\(\int \frac {x^7}{(1+x^2) (1+2 x^2)^{3/4}} \, dx\) [1464]

Optimal result

Integrand size = 22, antiderivative size = 125 \[ \int \frac {x^7}{\left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\frac {7}{4} \sqrt [4]{1+2 x^2}-\frac {1}{5} \left (1+2 x^2\right )^{5/4}+\frac {1}{36} \left (1+2 x^2\right )^{9/4}+\frac {\arctan \left (\frac {1-\sqrt {1+2 x^2}}{\sqrt {2} \sqrt [4]{1+2 x^2}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1+2 x^2}}{1+\sqrt {1+2 x^2}}\right )}{\sqrt {2}} \] Output:

7/4*(2*x^2+1)^(1/4)-1/5*(2*x^2+1)^(5/4)+1/36*(2*x^2+1)^(9/4)+1/2*arctan(1/ 
2*(1-(2*x^2+1)^(1/2))*2^(1/2)/(2*x^2+1)^(1/4))*2^(1/2)-1/2*arctanh(2^(1/2) 
*(2*x^2+1)^(1/4)/(1+(2*x^2+1)^(1/2)))*2^(1/2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.85 \[ \int \frac {x^7}{\left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\frac {1}{45} \sqrt [4]{1+2 x^2} \left (71-13 x^2+5 x^4\right )-\frac {\arctan \left (\frac {-1+\sqrt {1+2 x^2}}{\sqrt {2} \sqrt [4]{1+2 x^2}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1+2 x^2}}{1+\sqrt {1+2 x^2}}\right )}{\sqrt {2}} \] Input:

Integrate[x^7/((1 + x^2)*(1 + 2*x^2)^(3/4)),x]
 

Output:

((1 + 2*x^2)^(1/4)*(71 - 13*x^2 + 5*x^4))/45 - ArcTan[(-1 + Sqrt[1 + 2*x^2 
])/(Sqrt[2]*(1 + 2*x^2)^(1/4))]/Sqrt[2] - ArcTanh[(Sqrt[2]*(1 + 2*x^2)^(1/ 
4))/(1 + Sqrt[1 + 2*x^2])]/Sqrt[2]
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.45, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {352, 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.

\(\displaystyle \int \frac {x^7}{\left (x^2+1\right ) \left (2 x^2+1\right )^{3/4}} \, dx\)

\(\Big \downarrow \) 352

\(\displaystyle \int \left (-\frac {x}{\left (x^2+1\right ) \left (2 x^2+1\right )^{3/4}}+\frac {x}{\left (2 x^2+1\right )^{3/4}}+\frac {x^5}{\left (2 x^2+1\right )^{3/4}}-\frac {x^3}{\left (2 x^2+1\right )^{3/4}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\arctan \left (1-\sqrt {2} \sqrt [4]{2 x^2+1}\right )}{\sqrt {2}}-\frac {\arctan \left (\sqrt {2} \sqrt [4]{2 x^2+1}+1\right )}{\sqrt {2}}+\frac {1}{36} \left (2 x^2+1\right )^{9/4}-\frac {1}{5} \left (2 x^2+1\right )^{5/4}+\frac {7}{4} \sqrt [4]{2 x^2+1}+\frac {\log \left (\sqrt {2 x^2+1}-\sqrt {2} \sqrt [4]{2 x^2+1}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {2 x^2+1}+\sqrt {2} \sqrt [4]{2 x^2+1}+1\right )}{2 \sqrt {2}}\)

Input:

Int[x^7/((1 + x^2)*(1 + 2*x^2)^(3/4)),x]
 

Output:

(7*(1 + 2*x^2)^(1/4))/4 - (1 + 2*x^2)^(5/4)/5 + (1 + 2*x^2)^(9/4)/36 + Arc 
Tan[1 - Sqrt[2]*(1 + 2*x^2)^(1/4)]/Sqrt[2] - ArcTan[1 + Sqrt[2]*(1 + 2*x^2 
)^(1/4)]/Sqrt[2] + Log[1 - Sqrt[2]*(1 + 2*x^2)^(1/4) + Sqrt[1 + 2*x^2]]/(2 
*Sqrt[2]) - Log[1 + Sqrt[2]*(1 + 2*x^2)^(1/4) + Sqrt[1 + 2*x^2]]/(2*Sqrt[2 
])
 

