\(\int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx\) [1181]

Optimal result

Integrand size = 26, antiderivative size = 101 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx=\frac {4}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x}}{2541 (3+5 x)^{3/2}}+\frac {31030 \sqrt {1-2 x}}{27951 \sqrt {3+5 x}}-\frac {54 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7 \sqrt {7}} \] Output:

4/77/(1-2*x)^(1/2)/(3+5*x)^(3/2)-410/2541*(1-2*x)^(1/2)/(3+5*x)^(3/2)+3103 
0/27951*(1-2*x)^(1/2)/(3+5*x)^(1/2)-54/49*7^(1/2)*arctan(1/7*(1-2*x)^(1/2) 
*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx=-\frac {2 \left (-45016+11005 x+155150 x^2\right )}{27951 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {54 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7 \sqrt {7}} \] Input:

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

Output:

(-2*(-45016 + 11005*x + 155150*x^2))/(27951*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) 
 - (54*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {115, 27, 169, 27, 169, 27, 104, 217}

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.

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

\(\Big \downarrow \) 115

\(\displaystyle \frac {4}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {2}{77} \int -\frac {120 x+113}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{77} \int \frac {120 x+113}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {4}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{77} \left (-\frac {2}{33} \int \frac {1627-2460 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{77} \left (-\frac {1}{33} \int \frac {1627-2460 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{77} \left (\frac {1}{33} \left (\frac {2}{11} \int \frac {107811}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {31030 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{77} \left (\frac {1}{33} \left (9801 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {31030 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{77} \left (\frac {1}{33} \left (19602 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {31030 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{77} \left (\frac {1}{33} \left (\frac {31030 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {19602 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\right )-\frac {410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\)

Input:

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

Output:

4/(77*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((-410*Sqrt[1 - 2*x])/(33*(3 + 5*x) 
^(3/2)) + ((31030*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) - (19602*ArcTan[Sqrt[1 
 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7])/33)/77
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs. \(2(74)=148\).

Time = 0.26 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.93

method	result	size
default	\(-\frac {5390550 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+3773385 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-1293732 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +2172100 x^{2} \sqrt {-10 x^{2}-x +3}-970299 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+154070 x \sqrt {-10 x^{2}-x +3}-630224 \sqrt {-10 x^{2}-x +3}}{195657 \sqrt {1-2 x}\, \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(195\)

Input:

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

Output:

-1/195657*(5390550*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/ 
2))*x^3+3773385*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) 
*x^2-1293732*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+ 
2172100*x^2*(-10*x^2-x+3)^(1/2)-970299*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/ 
2)/(-10*x^2-x+3)^(1/2))+154070*x*(-10*x^2-x+3)^(1/2)-630224*(-10*x^2-x+3)^ 
(1/2))/(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx=-\frac {107811 \, \sqrt {7} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (155150 \, x^{2} + 11005 \, x - 45016\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{195657 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \] Input:

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

Output:

-1/195657*(107811*sqrt(7)*(50*x^3 + 35*x^2 - 12*x - 9)*arctan(1/14*sqrt(7) 
*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(155150*x 
^2 + 11005*x - 45016)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(50*x^3 + 35*x^2 - 12* 
x - 9)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (74) = 148\).

Time = 0.18 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.18 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx=-\frac {5}{63888} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {27}{490} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {145}{2662} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {16 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{46585 \, {\left (2 \, x - 1\right )}} \] Input:

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

Output:

-5/63888*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4* 
sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 27/490*sqrt(70)*sq 
rt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 
 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 
145/2662*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4* 
sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 16/46585*sqrt(5)*sqr 
t(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.17 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx=\frac {\frac {270 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x}{49}+\frac {162 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )}{49}-\frac {270 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x}{49}-\frac {162 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )}{49}-\frac {310300 x^{2}}{27951}-\frac {22010 x}{27951}+\frac {90032}{27951}}{\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (5 x +3\right )} \] Input:

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

Output:

(2*(539055*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35 
)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 323433*sq 
rt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((s 
qrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 539055*sqrt(5*x + 3)*sqr 
t( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1) 
*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 323433*sqrt(5*x + 3)*sqrt( - 2*x + 1) 
*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqr 
t(11))/2))/sqrt(2)) - 1086050*x**2 - 77035*x + 315112))/(195657*sqrt(5*x + 
 3)*sqrt( - 2*x + 1)*(5*x + 3))