Integrand size = 19, antiderivative size = 113 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx=-\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {52, 65, 223, 212} \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx=\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{5/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d} \]
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {(3 (b c-a d)) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 d} \\ & = -\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\left (3 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d^2} \\ & = -\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\left (3 (b c-a d)^2\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 )}{4 b d^2} \\ & = -\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\left (3 (b c-a d)^2\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 )}{4 b d^2} \\ & = -\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{5/2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} (-3 b c+5 a d+2 b d x)}{4 d^2}+\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 \sqrt {b} d^{5/2}} \]
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Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}}{2 d}-\frac {3 \left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{d}-\frac {\left (-a d +b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 d \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}}\right )}{4 d}\) | \(140\) |
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Time = 0.25 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.71 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx=\left [\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x - 3 \, b^{2} c d + 5 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d^{3}}, -\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x - 3 \, b^{2} c d + 5 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d^{3}}\right ] \]
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\[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}}}{\sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.34 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )}}{b d} - \frac {3 \, {\left (b c d - a d^{2}\right )}}{b d^{3}}\right )} - \frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{2}}\right )} b}{4 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{\sqrt {c+d\,x}} \,d x \]
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