Defintions of rubi rules used

rule 352

Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol 
] :> Int[ExpandIntegrand[x^m/((a + b*x^2)^(3/4)*(c + d*x^2)), x], x] /; Fre 
eQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && IntegerQ[m] && (PosQ[a] || In 
tegerQ[m/2])
 

rule 2009

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

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(\frac {\left (-\ln \left (\frac {-\left (2 x^{2}+1\right )^{\frac {1}{4}} \sqrt {2}-\sqrt {2 x^{2}+1}-1}{\left (2 x^{2}+1\right )^{\frac {1}{4}} \sqrt {2}-\sqrt {2 x^{2}+1}-1}\right )-2 \arctan \left (\left (2 x^{2}+1\right )^{\frac {1}{4}} \sqrt {2}-1\right )-2 \arctan \left (\left (2 x^{2}+1\right )^{\frac {1}{4}} \sqrt {2}+1\right )\right ) \sqrt {2}}{4}+\frac {\left (5 x^{4}-13 x^{2}+71\right ) \left (2 x^{2}+1\right )^{\frac {1}{4}}}{45}\)	\(126\)
trager	\(\left (\frac {1}{9} x^{4}-\frac {13}{45} x^{2}+\frac {71}{45}\right ) \left (2 x^{2}+1\right )^{\frac {1}{4}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (2 x^{2}+1\right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {2 x^{2}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (2 x^{2}+1\right )^{\frac {1}{4}}-x^{2}}{x^{2}+1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (2 x^{2}+1\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {2 x^{2}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (2 x^{2}+1\right )^{\frac {1}{4}}-x^{2}}{x^{2}+1}\right )}{2}\)	\(179\)
risch	\(\frac {\left (5 x^{4}-13 x^{2}+71\right ) \left (2 x^{2}+1\right )^{\frac {1}{4}}}{45}+\frac {\left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (8 x^{6}+12 x^{4}+6 x^{2}+1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {8 x^{6}+12 x^{4}+6 x^{2}+1}\, x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (8 x^{6}+12 x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{4}+4 x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {8 x^{6}+12 x^{4}+6 x^{2}+1}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (8 x^{6}+12 x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{2}+4 x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (8 x^{6}+12 x^{4}+6 x^{2}+1\right )^{\frac {1}{4}}+x^{2}}{\left (x^{2}+1\right ) \left (2 x^{2}+1\right )^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (8 x^{6}+12 x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (8 x^{6}+12 x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {8 x^{6}+12 x^{4}+6 x^{2}+1}\, x^{2}-4 x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (8 x^{6}+12 x^{4}+6 x^{2}+1\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {8 x^{6}+12 x^{4}+6 x^{2}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (8 x^{6}+12 x^{4}+6 x^{2}+1\right )^{\frac {3}{4}}-4 x^{4}-x^{2}}{\left (x^{2}+1\right ) \left (2 x^{2}+1\right )^{2}}\right )}{2}\right ) {\left (\left (2 x^{2}+1\right )^{3}\right )}^{\frac {1}{4}}}{\left (2 x^{2}+1\right )^{\frac {3}{4}}}\)	\(484\)

Input:

int(x^7/(x^2+1)/(2*x^2+1)^(3/4),x,method=_RETURNVERBOSE)
 

Output:

1/4*(-ln((-(2*x^2+1)^(1/4)*2^(1/2)-(2*x^2+1)^(1/2)-1)/((2*x^2+1)^(1/4)*2^( 
1/2)-(2*x^2+1)^(1/2)-1))-2*arctan((2*x^2+1)^(1/4)*2^(1/2)-1)-2*arctan((2*x 
^2+1)^(1/4)*2^(1/2)+1))*2^(1/2)+1/45*(5*x^4-13*x^2+71)*(2*x^2+1)^(1/4)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {x^7}{\left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} - 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {2 \, x^{2} + 1} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {2 \, x^{2} + 1} + 1\right ) + \frac {1}{45} \, {\left (5 \, x^{4} - 13 \, x^{2} + 71\right )} {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} \] Input:

integrate(x^7/(x^2+1)/(2*x^2+1)^(3/4),x, algorithm="fricas")
 

Output:

-1/2*sqrt(2)*arctan(sqrt(2)*(2*x^2 + 1)^(1/4) + 1) - 1/2*sqrt(2)*arctan(sq 
rt(2)*(2*x^2 + 1)^(1/4) - 1) - 1/4*sqrt(2)*log(sqrt(2)*(2*x^2 + 1)^(1/4) + 
 sqrt(2*x^2 + 1) + 1) + 1/4*sqrt(2)*log(-sqrt(2)*(2*x^2 + 1)^(1/4) + sqrt( 
2*x^2 + 1) + 1) + 1/45*(5*x^4 - 13*x^2 + 71)*(2*x^2 + 1)^(1/4)
 

