Integrand size = 26, antiderivative size = 59 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x}-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{\sqrt {7}} \] Output:
(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)-11/7*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)* 7^(1/2)/(3+5*x)^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(59)=118\).
Time = 1.19 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x}+\frac {11 \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )}{\sqrt {7}}+\frac {11 \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )}{\sqrt {7}} \] Input:
Integrate[Sqrt[1 - 2*x]/((2 + 3*x)^2*Sqrt[3 + 5*x]),x]
Output:
(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x) + (11*ArcTan[(Sqrt[2*(34 + Sqrt[11 55])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])])/Sqrt[7] + (11*ArcTan[S qrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))])/Sqrt[ 7]
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {105, 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.
\(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^2 \sqrt {5 x+3}} \, dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {11}{2} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 x+2}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle 11 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 x+2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 x+2}-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\) |
Input:
Int[Sqrt[1 - 2*x]/((2 + 3*x)^2*Sqrt[3 + 5*x]),x]
Output:
(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x) - (11*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7 ]*Sqrt[3 + 5*x])])/Sqrt[7]
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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(107\) vs. \(2(48)=96\).
Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.83
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (33 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +22 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+14 \sqrt {-10 x^{2}-x +3}\right )}{14 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\) | \(108\) |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\left (2+3 x \right ) \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {11 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{14 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(114\) |
Input:
int((1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/14*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(33*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2) /(-10*x^2-x+3)^(1/2))*x+22*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2- x+3)^(1/2))+14*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {11 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(1/2),x, algorithm="fricas")
Output:
-1/14*(11*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)* sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)
\[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\int \frac {\sqrt {1 - 2 x}}{\left (3 x + 2\right )^{2} \sqrt {5 x + 3}}\, dx \] Input:
integrate((1-2*x)**(1/2)/(2+3*x)**2/(3+5*x)**(1/2),x)
Output:
Integral(sqrt(1 - 2*x)/((3*x + 2)**2*sqrt(5*x + 3)), x)
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {11}{14} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{3 \, x + 2} \] Input:
integrate((1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(1/2),x, algorithm="maxima")
Output:
11/14*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + sqrt(-10 *x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (48) = 96\).
Time = 0.15 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.34 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {11}{140} \, \sqrt {5} {\left (\sqrt {70} \sqrt {2} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {280 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280}\right )} \] Input:
integrate((1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(1/2),x, algorithm="giac")
Output:
11/140*sqrt(5)*(sqrt(70)*sqrt(2)*(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)))) + 280*sqrt(2)*((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)))/(( (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)))^2 + 280))
Time = 4.51 (sec) , antiderivative size = 716, normalized size of antiderivative = 12.14 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx =\text {Too large to display} \] Input:
int((1 - 2*x)^(1/2)/((3*x + 2)^2*(5*x + 3)^(1/2)),x)
Output:
(2*((1 - 2*x)^(1/2) - 1)^3)/((3^(1/2) - (5*x + 3)^(1/2))^3*((14*((1 - 2*x) ^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + ((1 - 2*x)^(1/2) - 1)^ 4/(3^(1/2) - (5*x + 3)^(1/2))^4 + (6*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(5*( 3^(1/2) - (5*x + 3)^(1/2))^3) - (12*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(25*(3^ (1/2) - (5*x + 3)^(1/2))) + 4/25)) - (11*7^(1/2)*atan((2904*3^(1/2)*7^(1/2 ))/(875*((2904*((1 - 2*x)^(1/2) - 1)^2)/(35*(3^(1/2) - (5*x + 3)^(1/2))^2) + (53724*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(875*(3^(1/2) - (5*x + 3)^(1/2))) - 5808/175)) + (1452*7^(1/2)*((1 - 2*x)^(1/2) - 1))/(875*(3^(1/2) - (5*x + 3)^(1/2))*((2904*((1 - 2*x)^(1/2) - 1)^2)/(35*(3^(1/2) - (5*x + 3)^(1/2) )^2) + (53724*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(875*(3^(1/2) - (5*x + 3)^(1/ 2))) - 5808/175)) - (1452*3^(1/2)*7^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(175*(3 ^(1/2) - (5*x + 3)^(1/2))^2*((2904*((1 - 2*x)^(1/2) - 1)^2)/(35*(3^(1/2) - (5*x + 3)^(1/2))^2) + (53724*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(875*(3^(1/2) - (5*x + 3)^(1/2))) - 5808/175))))/7 - (4*((1 - 2*x)^(1/2) - 1))/(5*(3^(1 /2) - (5*x + 3)^(1/2))*((14*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + ((1 - 2*x)^(1/2) - 1)^4/(3^(1/2) - (5*x + 3)^(1/2))^4 + (6 *3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(5*(3^(1/2) - (5*x + 3)^(1/2))^3) - (12* 3^(1/2)*((1 - 2*x)^(1/2) - 1))/(25*(3^(1/2) - (5*x + 3)^(1/2))) + 4/25)) + (37*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2*(( 14*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + ((1 - ...
Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.69 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {33 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +22 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-33 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -22 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )+7 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{21 x +14} \] Input:
int((1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(1/2),x)
Output:
(33*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/ sqrt(11))/2))/sqrt(2))*x + 22*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin(( sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 33*sqrt(7)*atan((sqrt(3 3) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 22*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5) )/sqrt(11))/2))/sqrt(2)) + 7*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(7*(3*x + 2))