Integrand size = 17, antiderivative size = 163 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\frac {45 a^6 (a+2 b x) \sqrt {a x+b x^2}}{16384 b^5}-\frac {15 a^4 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{2048 b^4}+\frac {3 a^2 (a+2 b x) \left (a x+b x^2\right )^{5/2}}{128 b^3}-\frac {9 a \left (a x+b x^2\right )^{7/2}}{112 b^2}+\frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {45 a^8 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{16384 b^{11/2}} \]
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Time = 0.06 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {684, 654, 626, 634, 212} \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=-\frac {45 a^8 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{16384 b^{11/2}}+\frac {45 a^6 (a+2 b x) \sqrt {a x+b x^2}}{16384 b^5}-\frac {15 a^4 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{2048 b^4}+\frac {3 a^2 (a+2 b x) \left (a x+b x^2\right )^{5/2}}{128 b^3}-\frac {9 a \left (a x+b x^2\right )^{7/2}}{112 b^2}+\frac {x \left (a x+b x^2\right )^{7/2}}{8 b} \]
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Rule 212
Rule 626
Rule 634
Rule 654
Rule 684
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {(9 a) \int x \left (a x+b x^2\right )^{5/2} \, dx}{16 b} \\ & = -\frac {9 a \left (a x+b x^2\right )^{7/2}}{112 b^2}+\frac {x \left (a x+b x^2\right )^{7/2}}{8 b}+\frac {\left (9 a^2\right ) \int \left (a x+b x^2\right )^{5/2} \, dx}{32 b^2} \\ & = \frac {3 a^2 (a+2 b x) \left (a x+b x^2\right )^{5/2}}{128 b^3}-\frac {9 a \left (a x+b x^2\right )^{7/2}}{112 b^2}+\frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {\left (15 a^4\right ) \int \left (a x+b x^2\right )^{3/2} \, dx}{256 b^3} \\ & = -\frac {15 a^4 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{2048 b^4}+\frac {3 a^2 (a+2 b x) \left (a x+b x^2\right )^{5/2}}{128 b^3}-\frac {9 a \left (a x+b x^2\right )^{7/2}}{112 b^2}+\frac {x \left (a x+b x^2\right )^{7/2}}{8 b}+\frac {\left (45 a^6\right ) \int \sqrt {a x+b x^2} \, dx}{4096 b^4} \\ & = \frac {45 a^6 (a+2 b x) \sqrt {a x+b x^2}}{16384 b^5}-\frac {15 a^4 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{2048 b^4}+\frac {3 a^2 (a+2 b x) \left (a x+b x^2\right )^{5/2}}{128 b^3}-\frac {9 a \left (a x+b x^2\right )^{7/2}}{112 b^2}+\frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {\left (45 a^8\right ) \int \frac {1}{\sqrt {a x+b x^2}} \, dx}{32768 b^5} \\ & = \frac {45 a^6 (a+2 b x) \sqrt {a x+b x^2}}{16384 b^5}-\frac {15 a^4 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{2048 b^4}+\frac {3 a^2 (a+2 b x) \left (a x+b x^2\right )^{5/2}}{128 b^3}-\frac {9 a \left (a x+b x^2\right )^{7/2}}{112 b^2}+\frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {\left (45 a^8\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )}{16384 b^5} \\ & = \frac {45 a^6 (a+2 b x) \sqrt {a x+b x^2}}{16384 b^5}-\frac {15 a^4 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{2048 b^4}+\frac {3 a^2 (a+2 b x) \left (a x+b x^2\right )^{5/2}}{128 b^3}-\frac {9 a \left (a x+b x^2\right )^{7/2}}{112 b^2}+\frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {45 a^8 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{16384 b^{11/2}} \\ \end{align*}
Time = 1.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\frac {\sqrt {x (a+b x)} \left (\sqrt {b} \left (315 a^7-210 a^6 b x+168 a^5 b^2 x^2-144 a^4 b^3 x^3+128 a^3 b^4 x^4+20736 a^2 b^5 x^5+33792 a b^6 x^6+14336 b^7 x^7\right )+\frac {630 a^8 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{\sqrt {x} \sqrt {a+b x}}\right )}{114688 b^{11/2}} \]
