Integrand size = 19, antiderivative size = 74 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=-\frac {5 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {(3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {5}{2} \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]
1/3*(3+5*x)^(3/2)/(1-2*x)^(3/2)+5/4*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10 ^(1/2)-5/2*(3+5*x)^(1/2)/(1-2*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.82 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\frac {\sqrt {3+5 x} (-9+40 x)}{6 (1-2 x)^{3/2}}+\frac {5}{2} \sqrt {\frac {5}{2}} \arctan \left (\frac {\sqrt {\frac {6}{5}+2 x}}{\sqrt {1-2 x}}\right ) \]
(Sqrt[3 + 5*x]*(-9 + 40*x))/(6*(1 - 2*x)^(3/2)) + (5*Sqrt[5/2]*ArcTan[Sqrt [6/5 + 2*x]/Sqrt[1 - 2*x]])/2
Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {57, 57, 64, 223}
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.
\(\displaystyle \int \frac {(5 x+3)^{3/2}}{(1-2 x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {(5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {5}{2} \int \frac {\sqrt {5 x+3}}{(1-2 x)^{3/2}}dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {(5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {5}{2} \left (\frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {(5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {5}{2} \left (\frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}-\int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {(5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {5}{2} \left (\frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}-\sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )\right )\) |
(3 + 5*x)^(3/2)/(3*(1 - 2*x)^(3/2)) - (5*(Sqrt[3 + 5*x]/Sqrt[1 - 2*x] - Sq rt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]))/2
3.26.93.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [2/b Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] || PosQ[b])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
\[\int \frac {\left (3+5 x \right )^{\frac {3}{2}}}{\left (1-2 x \right )^{\frac {5}{2}}}d x\]
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=-\frac {15 \, \sqrt {5} \sqrt {2} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 4 \, {\left (40 \, x - 9\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{24 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/24*(15*sqrt(5)*sqrt(2)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(2 0*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 4*(40*x - 9)*sqr t(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)
Result contains complex when optimal does not.
Time = 1.92 (sec) , antiderivative size = 634, normalized size of antiderivative = 8.57 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\begin {cases} - \frac {300 \sqrt {10} i \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} + \frac {150 \sqrt {10} \pi \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} + \frac {330 \sqrt {10} i \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5} \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} - \frac {165 \sqrt {10} \pi \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} + \frac {4000 i \left (x + \frac {3}{5}\right )^{8}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} - \frac {3300 i \left (x + \frac {3}{5}\right )^{7}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {150 \sqrt {10} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} - \frac {165 \sqrt {10} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} - \frac {2000 \left (x + \frac {3}{5}\right )^{8}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} + \frac {1650 \left (x + \frac {3}{5}\right )^{7}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((-300*sqrt(10)*I*(x + 3/5)**(15/2)*sqrt(10*x - 5)*acosh(sqrt(110 )*sqrt(x + 3/5)/11)/(240*(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)* *(13/2)*sqrt(10*x - 5)) + 150*sqrt(10)*pi*(x + 3/5)**(15/2)*sqrt(10*x - 5) /(240*(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) + 330*sqrt(10)*I*(x + 3/5)**(13/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqr t(x + 3/5)/11)/(240*(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**(13/ 2)*sqrt(10*x - 5)) - 165*sqrt(10)*pi*(x + 3/5)**(13/2)*sqrt(10*x - 5)/(240 *(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) + 4000*I*(x + 3/5)**8/(240*(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x + 3/5 )**(13/2)*sqrt(10*x - 5)) - 3300*I*(x + 3/5)**7/(240*(x + 3/5)**(15/2)*sqr t(10*x - 5) - 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)), Abs(x + 3/5) > 11/10) , (150*sqrt(10)*sqrt(5 - 10*x)*(x + 3/5)**(15/2)*asin(sqrt(110)*sqrt(x + 3 /5)/11)/(120*sqrt(5 - 10*x)*(x + 3/5)**(15/2) - 132*sqrt(5 - 10*x)*(x + 3/ 5)**(13/2)) - 165*sqrt(10)*sqrt(5 - 10*x)*(x + 3/5)**(13/2)*asin(sqrt(110) *sqrt(x + 3/5)/11)/(120*sqrt(5 - 10*x)*(x + 3/5)**(15/2) - 132*sqrt(5 - 10 *x)*(x + 3/5)**(13/2)) - 2000*(x + 3/5)**8/(120*sqrt(5 - 10*x)*(x + 3/5)** (15/2) - 132*sqrt(5 - 10*x)*(x + 3/5)**(13/2)) + 1650*(x + 3/5)**7/(120*sq rt(5 - 10*x)*(x + 3/5)**(15/2) - 132*sqrt(5 - 10*x)*(x + 3/5)**(13/2)), Tr ue))
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.26 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\frac {5}{8} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{6 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {11 \, \sqrt {-10 \, x^{2} - x + 3}}{12 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {35 \, \sqrt {-10 \, x^{2} - x + 3}}{12 \, {\left (2 \, x - 1\right )}} \]
5/8*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/6*(-10*x^2 - x + 3)^(3/2)/( 8*x^3 - 12*x^2 + 6*x - 1) + 11/12*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 35/12*sqrt(-10*x^2 - x + 3)/(2*x - 1)
Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\frac {5}{4} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{30 \, {\left (2 \, x - 1\right )}^{2}} \]
5/4*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/30*(8*sqrt(5)*(5*x + 3) - 33*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]