^{3/2}} \, dx \]

[In]

Int[(d - c^2*d*x^2)^2/(x*(a + b*ArcSin[c*x])^(3/2)),x]

[Out]

(-2*d^2*(1 - c^2*x^2)^(5/2))/(b*c*x*Sqrt[a + b*ArcSin[c*x]]) - (d^2*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2
/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/b^(3/2) - (3*d^2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c
*x]])/(Sqrt[b]*Sqrt[Pi])])/b^(3/2) - (3*d^2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*
Sin[(2*a)/b])/b^(3/2) - (d^2*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(4*a)/b])
/b^(3/2) - (2*d^2*Defer[Int][1/(x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]]), x])/(b*c)

Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arcsin (c x)}}-\frac {\left (2 d^2\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx}{b c}-\frac {\left (8 c d^2\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}} \, dx}{b} \\ & = -\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arcsin (c x)}}-\frac {\left (8 d^2\right ) \text {Subst}\left (\int \frac {\cos ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\left (2 d^2\right ) \int \left (-\frac {2 c^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}+\frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}+\frac {c^4 x^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}\right ) \, dx}{b c} \\ & = -\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arcsin (c x)}}-\frac {\left (8 d^2\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \sqrt {x}}+\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx}{b c}+\frac {\left (4 c d^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx}{b}-\frac {\left (2 c^3 d^2\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx}{b} \\ & = -\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arcsin (c x)}}+\frac {2 d^2 \sqrt {a+b \arcsin (c x)}}{b^2}-\frac {d^2 \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \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 \sqrt {a+b \arcsin (c x)}}+\frac {2 d^2 \sqrt {a+b \arcsin (c x)}}{b^2}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx}{b c}-\frac {\left (4 d^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\left (d^2 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\left (4 d^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\left (d^2 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2} \\ & = -\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arcsin (c x)}}+\frac {d^2 \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx}{b c}-\frac {\left (8 d^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2}-\frac {\left (2 d^2 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2}-\frac {\left (8 d^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2}-\frac {\left (2 d^2 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2} \\ & = -\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arcsin (c x)}}-\frac {d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {4 d^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {4 d^2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2}}-\frac {d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{b^{3/2}}-\frac {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx}{b c}+\frac {\left (d^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2}+\frac {\left (d^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2} \\ & = -\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arcsin (c x)}}-\frac {d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {4 d^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {4 d^2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2}}-\frac {d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{b^{3/2}}-\frac {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx}{b c}+\frac {\left (2 d^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2}+\frac {\left (2 d^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2} \\ & = -\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arcsin (c x)}}-\frac {d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {3 d^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {3 d^2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2}}-\frac {d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{b^{3/2}}-\frac {\left (2 d^2\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx}{b c} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arcsin (c x))^{3/2}} \, dx \]

[In]

Integrate[(d - c^2*d*x^2)^2/(x*(a + b*ArcSin[c*x])^(3/2)),x]

[Out]

Integrate[(d - c^2*d*x^2)^2/(x*(a + b*ArcSin[c*x])^(3/2)), x]

Maple [N/A] (verified)

Not integrable

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{2}}{x \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}d x\]

[In]

int((-c^2*d*x^2+d)^2/x/(a+b*arcsin(c*x))^(3/2),x)

[Out]

int((-c^2*d*x^2+d)^2/x/(a+b*arcsin(c*x))^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*d*x^2+d)^2/x/(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 5.70 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.59 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arcsin (c x))^{3/2}} \, dx=d^{2} \left (\int \left (- \frac {2 c^{2} x^{2}}{a x \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{4}}{a x \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \frac {1}{a x \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx\right ) \]

[In]

integrate((-c**2*d*x**2+d)**2/x/(a+b*asin(c*x))**(3/2),x)

[Out]

d**2*(Integral(-2*c**2*x**2/(a*x*sqrt(a + b*asin(c*x)) + b*x*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(c
**4*x**4/(a*x*sqrt(a + b*asin(c*x)) + b*x*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(1/(a*x*sqrt(a + b*as
in(c*x)) + b*x*sqrt(a + b*asin(c*x))*asin(c*x)), x))

Maxima [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

integrate((-c^2*d*x^2+d)^2/x/(a+b*arcsin(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate((c^2*d*x^2 - d)^2/((b*arcsin(c*x) + a)^(3/2)*x), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((-c^2*d*x^2+d)^2/x/(a+b*arcsin(c*x))^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {{\left (d-c^2\,d\,x^2\right )}^2}{x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \]

[In]

int((d - c^2*d*x^2)^2/(x*(a + b*asin(c*x))^(3/2)),x)

[Out]

int((d - c^2*d*x^2)^2/(x*(a + b*asin(c*x))^(3/2)), x)