Integrand size = 22, antiderivative size = 314 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^3}-\frac {3 (3 b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d^2}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{11/2}} \]
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Time = 0.20 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {92, 81, 52, 65, 223, 212} \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=-\frac {(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{11/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{128 b^2 d^5}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{192 b^2 d^4}+\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{240 b^2 d^3}-\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{40 b^2 d^2}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d} \]
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}+\frac {\int \frac {(a+b x)^{5/2} \left (-a c-\frac {3}{2} (3 b c+a d) x\right )}{\sqrt {c+d x}} \, dx}{5 b d} \\ & = -\frac {3 (3 b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d^2}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{80 b^2 d^2} \\ & = \frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^3}-\frac {3 (3 b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d^2}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\left ((b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{96 b^2 d^3} \\ & = -\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^3}-\frac {3 (3 b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d^2}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}+\frac {\left ((b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^2 d^4} \\ & = \frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^3}-\frac {3 (3 b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d^2}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^2 d^5} \\ & = \frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^3}-\frac {3 (3 b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d^2}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{128 b^3 d^5} \\ & = \frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^3}-\frac {3 (3 b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d^2}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 b^3 d^5} \\ & = \frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^3}-\frac {3 (3 b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d^2}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{11/2}} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b d^3 (-3 c+d x)+2 a^2 b^2 d^2 \left (782 c^2-481 c d x+372 d^2 x^2\right )+2 a b^3 d \left (-1155 c^3+749 c^2 d x-592 c d^2 x^2+504 d^3 x^3\right )+b^4 \left (945 c^4-630 c^3 d x+504 c^2 d^2 x^2-432 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^2 d^5}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{5/2} d^{11/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(270)=540\).
Time = 1.78 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.51
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (768 b^{4} d^{4} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+2016 a \,b^{3} d^{4} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-864 b^{4} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+1488 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-2368 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+1008 b^{4} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{5} d^{5}+75 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} b c \,d^{4}+450 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b^{2} c^{2} d^{3}-2250 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{3} c^{3} d^{2}+2625 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{4} c^{4} d -945 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{5} c^{5}+60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b \,d^{4} x -1924 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x +2996 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{2} d^{2} x -1260 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{3} d x -90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{4} d^{4}-180 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b c \,d^{3}+3128 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}-4620 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{3} d +1890 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{4}\right )}{3840 b^{2} d^{5} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}\) | \(788\) |
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Time = 0.30 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.25 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\left [-\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{3} d^{6}}, \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (384 \, b^{5} d^{5} x^{4} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{3} d^{6}}\right ] \]
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\[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\int \frac {x^{2} \left (a + b x\right )^{\frac {5}{2}}}{\sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.22 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (6 \, {\left (b x + a\right )} {\left (\frac {8 \, {\left (b x + a\right )}}{b^{3} d} - \frac {9 \, b^{7} c d^{7} + 11 \, a b^{6} d^{8}}{b^{9} d^{9}}\right )} + \frac {63 \, b^{8} c^{2} d^{6} + 14 \, a b^{7} c d^{7} + 3 \, a^{2} b^{6} d^{8}}{b^{9} d^{9}}\right )} - \frac {5 \, {\left (63 \, b^{9} c^{3} d^{5} - 49 \, a b^{8} c^{2} d^{6} - 11 \, a^{2} b^{7} c d^{7} - 3 \, a^{3} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (63 \, b^{10} c^{4} d^{4} - 112 \, a b^{9} c^{3} d^{5} + 38 \, a^{2} b^{8} c^{2} d^{6} + 8 \, a^{3} b^{7} c d^{7} + 3 \, a^{4} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} \sqrt {b x + a} + \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{5}}\right )} b}{1920 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\int \frac {x^2\,{\left (a+b\,x\right )}^{5/2}}{\sqrt {c+d\,x}} \,d x \]
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