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Time = 2.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {\left (14336 b^{7} x^{7}+33792 a \,b^{6} x^{6}+20736 a^{2} b^{5} x^{5}+128 a^{3} b^{4} x^{4}-144 a^{4} b^{3} x^{3}+168 a^{5} b^{2} x^{2}-210 a^{6} b x +315 a^{7}\right ) x \left (b x +a \right )}{114688 b^{5} \sqrt {x \left (b x +a \right )}}-\frac {45 a^{8} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{32768 b^{\frac {11}{2}}}\) | \(128\) |
default | \(\frac {x \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{8 b}-\frac {9 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{7 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{12 b}-\frac {5 a^{2} \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{24 b}\right )}{2 b}\right )}{16 b}\) | \(165\) |
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Time = 0.27 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.58 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\left [\frac {315 \, a^{8} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (14336 \, b^{8} x^{7} + 33792 \, a b^{7} x^{6} + 20736 \, a^{2} b^{6} x^{5} + 128 \, a^{3} b^{5} x^{4} - 144 \, a^{4} b^{4} x^{3} + 168 \, a^{5} b^{3} x^{2} - 210 \, a^{6} b^{2} x + 315 \, a^{7} b\right )} \sqrt {b x^{2} + a x}}{229376 \, b^{6}}, \frac {315 \, a^{8} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right ) + {\left (14336 \, b^{8} x^{7} + 33792 \, a b^{7} x^{6} + 20736 \, a^{2} b^{6} x^{5} + 128 \, a^{3} b^{5} x^{4} - 144 \, a^{4} b^{4} x^{3} + 168 \, a^{5} b^{3} x^{2} - 210 \, a^{6} b^{2} x + 315 \, a^{7} b\right )} \sqrt {b x^{2} + a x}}{114688 \, b^{6}}\right ] \]
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Time = 0.46 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.15 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\begin {cases} - \frac {45 a^{8} \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{32768 b^{5}} + \sqrt {a x + b x^{2}} \cdot \left (\frac {45 a^{7}}{16384 b^{5}} - \frac {15 a^{6} x}{8192 b^{4}} + \frac {3 a^{5} x^{2}}{2048 b^{3}} - \frac {9 a^{4} x^{3}}{7168 b^{2}} + \frac {a^{3} x^{4}}{896 b} + \frac {81 a^{2} x^{5}}{448} + \frac {33 a b x^{6}}{112} + \frac {b^{2} x^{7}}{8}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (a x\right )^{\frac {11}{2}}}{11 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.12 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\frac {45 \, \sqrt {b x^{2} + a x} a^{6} x}{8192 \, b^{4}} - \frac {15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4} x}{1024 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} a^{2} x}{64 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {7}{2}} x}{8 \, b} - \frac {45 \, a^{8} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{32768 \, b^{\frac {11}{2}}} + \frac {45 \, \sqrt {b x^{2} + a x} a^{7}}{16384 \, b^{5}} - \frac {15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{5}}{2048 \, b^{4}} + \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} a^{3}}{128 \, b^{3}} - \frac {9 \, {\left (b x^{2} + a x\right )}^{\frac {7}{2}} a}{112 \, b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.79 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\frac {45 \, a^{8} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{32768 \, b^{\frac {11}{2}}} + \frac {1}{114688} \, \sqrt {b x^{2} + a x} {\left (\frac {315 \, a^{7}}{b^{5}} - 2 \, {\left (\frac {105 \, a^{6}}{b^{4}} - 4 \, {\left (\frac {21 \, a^{5}}{b^{3}} - 2 \, {\left (\frac {9 \, a^{4}}{b^{2}} - 8 \, {\left (\frac {a^{3}}{b} + 2 \, {\left (81 \, a^{2} + 4 \, {\left (14 \, b^{2} x + 33 \, a b\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
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Timed out. \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\int x^2\,{\left (b\,x^2+a\,x\right )}^{5/2} \,d x \]
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