Integrand size = 20, antiderivative size = 258 \[ \int x (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^3}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {3 (b c-a d)^4 (b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int x (a+b x)^{3/2} (c+d x)^{3/2} \, dx=-\frac {3 (a d+b c) (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{7/2}}+\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c) (b c-a d)^3}{128 b^3 d^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (a d+b c) (b c-a d)^2}{64 b^3 d^2}-\frac {(a+b x)^{5/2} \sqrt {c+d x} (a d+b c) (b c-a d)}{16 b^3 d}-\frac {(a+b x)^{5/2} (c+d x)^{3/2} (a d+b c)}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d} \]
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {(b c+a d) \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx}{2 b d} \\ & = -\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {\left (3 \left (c^2-\frac {a^2 d^2}{b^2}\right )\right ) \int (a+b x)^{3/2} \sqrt {c+d x} \, dx}{16 d} \\ & = -\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {\left ((b c-a d)^2 (b c+a d)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{32 b^3 d} \\ & = -\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}+\frac {\left (3 (b c-a d)^3 (b c+a d)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^3 d^2} \\ & = \frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^3}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {\left (3 (b c-a d)^4 (b c+a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^3 d^3} \\ & = \frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^3}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {\left (3 (b c-a d)^4 (b c+a d)\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^4 d^3} \\ & = \frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^3}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {\left (3 (b c-a d)^4 (b c+a d)\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^4 d^3} \\ & = \frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^3}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {3 (b c-a d)^4 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{7/2}} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.88 \[ \int x (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^4 d^4-10 a^3 b d^3 (4 c+d x)+2 a^2 b^2 d^2 \left (9 c^2+13 c d x+4 d^2 x^2\right )+2 a b^3 d \left (-20 c^3+13 c^2 d x+136 c d^2 x^2+88 d^3 x^3\right )+b^4 \left (15 c^4-10 c^3 d x+8 c^2 d^2 x^2+176 c d^3 x^3+128 d^4 x^4\right )\right )}{640 b^3 d^3}-\frac {3 (b c-a d)^4 (b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{7/2} d^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(214)=428\).
Time = 1.54 (sec) , antiderivative size = 788, normalized size of antiderivative = 3.05
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-256 b^{4} d^{4} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-352 a \,b^{3} d^{4} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-352 b^{4} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-16 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-544 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-16 b^{4} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+15 \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}-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^{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}-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 \,b^{4} c^{4} d +15 \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}+20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b \,d^{4} x -52 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x -52 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{2} d^{2} x +20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{3} d x -30 \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}-36 \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 -30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{4}\right )}{1280 b^{3} d^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}\) | \(788\) |
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Time = 0.27 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.69 \[ \int x (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\left [\frac {15 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (128 \, b^{5} d^{5} x^{4} + 15 \, b^{5} c^{4} d - 40 \, a b^{4} c^{3} d^{2} + 18 \, a^{2} b^{3} c^{2} d^{3} - 40 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5} + 176 \, {\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (b^{5} c^{2} d^{3} + 34 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{5} c^{3} d^{2} - 13 \, a b^{4} c^{2} d^{3} - 13 \, a^{2} b^{3} c d^{4} + 5 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, b^{4} d^{4}}, \frac {15 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (128 \, b^{5} d^{5} x^{4} + 15 \, b^{5} c^{4} d - 40 \, a b^{4} c^{3} d^{2} + 18 \, a^{2} b^{3} c^{2} d^{3} - 40 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5} + 176 \, {\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (b^{5} c^{2} d^{3} + 34 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{5} c^{3} d^{2} - 13 \, a b^{4} c^{2} d^{3} - 13 \, a^{2} b^{3} c d^{4} + 5 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, b^{4} d^{4}}\right ] \]
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\[ \int x (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\int x \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \]
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Exception generated. \[ \int x (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1511 vs. \(2 (214) = 428\).
Time = 0.50 (sec) , antiderivative size = 1511, normalized size of antiderivative = 5.86 \[ \int x (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int x (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\int x\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \]
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