Integrand size = 74, antiderivative size = 26 \[ \int \left (\frac {x^4}{b \sqrt {x^3+3 \sin (a+b x)}}+\frac {x^2 \cos (a+b x)}{\sqrt {x^3+3 \sin (a+b x)}}+\frac {4 x \sqrt {x^3+3 \sin (a+b x)}}{3 b}\right ) \, dx=\frac {2 x^2 \sqrt {x^3+3 \sin (a+b x)}}{3 b} \] Output:
2/3*x^2*(x^3+3*sin(b*x+a))^(1/2)/b
Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \left (\frac {x^4}{b \sqrt {x^3+3 \sin (a+b x)}}+\frac {x^2 \cos (a+b x)}{\sqrt {x^3+3 \sin (a+b x)}}+\frac {4 x \sqrt {x^3+3 \sin (a+b x)}}{3 b}\right ) \, dx=\frac {2 x^2 \sqrt {x^3+3 \sin (a+b x)}}{3 b} \] Input:
Integrate[x^4/(b*Sqrt[x^3 + 3*Sin[a + b*x]]) + (x^2*Cos[a + b*x])/Sqrt[x^3 + 3*Sin[a + b*x]] + (4*x*Sqrt[x^3 + 3*Sin[a + b*x]])/(3*b),x]
Output:
(2*x^2*Sqrt[x^3 + 3*Sin[a + b*x]])/(3*b)
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.
| \(\displaystyle \int \left (\frac {4 x \sqrt {3 \sin (a+b x)+x^3}}{3 b}+\frac {x^4}{b \sqrt {3 \sin (a+b x)+x^3}}+\frac {x^2 \cos (a+b x)}{\sqrt {3 \sin (a+b x)+x^3}}\right ) \, dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \int x \sqrt {x^3+3 \sin (a+b x)}dx}{3 b}+\frac {\int \frac {x^4}{\sqrt {x^3+3 \sin (a+b x)}}dx}{b}+\int \frac {x^2 \cos (a+b x)}{\sqrt {x^3+3 \sin (a+b x)}}dx\) |
Input:
Int[x^4/(b*Sqrt[x^3 + 3*Sin[a + b*x]]) + (x^2*Cos[a + b*x])/Sqrt[x^3 + 3*S in[a + b*x]] + (4*x*Sqrt[x^3 + 3*Sin[a + b*x]])/(3*b),x]
Output:
$Aborted
Time = 0.95 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08
| method | result | size |
| risch | \(\frac {\sqrt {2 x^{3}+6 \sin \left (b x +a \right )}\, \sqrt {2}\, x^{2}}{3 b}\) | \(28\) |
Input:
int(x^4/b/(x^3+3*sin(b*x+a))^(1/2)+x^2*cos(b*x+a)/(x^3+3*sin(b*x+a))^(1/2) +4/3*x*(x^3+3*sin(b*x+a))^(1/2)/b,x,method=_RETURNVERBOSE)
Output:
1/3*(2*x^3+6*sin(b*x+a))^(1/2)/b*2^(1/2)*x^2
Exception generated. \[ \int \left (\frac {x^4}{b \sqrt {x^3+3 \sin (a+b x)}}+\frac {x^2 \cos (a+b x)}{\sqrt {x^3+3 \sin (a+b x)}}+\frac {4 x \sqrt {x^3+3 \sin (a+b x)}}{3 b}\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^4/b/(x^3+3*sin(b*x+a))^(1/2)+x^2*cos(b*x+a)/(x^3+3*sin(b*x+a)) ^(1/2)+4/3*x*(x^3+3*sin(b*x+a))^(1/2)/b,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \left (\frac {x^4}{b \sqrt {x^3+3 \sin (a+b x)}}+\frac {x^2 \cos (a+b x)}{\sqrt {x^3+3 \sin (a+b x)}}+\frac {4 x \sqrt {x^3+3 \sin (a+b x)}}{3 b}\right ) \, dx=\frac {\int \frac {7 x^{4}}{\sqrt {x^{3} + 3 \sin {\left (a + b x \right )}}}\, dx + \int \frac {12 x \sin {\left (a + b x \right )}}{\sqrt {x^{3} + 3 \sin {\left (a + b x \right )}}}\, dx + \int \frac {3 b x^{2} \cos {\left (a + b x \right )}}{\sqrt {x^{3} + 3 \sin {\left (a + b x \right )}}}\, dx}{3 b} \] Input:
integrate(x**4/b/(x**3+3*sin(b*x+a))**(1/2)+x**2*cos(b*x+a)/(x**3+3*sin(b* x+a))**(1/2)+4/3*x*(x**3+3*sin(b*x+a))**(1/2)/b,x)
Output:
