Integrand size = 22, antiderivative size = 251 \[ \int \frac {x^3 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=-\frac {\left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d^4}+\frac {x^2 (a+b x)^{3/2} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (35 b^2 c^2+22 a b c d+15 a^2 d^2-4 b d (7 b c+5 a d) x\right )}{96 b^3 d^3}+\frac {(b c-a d) \left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{9/2}} \]
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Time = 0.12 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {102, 152, 52, 65, 223, 212} \[ \int \frac {x^3 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (15 a^2 d^2-4 b d x (5 a d+7 b c)+22 a b c d+35 b^2 c^2\right )}{96 b^3 d^3}+\frac {(b c-a d) \left (5 a^3 d^3+9 a^2 b c d^2+15 a b^2 c^2 d+35 b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3+9 a^2 b c d^2+15 a b^2 c^2 d+35 b^3 c^3\right )}{64 b^3 d^4}+\frac {x^2 (a+b x)^{3/2} \sqrt {c+d x}}{4 b d} \]
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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} \sqrt {c+d x}}{4 b d}+\frac {\int \frac {x \sqrt {a+b x} \left (-2 a c+\frac {1}{2} (-7 b c-5 a d) x\right )}{\sqrt {c+d x}} \, dx}{4 b d} \\ & = \frac {x^2 (a+b x)^{3/2} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (35 b^2 c^2+22 a b c d+15 a^2 d^2-4 b d (7 b c+5 a d) x\right )}{96 b^3 d^3}-\frac {\left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{64 b^3 d^3} \\ & = -\frac {\left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d^4}+\frac {x^2 (a+b x)^{3/2} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (35 b^2 c^2+22 a b c d+15 a^2 d^2-4 b d (7 b c+5 a d) x\right )}{96 b^3 d^3}+\frac {\left ((b c-a d) \left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^3 d^4} \\ & = -\frac {\left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d^4}+\frac {x^2 (a+b x)^{3/2} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (35 b^2 c^2+22 a b c d+15 a^2 d^2-4 b d (7 b c+5 a d) x\right )}{96 b^3 d^3}+\frac {\left ((b c-a d) \left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\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 )}{64 b^4 d^4} \\ & = -\frac {\left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d^4}+\frac {x^2 (a+b x)^{3/2} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (35 b^2 c^2+22 a b c d+15 a^2 d^2-4 b d (7 b c+5 a d) x\right )}{96 b^3 d^3}+\frac {\left ((b c-a d) \left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\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 )}{64 b^4 d^4} \\ & = -\frac {\left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d^4}+\frac {x^2 (a+b x)^{3/2} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (35 b^2 c^2+22 a b c d+15 a^2 d^2-4 b d (7 b c+5 a d) x\right )}{96 b^3 d^3}+\frac {(b c-a d) \left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{9/2}} \\ \end{align*}
Time = 11.01 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.90 \[ \int \frac {x^3 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {b \sqrt {d} \sqrt {a+b x} (c+d x) \left (15 a^3 d^3+a^2 b d^2 (17 c-10 d x)+a b^2 d \left (25 c^2-12 c d x+8 d^2 x^2\right )+b^3 \left (-105 c^3+70 c^2 d x-56 c d^2 x^2+48 d^3 x^3\right )\right )+3 (b c-a d)^{3/2} \left (35 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{192 b^4 d^{9/2} \sqrt {c+d x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(573\) vs. \(2(219)=438\).
Time = 1.51 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.29
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-96 b^{3} d^{3} x^{3} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-16 a \,b^{2} d^{3} x^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+112 b^{3} c \,d^{2} x^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} d^{4}+12 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b c \,d^{3}+18 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c^{2} d^{2}+60 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{3} d -105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{4} c^{4}+20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x +24 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c \,d^{2} x -140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{2} d x -30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} d^{3}-34 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2}-50 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2} d +210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{3}\right )}{384 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{4} b^{3} \sqrt {b d}}\) | \(574\) |
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Time = 0.27 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.18 \[ \int \frac {x^3 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\left [-\frac {3 \, {\left (35 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 25 \, a b^{3} c^{2} d^{2} + 17 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 6 \, a b^{3} c d^{3} - 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{4} d^{5}}, -\frac {3 \, {\left (35 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 25 \, a b^{3} c^{2} d^{2} + 17 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 6 \, a b^{3} c d^{3} - 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{4} d^{5}}\right ] \]
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\[ \int \frac {x^3 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\int \frac {x^{3} \sqrt {a + b x}}{\sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {x^3 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.31 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.16 \[ \int \frac {x^3 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{4} d} - \frac {7 \, b^{13} c d^{5} + 17 \, a b^{12} d^{6}}{b^{16} d^{7}}\right )} + \frac {35 \, b^{14} c^{2} d^{4} + 50 \, a b^{13} c d^{5} + 59 \, a^{2} b^{12} d^{6}}{b^{16} d^{7}}\right )} - \frac {3 \, {\left (35 \, b^{15} c^{3} d^{3} + 15 \, a b^{14} c^{2} d^{4} + 9 \, a^{2} b^{13} c d^{5} + 5 \, a^{3} b^{12} d^{6}\right )}}{b^{16} d^{7}}\right )} \sqrt {b x + a} - \frac {3 \, {\left (35 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{3} d^{4}}\right )} b}{192 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {x^3 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\text {Hanged} \]
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