Integrand size = 22, antiderivative size = 302 \[ \int x^3 \sqrt {a+b x} \sqrt {c+d x} \, dx=-\frac {\left (7 b^4 c^4+2 a b^3 c^3 d-2 a^3 b c d^3-7 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^4}-\frac {(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^3}+\frac {x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}+\frac {(b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{9/2}} \]
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Time = 0.18 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {102, 152, 52, 65, 223, 212} \[ \int x^3 \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{9/2}}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right )}{64 b^4 d^3}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (35 a^2 d^2-42 b d x (a d+b c)+38 a b c d+35 b^2 c^2\right )}{240 b^3 d^3}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-7 a^4 d^4-2 a^3 b c d^3+2 a b^3 c^3 d+7 b^4 c^4\right )}{128 b^4 d^4}+\frac {x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d} \]
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Rule 52
Rule 65
Rule 102
Rule 152
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac {\int x \sqrt {a+b x} \sqrt {c+d x} \left (-2 a c-\frac {7}{2} (b c+a d) x\right ) \, dx}{5 b d} \\ & = \frac {x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}-\frac {\left ((b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right )\right ) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{32 b^3 d^3} \\ & = -\frac {(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^3}+\frac {x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}-\frac {\left ((b c-a d) (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^4 d^3} \\ & = -\frac {(b c-a d) (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^4}-\frac {(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^3}+\frac {x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}+\frac {\left ((b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^4 d^4} \\ & = -\frac {(b c-a d) (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^4}-\frac {(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^3}+\frac {x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}+\frac {\left ((b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 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^5 d^4} \\ & = -\frac {(b c-a d) (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^4}-\frac {(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^3}+\frac {x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}+\frac {\left ((b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 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^5 d^4} \\ & = -\frac {(b c-a d) (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^4}-\frac {(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^3}+\frac {x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}+\frac {(b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{9/2}} \\ \end{align*}
Time = 1.26 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.85 \[ \int x^3 \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+10 a^3 b d^3 (4 c+7 d x)-2 a^2 b^2 d^2 \left (-17 c^2+11 c d x+28 d^2 x^2\right )+2 a b^3 d \left (20 c^3-11 c^2 d x+8 c d^2 x^2+24 d^3 x^3\right )+b^4 \left (-105 c^4+70 c^3 d x-56 c^2 d^2 x^2+48 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^4 d^4}+\frac {(b c-a d)^2 \left (7 b^3 c^3+9 a b^2 c^2 d+9 a^2 b c d^2+7 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{9/2} d^{9/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(264)=528\).
Time = 1.49 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.61
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}+96 a \,b^{3} d^{4} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+96 b^{4} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-112 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+32 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-112 b^{4} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+105 \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}-30 \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}-30 \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}-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 \,b^{4} c^{4} d +105 \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}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b \,d^{4} x -44 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x -44 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{2} d^{2} x +140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{3} d x -210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{4} d^{4}+80 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b c \,d^{3}+68 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}+80 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{3} d -210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{4}\right )}{3840 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{4} d^{4} \sqrt {b d}}\) | \(788\) |
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Time = 0.27 (sec) , antiderivative size = 702, normalized size of antiderivative = 2.32 \[ \int x^3 \sqrt {a+b x} \sqrt {c+d x} \, dx=\left [\frac {15 \, {\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 40 \, a b^{4} c^{3} d^{2} + 34 \, a^{2} b^{3} c^{2} d^{3} + 40 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 11 \, a b^{4} c^{2} d^{3} - 11 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{5} d^{5}}, -\frac {15 \, {\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 40 \, a b^{4} c^{3} d^{2} + 34 \, a^{2} b^{3} c^{2} d^{3} + 40 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 11 \, a b^{4} c^{2} d^{3} - 11 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{5} d^{5}}\right ] \]
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\[ \int x^3 \sqrt {a+b x} \sqrt {c+d x} \, dx=\int x^{3} \sqrt {a + b x} \sqrt {c + d x}\, dx \]
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Exception generated. \[ \int x^3 \sqrt {a+b x} \sqrt {c+d x} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (264) = 528\).
Time = 0.36 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.22 \[ \int x^3 \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\frac {10 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\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^{3}}\right )} a {\left | b \right |}}{b^{2}} + \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^{4}} + \frac {b^{20} c d^{7} - 41 \, a b^{19} d^{8}}{b^{23} d^{8}}\right )} - \frac {7 \, b^{21} c^{2} d^{6} + 26 \, a b^{20} c d^{7} - 513 \, a^{2} b^{19} d^{8}}{b^{23} d^{8}}\right )} + \frac {5 \, {\left (7 \, b^{22} c^{3} d^{5} + 19 \, a b^{21} c^{2} d^{6} + 37 \, a^{2} b^{20} c d^{7} - 447 \, a^{3} b^{19} d^{8}\right )}}{b^{23} d^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (7 \, b^{23} c^{4} d^{4} + 12 \, a b^{22} c^{3} d^{5} + 18 \, a^{2} b^{21} c^{2} d^{6} + 28 \, a^{3} b^{20} c d^{7} - 193 \, a^{4} b^{19} d^{8}\right )}}{b^{23} d^{8}}\right )} \sqrt {b x + a} - \frac {15 \, {\left (7 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 10 \, a^{3} b^{2} c^{2} d^{3} + 35 \, a^{4} b c d^{4} - 63 \, 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^{3} d^{4}}\right )} {\left | b \right |}}{b}}{1920 \, b} \]
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Timed out. \[ \int x^3 \sqrt {a+b x} \sqrt {c+d x} \, dx=\text {Hanged} \]
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