(Integral(7*x**4/sqrt(x**3 + 3*sin(a + b*x)), x) + Integral(12*x*sin(a + b *x)/sqrt(x**3 + 3*sin(a + b*x)), x) + Integral(3*b*x**2*cos(a + b*x)/sqrt( x**3 + 3*sin(a + b*x)), x))/(3*b)
\[ \int \left (\frac {x^4}{b \sqrt {x^3+3 \sin (a+b x)}}+\frac {x^2 \cos (a+b x)}{\sqrt {x^3+3 \sin (a+b x)}}+\frac {4 x \sqrt {x^3+3 \sin (a+b x)}}{3 b}\right ) \, dx=\int { \frac {x^{4}}{\sqrt {x^{3} + 3 \, \sin \left (b x + a\right )} b} + \frac {x^{2} \cos \left (b x + a\right )}{\sqrt {x^{3} + 3 \, \sin \left (b x + a\right )}} + \frac {4 \, \sqrt {x^{3} + 3 \, \sin \left (b x + a\right )} x}{3 \, b} \,d x } \] Input:
integrate(x^4/b/(x^3+3*sin(b*x+a))^(1/2)+x^2*cos(b*x+a)/(x^3+3*sin(b*x+a)) ^(1/2)+4/3*x*(x^3+3*sin(b*x+a))^(1/2)/b,x, algorithm="maxima")
Output:
integrate(x^4/(sqrt(x^3 + 3*sin(b*x + a))*b) + x^2*cos(b*x + a)/sqrt(x^3 + 3*sin(b*x + a)) + 4/3*sqrt(x^3 + 3*sin(b*x + a))*x/b, x)
\[ \int \left (\frac {x^4}{b \sqrt {x^3+3 \sin (a+b x)}}+\frac {x^2 \cos (a+b x)}{\sqrt {x^3+3 \sin (a+b x)}}+\frac {4 x \sqrt {x^3+3 \sin (a+b x)}}{3 b}\right ) \, dx=\int { \frac {x^{4}}{\sqrt {x^{3} + 3 \, \sin \left (b x + a\right )} b} + \frac {x^{2} \cos \left (b x + a\right )}{\sqrt {x^{3} + 3 \, \sin \left (b x + a\right )}} + \frac {4 \, \sqrt {x^{3} + 3 \, \sin \left (b x + a\right )} x}{3 \, b} \,d x } \] Input:
integrate(x^4/b/(x^3+3*sin(b*x+a))^(1/2)+x^2*cos(b*x+a)/(x^3+3*sin(b*x+a)) ^(1/2)+4/3*x*(x^3+3*sin(b*x+a))^(1/2)/b,x, algorithm="giac")
Output:
integrate(x^4/(sqrt(x^3 + 3*sin(b*x + a))*b) + x^2*cos(b*x + a)/sqrt(x^3 + 3*sin(b*x + a)) + 4/3*sqrt(x^3 + 3*sin(b*x + a))*x/b, x)
Time = 16.92 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \left (\frac {x^4}{b \sqrt {x^3+3 \sin (a+b x)}}+\frac {x^2 \cos (a+b x)}{\sqrt {x^3+3 \sin (a+b x)}}+\frac {4 x \sqrt {x^3+3 \sin (a+b x)}}{3 b}\right ) \, dx=\frac {2\,x^2\,\sqrt {3\,\sin \left (a+b\,x\right )+x^3}}{3\,b} \] Input:
int(x^4/(b*(3*sin(a + b*x) + x^3)^(1/2)) + (x^2*cos(a + b*x))/(3*sin(a + b *x) + x^3)^(1/2) + (4*x*(3*sin(a + b*x) + x^3)^(1/2))/(3*b),x)
Output:
(2*x^2*(3*sin(a + b*x) + x^3)^(1/2))/(3*b)
\[ \int \left (\frac {x^4}{b \sqrt {x^3+3 \sin (a+b x)}}+\frac {x^2 \cos (a+b x)}{\sqrt {x^3+3 \sin (a+b x)}}+\frac {4 x \sqrt {x^3+3 \sin (a+b x)}}{3 b}\right ) \, dx=\frac {7 \left (\int \frac {x^{4}}{\sqrt {3 \sin \left (b x +a \right )+x^{3}}}d x \right )+3 \left (\int \frac {\cos \left (b x +a \right ) x^{2}}{\sqrt {3 \sin \left (b x +a \right )+x^{3}}}d x \right ) b +12 \left (\int \frac {\sin \left (b x +a \right ) x}{\sqrt {3 \sin \left (b x +a \right )+x^{3}}}d x \right )}{3 b} \] Input:
int(x^4/b/(x^3+3*sin(b*x+a))^(1/2)+x^2*cos(b*x+a)/(x^3+3*sin(b*x+a))^(1/2) +4/3*x*(x^3+3*sin(b*x+a))^(1/2)/b,x)
Output:
(7*int(x**4/sqrt(3*sin(a + b*x) + x**3),x) + 3*int((cos(a + b*x)*x**2)/sqr t(3*sin(a + b*x) + x**3),x)*b + 12*int((sin(a + b*x)*x)/sqrt(3*sin(a + b*x ) + x**3),x))/(3*b)