\(\int \frac {(5-2 x)^{3/2}}{(4+3 x)^{5/2} (2+3 x+x^2)} \, dx\) [373]

Optimal result

Integrand size = 29, antiderivative size = 112 \[ \int \frac {(5-2 x)^{3/2}}{(4+3 x)^{5/2} \left (2+3 x+x^2\right )} \, dx=\frac {23 \sqrt {5-2 x}}{3 (4+3 x)^{3/2}}+\frac {53 \sqrt {5-2 x}}{6 \sqrt {4+3 x}}-\frac {27 \arctan \left (\frac {3 \sqrt {4+3 x}}{\sqrt {2} \sqrt {5-2 x}}\right )}{2 \sqrt {2}}-14 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {7} \sqrt {4+3 x}}{\sqrt {5-2 x}}\right ) \] Output:

23/3*(5-2*x)^(1/2)/(4+3*x)^(3/2)+53/6*(5-2*x)^(1/2)/(4+3*x)^(1/2)-27/4*2^( 
1/2)*arctan(3/2*(4+3*x)^(1/2)*2^(1/2)/(5-2*x)^(1/2))-14*7^(1/2)*arctanh(7^ 
(1/2)*(4+3*x)^(1/2)/(5-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.82 \[ \int \frac {(5-2 x)^{3/2}}{(4+3 x)^{5/2} \left (2+3 x+x^2\right )} \, dx=\frac {1}{4} \left (\frac {2 \sqrt {5-2 x} (86+53 x)}{(4+3 x)^{3/2}}+27 \sqrt {2} \arctan \left (\frac {\sqrt {10-4 x}}{3 \sqrt {4+3 x}}\right )-56 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7} \sqrt {4+3 x}}\right )\right ) \] Input:

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

Output:

((2*Sqrt[5 - 2*x]*(86 + 53*x))/(4 + 3*x)^(3/2) + 27*Sqrt[2]*ArcTan[Sqrt[10 
 - 4*x]/(3*Sqrt[4 + 3*x])] - 56*Sqrt[7]*ArcTanh[Sqrt[5 - 2*x]/(Sqrt[7]*Sqr 
t[4 + 3*x])])/4
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1203, 25, 48, 2153, 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.

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

\(\Big \downarrow \) 1203

\(\displaystyle -\frac {1}{2} \int -\frac {\sqrt {5-2 x} (23 x+37)}{(3 x+4)^{3/2} \left (x^2+3 x+2\right )}dx-\frac {69}{2} \int \frac {\sqrt {5-2 x}}{(3 x+4)^{5/2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {5-2 x} (23 x+37)}{(3 x+4)^{3/2} \left (x^2+3 x+2\right )}dx-\frac {69}{2} \int \frac {\sqrt {5-2 x}}{(3 x+4)^{5/2}}dx\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {5-2 x} (23 x+37)}{(3 x+4)^{3/2} \left (x^2+3 x+2\right )}dx+\frac {(5-2 x)^{3/2}}{(3 x+4)^{3/2}}\)

\(\Big \downarrow \) 2153

\(\displaystyle \frac {1}{2} \int \left (\frac {28 \sqrt {5-2 x}}{(2 x+2) (3 x+4)^{3/2}}+\frac {18 \sqrt {5-2 x}}{(2 x+4) (3 x+4)^{3/2}}\right )dx+\frac {(5-2 x)^{3/2}}{(3 x+4)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {27 \arctan \left (\frac {3 \sqrt {3 x+4}}{\sqrt {2} \sqrt {5-2 x}}\right )}{\sqrt {2}}-28 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {7} \sqrt {3 x+4}}{\sqrt {5-2 x}}\right )+\frac {19 \sqrt {5-2 x}}{\sqrt {3 x+4}}\right )+\frac {(5-2 x)^{3/2}}{(3 x+4)^{3/2}}\)

Input:

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

Output:

(5 - 2*x)^(3/2)/(4 + 3*x)^(3/2) + ((19*Sqrt[5 - 2*x])/Sqrt[4 + 3*x] - (27* 
ArcTan[(3*Sqrt[4 + 3*x])/(Sqrt[2]*Sqrt[5 - 2*x])])/Sqrt[2] - 28*Sqrt[7]*Ar 
cTanh[(Sqrt[7]*Sqrt[4 + 3*x])/Sqrt[5 - 2*x]])/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 1203

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(-g)*((e*f - d*g)/(c*f^2 - b*f*g + a* 
g^2))   Int[(d + e*x)^(m - 1)*(f + g*x)^n, x], x] + Simp[1/(c*f^2 - b*f*g + 
 a*g^2)   Int[Simp[c*d*f - b*d*g + a*e*g + c*(e*f - d*g)*x, x]*(d + e*x)^(m 
 - 1)*((f + g*x)^(n + 1)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e 
, f, g}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[m, 0] && LtQ[n, -1]
 

rule 2009

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

rule 2153

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* 
x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ 
[m] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] && IGtQ[n, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(234\) vs. \(2(82)=164\).