Sympy [F]

\[ \int \frac {x^7}{\left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\int \frac {x^{7}}{\left (x^{2} + 1\right ) \left (2 x^{2} + 1\right )^{\frac {3}{4}}}\, dx \] Input:

integrate(x**7/(x**2+1)/(2*x**2+1)**(3/4),x)
 

Output:

Integral(x**7/((x**2 + 1)*(2*x**2 + 1)**(3/4)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.18 \[ \int \frac {x^7}{\left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\frac {1}{36} \, {\left (2 \, x^{2} + 1\right )}^{\frac {9}{4}} - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {2 \, x^{2} + 1} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {2 \, x^{2} + 1} + 1\right ) - \frac {1}{5} \, {\left (2 \, x^{2} + 1\right )}^{\frac {5}{4}} + \frac {7}{4} \, {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} \] Input:

integrate(x^7/(x^2+1)/(2*x^2+1)^(3/4),x, algorithm="maxima")
 

Output:

1/36*(2*x^2 + 1)^(9/4) - 1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(2*x^ 
2 + 1)^(1/4))) - 1/2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(2*x^2 + 1)^ 
(1/4))) - 1/4*sqrt(2)*log(sqrt(2)*(2*x^2 + 1)^(1/4) + sqrt(2*x^2 + 1) + 1) 
 + 1/4*sqrt(2)*log(-sqrt(2)*(2*x^2 + 1)^(1/4) + sqrt(2*x^2 + 1) + 1) - 1/5 
*(2*x^2 + 1)^(5/4) + 7/4*(2*x^2 + 1)^(1/4)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.18 \[ \int \frac {x^7}{\left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\frac {1}{36} \, {\left (2 \, x^{2} + 1\right )}^{\frac {9}{4}} - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {2 \, x^{2} + 1} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {2 \, x^{2} + 1} + 1\right ) - \frac {1}{5} \, {\left (2 \, x^{2} + 1\right )}^{\frac {5}{4}} + \frac {7}{4} \, {\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} \] Input:

integrate(x^7/(x^2+1)/(2*x^2+1)^(3/4),x, algorithm="giac")
 

Output:

1/36*(2*x^2 + 1)^(9/4) - 1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(2*x^ 
2 + 1)^(1/4))) - 1/2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(2*x^2 + 1)^ 
(1/4))) - 1/4*sqrt(2)*log(sqrt(2)*(2*x^2 + 1)^(1/4) + sqrt(2*x^2 + 1) + 1) 
 + 1/4*sqrt(2)*log(-sqrt(2)*(2*x^2 + 1)^(1/4) + sqrt(2*x^2 + 1) + 1) - 1/5 
*(2*x^2 + 1)^(5/4) + 7/4*(2*x^2 + 1)^(1/4)
 

Mupad [B] (verification not implemented)

Time = 1.49 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.66 \[ \int \frac {x^7}{\left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\frac {7\,{\left (2\,x^2+1\right )}^{1/4}}{4}-\frac {{\left (2\,x^2+1\right )}^{5/4}}{5}+\frac {{\left (2\,x^2+1\right )}^{9/4}}{36}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (2\,x^2+1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (2\,x^2+1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right ) \] Input:

int(x^7/((x^2 + 1)*(2*x^2 + 1)^(3/4)),x)
 

Output:

(7*(2*x^2 + 1)^(1/4))/4 - 2^(1/2)*atan(2^(1/2)*(2*x^2 + 1)^(1/4)*(1/2 + 1i 
/2))*(1/2 - 1i/2) - 2^(1/2)*atan(2^(1/2)*(2*x^2 + 1)^(1/4)*(1/2 - 1i/2))*( 
1/2 + 1i/2) - (2*x^2 + 1)^(5/4)/5 + (2*x^2 + 1)^(9/4)/36
 

Reduce [F]

\[ \int \frac {x^7}{\left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\int \frac {x^{7}}{\left (2 x^{2}+1\right )^{\frac {3}{4}} x^{2}+\left (2 x^{2}+1\right )^{\frac {3}{4}}}d x \] Input:

int(x^7/(x^2+1)/(2*x^2+1)^(3/4),x)
 

Output:

int(x**7/((2*x**2 + 1)**(3/4)*x**2 + (2*x**2 + 1)**(3/4)),x)