\(\int \frac {1}{\sqrt {5-2 x} (4+3 x)^{3/2} (2+3 x+x^2)} \, dx\) [369]

Optimal result

Integrand size = 29, antiderivative size = 90 \[ \int \frac {1}{\sqrt {5-2 x} (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {9 \sqrt {5-2 x}}{23 \sqrt {4+3 x}}+\frac {\arctan \left (\frac {3 \sqrt {4+3 x}}{\sqrt {2} \sqrt {5-2 x}}\right )}{3 \sqrt {2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {7} \sqrt {4+3 x}}{\sqrt {5-2 x}}\right )}{\sqrt {7}} \] Output:

9/23*(5-2*x)^(1/2)/(4+3*x)^(1/2)+1/6*2^(1/2)*arctan(3/2*(4+3*x)^(1/2)*2^(1 
/2)/(5-2*x)^(1/2))-2/7*7^(1/2)*arctanh(7^(1/2)*(4+3*x)^(1/2)/(5-2*x)^(1/2) 
)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {5-2 x} (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {9 \sqrt {5-2 x}}{23 \sqrt {4+3 x}}-\frac {\arctan \left (\frac {\sqrt {10-4 x}}{3 \sqrt {4+3 x}}\right )}{3 \sqrt {2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7} \sqrt {4+3 x}}\right )}{\sqrt {7}} \] Input:

Integrate[1/(Sqrt[5 - 2*x]*(4 + 3*x)^(3/2)*(2 + 3*x + x^2)),x]
 

Output:

(9*Sqrt[5 - 2*x])/(23*Sqrt[4 + 3*x]) - ArcTan[Sqrt[10 - 4*x]/(3*Sqrt[4 + 3 
*x])]/(3*Sqrt[2]) - (2*ArcTanh[Sqrt[5 - 2*x]/(Sqrt[7]*Sqrt[4 + 3*x])])/Sqr 
t[7]
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1205, 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.

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

\(\Big \downarrow \) 1205

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\arctan \left (\frac {3 \sqrt {3 x+4}}{\sqrt {2} \sqrt {5-2 x}}\right )}{3 \sqrt {2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {7} \sqrt {3 x+4}}{\sqrt {5-2 x}}\right )}{\sqrt {7}}+\frac {9 \sqrt {5-2 x}}{23 \sqrt {3 x+4}}\)

Input:

Int[1/(Sqrt[5 - 2*x]*(4 + 3*x)^(3/2)*(2 + 3*x + x^2)),x]
 

Output:

(9*Sqrt[5 - 2*x])/(23*Sqrt[4 + 3*x]) + ArcTan[(3*Sqrt[4 + 3*x])/(Sqrt[2]*S 
qrt[5 - 2*x])]/(3*Sqrt[2]) - (2*ArcTanh[(Sqrt[7]*Sqrt[4 + 3*x])/Sqrt[5 - 2 
*x]])/Sqrt[7]
 

Defintions of rubi rules used

rule 1205

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^ 
n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && 
 !IntegerQ[m] &&  !IntegerQ[n]
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(66)=132\).

Time = 1.94 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.76

method	result	size
default	\(\frac {\left (483 \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 x^{2}+7 x +20}}\right ) \sqrt {2}\, x -828 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 x^{2}+7 x +20}}\right ) x +644 \sqrt {2}\, \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 x^{2}+7 x +20}}\right )-1104 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 x^{2}+7 x +20}}\right )+756 \sqrt {-6 x^{2}+7 x +20}\right ) \sqrt {5-2 x}}{1932 \sqrt {-6 x^{2}+7 x +20}\, \sqrt {3 x +4}}\)	\(158\)

Input:

int(1/(5-2*x)^(1/2)/(3*x+4)^(3/2)/(x^2+3*x+2),x,method=_RETURNVERBOSE)
 

Output:

1/1932*(483*arctan(1/12*(26+31*x)*2^(1/2)/(-6*x^2+7*x+20)^(1/2))*2^(1/2)*x 
-828*7^(1/2)*arctanh(1/14*(33+19*x)*7^(1/2)/(-6*x^2+7*x+20)^(1/2))*x+644*2 
^(1/2)*arctan(1/12*(26+31*x)*2^(1/2)/(-6*x^2+7*x+20)^(1/2))-1104*7^(1/2)*a 
rctanh(1/14*(33+19*x)*7^(1/2)/(-6*x^2+7*x+20)^(1/2))+756*(-6*x^2+7*x+20)^( 
1/2))*(5-2*x)^(1/2)/(-6*x^2+7*x+20)^(1/2)/(3*x+4)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt {5-2 x} (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=-\frac {161 \, \sqrt {2} {\left (3 \, x + 4\right )} \arctan \left (\frac {\sqrt {2} {\left (31 \, x + 26\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{12 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) - 138 \, \sqrt {7} {\left (3 \, x + 4\right )} \log \left (-\frac {4 \, \sqrt {7} {\left (19 \, x + 33\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5} - 193 \, x^{2} - 1450 \, x - 1649}{x^{2} + 2 \, x + 1}\right ) - 756 \, \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{1932 \, {\left (3 \, x + 4\right )}} \] Input:

integrate(1/(5-2*x)^(1/2)/(4+3*x)^(3/2)/(x^2+3*x+2),x, algorithm="fricas")
 

Output:

-1/1932*(161*sqrt(2)*(3*x + 4)*arctan(1/12*sqrt(2)*(31*x + 26)*sqrt(3*x + 
4)*sqrt(-2*x + 5)/(6*x^2 - 7*x - 20)) - 138*sqrt(7)*(3*x + 4)*log(-(4*sqrt 
(7)*(19*x + 33)*sqrt(3*x + 4)*sqrt(-2*x + 5) - 193*x^2 - 1450*x - 1649)/(x 
^2 + 2*x + 1)) - 756*sqrt(3*x + 4)*sqrt(-2*x + 5))/(3*x + 4)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {5-2 x} (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\int \frac {1}{\sqrt {5 - 2 x} \left (x + 1\right ) \left (x + 2\right ) \left (3 x + 4\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(5-2*x)**(1/2)/(4+3*x)**(3/2)/(x**2+3*x+2),x)
 

Output:

Integral(1/(sqrt(5 - 2*x)*(x + 1)*(x + 2)*(3*x + 4)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {5-2 x} (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} + 3 \, x + 2\right )} {\left (3 \, x + 4\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 5}} \,d x } \] Input:

integrate(1/(5-2*x)^(1/2)/(4+3*x)^(3/2)/(x^2+3*x+2),x, algorithm="maxima")
 

Output:

integrate(1/((x^2 + 3*x + 2)*(3*x + 4)^(3/2)*sqrt(-2*x + 5)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (66) = 132\).

Time = 0.42 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.94 \[ \int \frac {1}{\sqrt {5-2 x} (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {1}{36} \, \sqrt {6} \sqrt {3} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {3} \sqrt {3 \, x + 4} {\left (\frac {{\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{2}}{3 \, x + 4} - 4\right )}}{18 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}}\right )\right )} + \frac {1}{42} \, \sqrt {42} \sqrt {6} \log \left (\frac {{\left | -2 \, \sqrt {42} + \frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}} \right |}}{{\left | 2 \, \sqrt {42} + \frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}} \right |}}\right ) + \frac {3}{92} \, \sqrt {6} {\left (\frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}\right )} \] Input:

integrate(1/(5-2*x)^(1/2)/(4+3*x)^(3/2)/(x^2+3*x+2),x, algorithm="giac")
 

Output:

1/36*sqrt(6)*sqrt(3)*(pi + 2*arctan(-1/18*sqrt(3)*sqrt(3*x + 4)*((sqrt(2)* 
sqrt(-6*x + 15) - sqrt(46))^2/(3*x + 4) - 4)/(sqrt(2)*sqrt(-6*x + 15) - sq 
rt(46)))) + 1/42*sqrt(42)*sqrt(6)*log(abs(-2*sqrt(42) + (sqrt(2)*sqrt(-6*x 
 + 15) - sqrt(46))/sqrt(3*x + 4) - 4*sqrt(3*x + 4)/(sqrt(2)*sqrt(-6*x + 15 
) - sqrt(46)))/abs(2*sqrt(42) + (sqrt(2)*sqrt(-6*x + 15) - sqrt(46))/sqrt( 
3*x + 4) - 4*sqrt(3*x + 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46)))) + 3/92*s 
qrt(6)*((sqrt(2)*sqrt(-6*x + 15) - sqrt(46))/sqrt(3*x + 4) - 4*sqrt(3*x + 
4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46)))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {5-2 x} (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\int \frac {1}{\sqrt {5-2\,x}\,{\left (3\,x+4\right )}^{3/2}\,\left (x^2+3\,x+2\right )} \,d x \] Input:

int(1/((5 - 2*x)^(1/2)*(3*x + 4)^(3/2)*(3*x + x^2 + 2)),x)
 

Output:

int(1/((5 - 2*x)^(1/2)*(3*x + 4)^(3/2)*(3*x + x^2 + 2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.72 \[ \int \frac {1}{\sqrt {5-2 x} (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {161 \sqrt {3 x +4}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {23}}{2}-\frac {3 \sqrt {3}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )}{2}\right )-161 \sqrt {3 x +4}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {23}}{2}+\frac {3 \sqrt {3}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )}{2}\right )+378 \sqrt {-2 x +5}-138 \sqrt {3 x +4}\, \sqrt {7}\, \mathrm {log}\left (-\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )-\sqrt {2}\right )+138 \sqrt {3 x +4}\, \sqrt {7}\, \mathrm {log}\left (-\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )+\sqrt {2}\right )-138 \sqrt {3 x +4}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )-\sqrt {2}\right )+138 \sqrt {3 x +4}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )+\sqrt {2}\right )}{966 \sqrt {3 x +4}} \] Input:

int(1/(5-2*x)^(1/2)/(4+3*x)^(3/2)/(x^2+3*x+2),x)
 

Output:

(161*sqrt(3*x + 4)*sqrt(2)*atan((sqrt(23) - 3*sqrt(3)*tan(asin((sqrt( - 2* 
x + 5)*sqrt(3))/sqrt(23))/2))/2) - 161*sqrt(3*x + 4)*sqrt(2)*atan((sqrt(23 
) + 3*sqrt(3)*tan(asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23))/2))/2) + 378*s 
qrt( - 2*x + 5) - 138*sqrt(3*x + 4)*sqrt(7)*log( - sqrt(23) + sqrt(21)*tan 
(asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23))/2) - sqrt(2)) + 138*sqrt(3*x + 
4)*sqrt(7)*log( - sqrt(23) + sqrt(21)*tan(asin((sqrt( - 2*x + 5)*sqrt(3))/ 
sqrt(23))/2) + sqrt(2)) - 138*sqrt(3*x + 4)*sqrt(7)*log(sqrt(23) + sqrt(21 
)*tan(asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23))/2) - sqrt(2)) + 138*sqrt(3 
*x + 4)*sqrt(7)*log(sqrt(23) + sqrt(21)*tan(asin((sqrt( - 2*x + 5)*sqrt(3) 
)/sqrt(23))/2) + sqrt(2)))/(966*sqrt(3*x + 4))