Optimal. Leaf size=151 \[ \frac {65 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}+\frac {65 \sqrt {1-2 x} (3+5 x)^{3/2}}{3234 (2+3 x)^2}+\frac {26 (3+5 x)^{5/2}}{231 \sqrt {1-2 x} (2+3 x)^2}+\frac {4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}+\frac {715 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210}
\begin {gather*} \frac {715 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}}+\frac {4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}+\frac {26 (5 x+3)^{5/2}}{231 \sqrt {1-2 x} (3 x+2)^2}+\frac {65 \sqrt {1-2 x} (5 x+3)^{3/2}}{3234 (3 x+2)^2}+\frac {65 \sqrt {1-2 x} \sqrt {5 x+3}}{1372 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 95
Rule 96
Rule 98
Rule 210
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx &=\frac {4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}+\frac {13}{33} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\\ &=\frac {26 (3+5 x)^{5/2}}{231 \sqrt {1-2 x} (2+3 x)^2}+\frac {4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}-\frac {65}{231} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {65 \sqrt {1-2 x} (3+5 x)^{3/2}}{3234 (2+3 x)^2}+\frac {26 (3+5 x)^{5/2}}{231 \sqrt {1-2 x} (2+3 x)^2}+\frac {4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}-\frac {65}{196} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {65 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}+\frac {65 \sqrt {1-2 x} (3+5 x)^{3/2}}{3234 (2+3 x)^2}+\frac {26 (3+5 x)^{5/2}}{231 \sqrt {1-2 x} (2+3 x)^2}+\frac {4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}-\frac {715 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=\frac {65 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}+\frac {65 \sqrt {1-2 x} (3+5 x)^{3/2}}{3234 (2+3 x)^2}+\frac {26 (3+5 x)^{5/2}}{231 \sqrt {1-2 x} (2+3 x)^2}+\frac {4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}-\frac {715 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=\frac {65 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}+\frac {65 \sqrt {1-2 x} (3+5 x)^{3/2}}{3234 (2+3 x)^2}+\frac {26 (3+5 x)^{5/2}}{231 \sqrt {1-2 x} (2+3 x)^2}+\frac {4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}+\frac {715 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.19, size = 150, normalized size = 0.99 \begin {gather*} \frac {5 \left (-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (-6732-13627 x+1620 x^2+10260 x^3\right )}{5 \left (-2+x+6 x^2\right )^2}-429 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )-429 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{28812} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs.
\(2(118)=236\).
time = 0.08, size = 257, normalized size = 1.70
method | result | size |
default | \(-\frac {\left (77220 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+25740 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-49335 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+143640 x^{3} \sqrt {-10 x^{2}-x +3}-8580 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +22680 x^{2} \sqrt {-10 x^{2}-x +3}+8580 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-190778 x \sqrt {-10 x^{2}-x +3}-94248 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{57624 \left (2+3 x \right )^{2} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(257\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 172, normalized size = 1.14 \begin {gather*} -\frac {715}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {475 \, x}{686 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {215}{4116 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {17375 \, x}{2646 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{1134 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {1}{36 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {60695}{15876 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.41, size = 116, normalized size = 0.77 \begin {gather*} \frac {2145 \, \sqrt {7} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\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 (10260 \, x^{3} + 1620 \, x^{2} - 13627 \, x - 6732\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{57624 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 291 vs.
\(2 (118) = 236\).
time = 1.88, size = 291, normalized size = 1.93 \begin {gather*} -\frac {143}{38416} \, \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 {22 \, {\left (104 \, \sqrt {5} {\left (5 \, x + 3\right )} - 957 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{180075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {11 \, \sqrt {10} {\left (223 \, {\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 {80920 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {323680 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4802 \, {\left ({\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 )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________