Time = 1.94 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.10

method	result	size
default	\(-\frac {\left (243 \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 x^{2}+7 x +20}}\right ) \sqrt {2}\, x^{2}+504 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 x^{2}+7 x +20}}\right ) x^{2}+648 \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 x^{2}+7 x +20}}\right ) \sqrt {2}\, x +1344 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 x^{2}+7 x +20}}\right ) x +432 \sqrt {2}\, \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 x^{2}+7 x +20}}\right )+896 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 x^{2}+7 x +20}}\right )-212 x \sqrt {-6 x^{2}+7 x +20}-344 \sqrt {-6 x^{2}+7 x +20}\right ) \sqrt {5-2 x}}{8 \sqrt {-6 x^{2}+7 x +20}\, \left (3 x +4\right )^{\frac {3}{2}}}\)	\(235\)

Input:

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

Output:

-1/8*(243*arctan(1/12*(26+31*x)*2^(1/2)/(-6*x^2+7*x+20)^(1/2))*2^(1/2)*x^2 
+504*7^(1/2)*arctanh(1/14*(33+19*x)*7^(1/2)/(-6*x^2+7*x+20)^(1/2))*x^2+648 
*arctan(1/12*(26+31*x)*2^(1/2)/(-6*x^2+7*x+20)^(1/2))*2^(1/2)*x+1344*7^(1/ 
2)*arctanh(1/14*(33+19*x)*7^(1/2)/(-6*x^2+7*x+20)^(1/2))*x+432*2^(1/2)*arc 
tan(1/12*(26+31*x)*2^(1/2)/(-6*x^2+7*x+20)^(1/2))+896*7^(1/2)*arctanh(1/14 
*(33+19*x)*7^(1/2)/(-6*x^2+7*x+20)^(1/2))-212*x*(-6*x^2+7*x+20)^(1/2)-344* 
(-6*x^2+7*x+20)^(1/2))*(5-2*x)^(1/2)/(-6*x^2+7*x+20)^(1/2)/(3*x+4)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.34 \[ \int \frac {(5-2 x)^{3/2}}{(4+3 x)^{5/2} \left (2+3 x+x^2\right )} \, dx=\frac {27 \, \sqrt {2} {\left (9 \, x^{2} + 24 \, x + 16\right )} \arctan \left (\frac {\sqrt {2} {\left (31 \, x + 26\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{12 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) + 28 \, \sqrt {7} {\left (9 \, x^{2} + 24 \, x + 16\right )} \log \left (-\frac {4 \, \sqrt {7} {\left (19 \, x + 33\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5} - 193 \, x^{2} - 1450 \, x - 1649}{x^{2} + 2 \, x + 1}\right ) + 4 \, {\left (53 \, x + 86\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{8 \, {\left (9 \, x^{2} + 24 \, x + 16\right )}} \] Input:

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

Output:

1/8*(27*sqrt(2)*(9*x^2 + 24*x + 16)*arctan(1/12*sqrt(2)*(31*x + 26)*sqrt(3 
*x + 4)*sqrt(-2*x + 5)/(6*x^2 - 7*x - 20)) + 28*sqrt(7)*(9*x^2 + 24*x + 16 
)*log(-(4*sqrt(7)*(19*x + 33)*sqrt(3*x + 4)*sqrt(-2*x + 5) - 193*x^2 - 145 
0*x - 1649)/(x^2 + 2*x + 1)) + 4*(53*x + 86)*sqrt(3*x + 4)*sqrt(-2*x + 5)) 
/(9*x^2 + 24*x + 16)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (82) = 164\).

Time = 0.54 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.91 \[ \int \frac {(5-2 x)^{3/2}}{(4+3 x)^{5/2} \left (2+3 x+x^2\right )} \, dx=\frac {1}{288} \, \sqrt {6} {\left (\frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}\right )}^{3} - \frac {9}{8} \, \sqrt {6} \sqrt {3} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {3} \sqrt {3 \, x + 4} {\left (\frac {{\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{2}}{3 \, x + 4} - 4\right )}}{18 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}}\right )\right )} + \frac {7}{6} \, \sqrt {42} \sqrt {6} \log \left (\frac {{\left | -2 \, \sqrt {42} + \frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}} \right |}}{{\left | 2 \, \sqrt {42} + \frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}} \right |}}\right ) + \frac {19}{24} \, \sqrt {6} {\left (\frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}\right )} \] Input:

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

Output:

1/288*sqrt(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)))^3 - 9/8*sqrt(6)*sqrt(3)*(p 
i + 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) - sqrt(46)))) + 7/6*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*sqr 
t(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)))) + 19/24*sqrt(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. \[ \int \frac {(5-2 x)^{3/2}}{(4+3 x)^{5/2} \left (2+3 x+x^2\right )} \, dx=\int \frac {{\left (5-2\,x\right )}^{3/2}}{{\left (3\,x+4\right )}^{5/2}\,\left (x^2+3\,x+2\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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