\(\int (\frac {x^2}{\sqrt {\tan (a+b x^2)}}+\frac {\sqrt {\tan (a+b x^2)}}{b}+x^2 \tan ^{\frac {3}{2}}(a+b x^2)) \, dx\) [17]

Optimal result

Integrand size = 49, antiderivative size = 17 \[ \int \left (\frac {x^2}{\sqrt {\tan \left (a+b x^2\right )}}+\frac {\sqrt {\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac {3}{2}}\left (a+b x^2\right )\right ) \, dx=\frac {x \sqrt {\tan \left (a+b x^2\right )}}{b} \] Output:

x*tan(b*x^2+a)^(1/2)/b
 

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \left (\frac {x^2}{\sqrt {\tan \left (a+b x^2\right )}}+\frac {\sqrt {\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac {3}{2}}\left (a+b x^2\right )\right ) \, dx=\frac {x \sqrt {\tan \left (a+b x^2\right )}}{b} \] Input:

Integrate[x^2/Sqrt[Tan[a + b*x^2]] + Sqrt[Tan[a + b*x^2]]/b + x^2*Tan[a + 
b*x^2]^(3/2),x]
 

Output:

(x*Sqrt[Tan[a + b*x^2]])/b
 

Rubi [F]

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.

\(\displaystyle \int \left (x^2 \tan ^{\frac {3}{2}}\left (a+b x^2\right )+\frac {x^2}{\sqrt {\tan \left (a+b x^2\right )}}+\frac {\sqrt {\tan \left (a+b x^2\right )}}{b}\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int x^2 \tan ^{\frac {3}{2}}\left (b x^2+a\right )dx+\int \frac {x^2}{\sqrt {\tan \left (b x^2+a\right )}}dx+\frac {\int \sqrt {\tan \left (b x^2+a\right )}dx}{b}\)

Input:

Int[x^2/Sqrt[Tan[a + b*x^2]] + Sqrt[Tan[a + b*x^2]]/b + x^2*Tan[a + b*x^2] 
^(3/2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

int(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/2),x)
 

Output:

int(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \left (\frac {x^2}{\sqrt {\tan \left (a+b x^2\right )}}+\frac {\sqrt {\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac {3}{2}}\left (a+b x^2\right )\right ) \, dx=\frac {x \sqrt {\tan \left (b x^{2} + a\right )}}{b} \] Input:

integrate(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/ 
2),x, algorithm="fricas")
 

Output:

x*sqrt(tan(b*x^2 + a))/b
 

Sympy [F]

\[ \int \left (\frac {x^2}{\sqrt {\tan \left (a+b x^2\right )}}+\frac {\sqrt {\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac {3}{2}}\left (a+b x^2\right )\right ) \, dx=\frac {\int \frac {b x^{2}}{\sqrt {\tan {\left (a + b x^{2} \right )}}}\, dx + \int b x^{2} \tan ^{\frac {3}{2}}{\left (a + b x^{2} \right )}\, dx + \int \sqrt {\tan {\left (a + b x^{2} \right )}}\, dx}{b} \] Input:

integrate(x**2/tan(b*x**2+a)**(1/2)+tan(b*x**2+a)**(1/2)/b+x**2*tan(b*x**2 
+a)**(3/2),x)
 

Output:

(Integral(b*x**2/sqrt(tan(a + b*x**2)), x) + Integral(b*x**2*tan(a + b*x** 
2)**(3/2), x) + Integral(sqrt(tan(a + b*x**2)), x))/b
 

Maxima [F]

\[ \int \left (\frac {x^2}{\sqrt {\tan \left (a+b x^2\right )}}+\frac {\sqrt {\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac {3}{2}}\left (a+b x^2\right )\right ) \, dx=\int { x^{2} \tan \left (b x^{2} + a\right )^{\frac {3}{2}} + \frac {x^{2}}{\sqrt {\tan \left (b x^{2} + a\right )}} + \frac {\sqrt {\tan \left (b x^{2} + a\right )}}{b} \,d x } \] Input:

integrate(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/ 
2),x, algorithm="maxima")
 

Output:

integrate(x^2*tan(b*x^2 + a)^(3/2) + x^2/sqrt(tan(b*x^2 + a)) + sqrt(tan(b 
*x^2 + a))/b, x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (\frac {x^2}{\sqrt {\tan \left (a+b x^2\right )}}+\frac {\sqrt {\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac {3}{2}}\left (a+b x^2\right )\right ) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/ 
2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{-1,[0,2,1,0]%%%} / %%%{1,[0,0,0,1]%%%} Error: Bad Argument 
 Value
 

Mupad [B] (verification not implemented)

Time = 9.66 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.65 \[ \int \left (\frac {x^2}{\sqrt {\tan \left (a+b x^2\right )}}+\frac {\sqrt {\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac {3}{2}}\left (a+b x^2\right )\right ) \, dx=\frac {x\,\sqrt {-\frac {{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}}{{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}+1}}}{b} \] Input:

int(tan(a + b*x^2)^(1/2)/b + x^2/tan(a + b*x^2)^(1/2) + x^2*tan(a + b*x^2) 
^(3/2),x)
 

Output:

(x*(-(exp(a*2i + b*x^2*2i)*1i - 1i)/(exp(a*2i + b*x^2*2i) + 1))^(1/2))/b
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \left (\frac {x^2}{\sqrt {\tan \left (a+b x^2\right )}}+\frac {\sqrt {\tan \left (a+b x^2\right )}}{b}+x^2 \tan ^{\frac {3}{2}}\left (a+b x^2\right )\right ) \, dx=\frac {\sqrt {\tan \left (b \,x^{2}+a \right )}\, x}{b} \] Input:

int(x^2/tan(b*x^2+a)^(1/2)+tan(b*x^2+a)^(1/2)/b+x^2*tan(b*x^2+a)^(3/2),x)
 

Output:

(sqrt(tan(a + b*x**2))*x)/b