Integrand size = 22, antiderivative size = 119 \[ \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}} \, dx=-\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}-\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214} \[ \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}} \, dx=-\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{5/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 c^2 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2} \]
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Rule 95
Rule 96
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}+\frac {(3 (b c-a d)) \int \frac {\sqrt {a+b x}}{x^2 \sqrt {c+d x}} \, dx}{4 c} \\ & = -\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}+\frac {\left (3 (b c-a d)^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 c^2} \\ & = -\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}+\frac {\left (3 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 c^2} \\ & = -\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}-\frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{5/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} (-2 a c-5 b c x+3 a d x)}{4 c^2 x^2}-\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 \sqrt {a} c^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(254\) vs. \(2(93)=186\).
Time = 0.55 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.14
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x^{2}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x^{2}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x +10 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x +4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {a c}\right )}{8 c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {a c}}\) | \(255\) |
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Time = 0.35 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.79 \[ \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}} \, dx=\left [\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a c} x^{2} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} + {\left (5 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a c^{3} x^{2}}, \frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} + {\left (5 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a c^{3} x^{2}}\right ] \]
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\[ \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}}}{x^{3} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1037 vs. \(2 (93) = 186\).
Time = 0.65 (sec) , antiderivative size = 1037, normalized size of antiderivative = 8.71 \[ \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}} \, dx=-\frac {b {\left (\frac {3 \, {\left (\sqrt {b d} b^{3} c^{2} - 2 \, \sqrt {b d} a b^{2} c d + \sqrt {b d} a^{2} b d^{2}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c^{2}} + \frac {2 \, {\left (5 \, \sqrt {b d} b^{9} c^{5} - 23 \, \sqrt {b d} a b^{8} c^{4} d + 42 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{2} - 38 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{3} + 17 \, \sqrt {b d} a^{4} b^{5} c d^{4} - 3 \, \sqrt {b d} a^{5} b^{4} d^{5} - 15 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{7} c^{4} + 28 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{6} c^{3} d - 2 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{5} c^{2} d^{2} - 20 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{4} c d^{3} + 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{3} d^{4} + 15 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{5} c^{3} + \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{4} c^{2} d + 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{3} c d^{2} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{2} d^{3} - 5 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{3} c^{2} - 6 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{2} c d + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} b d^{2}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )}^{2} c^{2}}\right )}}{4 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{x^3\,\sqrt {c+d\,x}} \,d x \]
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