Optimal. Leaf size=54 \[ -\frac {407}{98 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {87, 63, 206} \begin {gather*} -\frac {407}{98 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 87
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)} \, dx &=\int \left (\frac {121}{14 (1-2 x)^{5/2}}-\frac {407}{98 (1-2 x)^{3/2}}+\frac {1}{49 \sqrt {1-2 x} (2+3 x)}\right ) \, dx\\ &=\frac {121}{42 (1-2 x)^{3/2}}-\frac {407}{98 \sqrt {1-2 x}}+\frac {1}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {121}{42 (1-2 x)^{3/2}}-\frac {407}{98 \sqrt {1-2 x}}-\frac {1}{49} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {121}{42 (1-2 x)^{3/2}}-\frac {407}{98 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 40, normalized size = 0.74 \begin {gather*} \frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+35 (45 x-7)}{189 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.07, size = 50, normalized size = 0.93 \begin {gather*} -\frac {11 (111 (1-2 x)-77)}{294 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.40, size = 68, normalized size = 1.26 \begin {gather*} \frac {\sqrt {21} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 77 \, {\left (111 \, x - 17\right )} \sqrt {-2 \, x + 1}}{1029 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.29, size = 61, normalized size = 1.13 \begin {gather*} \frac {1}{1029} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {11 \, {\left (111 \, x - 17\right )}}{147 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 38, normalized size = 0.70 \begin {gather*} -\frac {2 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{1029}+\frac {121}{42 \left (-2 x +1\right )^{\frac {3}{2}}}-\frac {407}{98 \sqrt {-2 x +1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 51, normalized size = 0.94 \begin {gather*} \frac {1}{1029} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {11 \, {\left (111 \, x - 17\right )}}{147 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.20, size = 32, normalized size = 0.59 \begin {gather*} \frac {\frac {407\,x}{49}-\frac {187}{147}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1029} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 40.43, size = 90, normalized size = 1.67 \begin {gather*} \frac {2 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{49} - \frac {407}{98 \sqrt {1 - 2 x}} + \frac {121}{42 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________